# bibtab

bibtype: article
filename: elefunt.bib
label: Kong:2006:IGA
author: Fanyu Kong and Zhun Cai and Jia Yu and Daxing Li
title: Improved generalized Atkin algorithm for computing square roots in finite fields
journal: j-INFO-PROC-LETT
volume: 98
number: 1
pages: 1--5
day: 15
month: apr
monthnumber: 4
year: 2006
CODEN: IFPLAT
ISSN: 0020-0190 (print), 1872-6119 (electronic) OR 00200190 (print), 18726119 (electronic)
ISSNL: 0020-0190 OR 00200190
bibdate: Thu Mar 31 18:41:08 MDT 2011
bibtimestamp: 2011.03.31 18:41:08 MDT
fjournal: Information Processing Letters
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00200190
rawauthor: Fanyu Kong and Zhun Cai and Jia Yu and Daxing Li
rawtitle: Improved generalized {Atkin} algorithm for computing square roots in finite fields
bibsource:
http://www.sciencedirect.com/science/journal/00200190; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib

entry:
@Article{Kong:2006:IGA,
  author =       "Fanyu Kong and Zhun Cai and Jia Yu and Daxing Li",
  title =        "Improved generalized {Atkin} algorithm for computing
                 square roots in finite fields",
  journal =      j-INFO-PROC-LETT,
  volume =       "98",
  number =       "1",
  pages =        "1--5",
  day =          "15",
  month =        apr,
  year =         "2006",
  CODEN =        "IFPLAT",
  ISSN =         "0020-0190 (print), 1872-6119 (electronic)",
  ISSN-L =       "0020-0190",
  bibdate =      "Thu Mar 31 18:41:08 MDT 2011",
  bibsource =    "http://www.sciencedirect.com/science/journal/00200190;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "Information Processing Letters",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00200190",
}


bibtype: article
filename: elefunt.bib
label: Kornerup:2006:CSV
author: Peter Kornerup and Jean-Michel Muller
title: Choosing starting values for certain Newton--Raphson iterations
journal: j-THEOR-COMP-SCI
volume: 351
number: 1
pages: 101--110
day: 14
month: feb
monthnumber: 2
year: 2006
CODEN: TCSCDI
ISSN: 0304-3975 (print), 1879-2294 (electronic) OR 03043975 (print), 18792294 (electronic)
ISSNL: 0304-3975 OR 03043975
bibdate: Tue Mar 29 06:48:55 MDT 2011
bibtimestamp: 2011.03.29 06:48:55 MDT
fjournal: Theoretical Computer Science
acknowledgement: ack-nhfb
ajournal: Theor. Comput. Sci.
journalURL: http://www.sciencedirect.com/science/journal/03043975/
rawauthor: Peter Kornerup and Jean-Michel Muller
rawtitle: Choosing starting values for certain {Newton--Raphson} iterations
bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/tcs2005.bib

abstract:
We aim at finding the best possible seed values when computing $ a^1 / p $ using the Newton--Raphson iteration in a given interval. A natural choice of the seed value would be the one that best approximates the expected result. It turns out that in most cases, the best seed value can be quite far from this natural choice. When we evaluate a monotone function $ f(a) $ in the interval $ [a_mathrm min, a_mathrm max] $, by building the sequence $ x_n $ defined by the Newton--Raphson iteration, the natural choice consists in choosing $ x_0 $ equal to the arithmetic mean of the endpoint values. This minimizes the maximum possible distance between $ x_0 $ and $ f(a) $. And yet, if we perform $n$ iterations, what matters is to minimize the maximum possible distance between $ x_n $ and $ f(a) $. In several examples, the value of the best starting point varies rather significantly with the number of iterations.

rawabstract:
We aim at finding the best possible seed values when computing $ a^{1 / p} $ using the Newton--Raphson iteration in a given interval. A natural choice of the seed value would be the one that best approximates the expected result. It turns out that in most cases, the best seed value can be quite far from this natural choice. When we evaluate a monotone function $ f(a) $ in the interval $ [a_\mathrm {min}, a_\mathrm {max}] $, by building the sequence $ x_n $ defined by the Newton--Raphson iteration, the natural choice consists in choosing $ x_0 $ equal to the arithmetic mean of the endpoint values. This minimizes the maximum possible distance between $ x_0 $ and $ f(a) $. And yet, if we perform $n$ iterations, what matters is to minimize the maximum possible distance between $ x_n $ and $ f(a) $. In several examples, the value of the best starting point varies rather significantly with the number of iterations.

entry:
@Article{Kornerup:2006:CSV,
  author =       "Peter Kornerup and Jean-Michel Muller",
  title =        "Choosing starting values for certain {Newton--Raphson}
                 iterations",
  journal =      j-THEOR-COMP-SCI,
  volume =       "351",
  number =       "1",
  pages =        "101--110",
  day =          "14",
  month =        feb,
  year =         "2006",
  CODEN =        "TCSCDI",
  ISSN =         "0304-3975 (print), 1879-2294 (electronic)",
  ISSN-L =       "0304-3975",
  bibdate =      "Tue Mar 29 06:48:55 MDT 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/tcs2005.bib",
  abstract =     "We aim at finding the best possible seed values when
                 computing $ a^{1 / p} $ using the Newton--Raphson
                 iteration in a given interval. A natural choice of the
                 seed value would be the one that best approximates the
                 expected result. It turns out that in most cases, the
                 best seed value can be quite far from this natural
                 choice. When we evaluate a monotone function $ f(a) $
                 in the interval $ [a_\mathrm {min}, a_\mathrm {max}] $,
                 by building the sequence $ x_n $ defined by the
                 Newton--Raphson iteration, the natural choice consists
                 in choosing $ x_0 $ equal to the arithmetic mean of the
                 endpoint values. This minimizes the maximum possible
                 distance between $ x_0 $ and $ f(a) $. And yet, if we
                 perform $n$ iterations, what matters is to minimize the
                 maximum possible distance between $ x_n $ and $ f(a) $.
                 In several examples, the value of the best starting
                 point varies rather significantly with the number of
                 iterations.",
  acknowledgement = ack-nhfb,
  ajournal =     "Theor. Comput. Sci.",
  fjournal =     "Theoretical Computer Science",
  journal-URL =  "http://www.sciencedirect.com/science/journal/03043975/",
}


bibtype: inproceedings
filename: elefunt.bib
label: Muller:2006:GFA
author: Jean-Michel Muller
editor: Michael B. Matthews
title: Generating function approximations at compile time
publisher: pub-IEEE
address: pub-IEEE:adr
pages: 328--331
year: 2006
DOI: https://doi.org/10.1109/ACSSC.2006.354761
ISBN: 1-4244-0785-0 OR 1424407850
ISBN13: 978-1-4244-0785-9 OR 9781424407859
bibdate: Fri Sep 29 10:57:58 2023
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2023.09.29 10:57:58 ???
acknowledgement: ack-nhfb
rawauthor: Jean-Michel Muller
raweditor: Michael B. Matthews
rawtitle: Generating function approximations at compile time
booktitle:
2006 Fortieth Asilomar Conference on Signals, Systems and Computers. October 29--November 1, 2006. Pacific Grove, California

rawbooktitle:
{2006 Fortieth Asilomar Conference on Signals, Systems and Computers. October 29--November 1, 2006. Pacific Grove, California}

entry:
@InProceedings{Muller:2006:GFA,
  author =       "Jean-Michel Muller",
  editor =       "Michael B. Matthews",
  booktitle =    "{2006 Fortieth Asilomar Conference on Signals, Systems
                 and Computers. October 29--November 1, 2006. Pacific
                 Grove, California}",
  title =        "Generating function approximations at compile time",
  publisher =    pub-IEEE,
  address =      pub-IEEE:adr,
  pages =        "328--331",
  year =         "2006",
  DOI =          "https://doi.org/10.1109/ACSSC.2006.354761",
  ISBN =         "1-4244-0785-0",
  ISBN-13 =      "978-1-4244-0785-9",
  bibdate =      "Fri Sep 29 10:57:58 2023",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: elefunt.bib
label: Nowak:2006:MCA
author: Rafal Nowak
title: A method of convergence acceleration of some continued fractions
journal: j-NUMER-ALGORITHMS
volume: 41
number: 3
pages: 297--317
month: mar
monthnumber: 3
year: 2006
CODEN: NUALEG
DOI: https://doi.org/10.1007/s11075-005-9013-3
ISSN: 1017-1398 (print), 1572-9265 (electronic) OR 10171398 (print), 15729265 (electronic)
ISSNL: 1017-1398 OR 10171398
MRclass: subject classification; 30B70; 40A15; 65B99
bibdate: Tue Jul 8 19:14:28 MDT 2008
bibtimestamp: 2008.07.08 19:14:28 MDT
URL: http://www.springerlink.com/openurl.asp?genre=article&issn=1017-1398&volume=41&issue=3&spage=297
fjournal: Numerical Algorithms
keywords: continued fraction; convergence acceleration; modified approximant; tail
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/11075
rawauthor: Rafal Nowak
rawtitle: A method of convergence acceleration of some continued fractions
bibsource:
http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1017-1398&volume=41&issue=3; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

entry:
@Article{Nowak:2006:MCA,
  author =       "Rafal Nowak",
  title =        "A method of convergence acceleration of some continued
                 fractions",
  journal =      j-NUMER-ALGORITHMS,
  volume =       "41",
  number =       "3",
  pages =        "297--317",
  month =        mar,
  year =         "2006",
  CODEN =        "NUALEG",
  DOI =          "https://doi.org/10.1007/s11075-005-9013-3",
  ISSN =         "1017-1398 (print), 1572-9265 (electronic)",
  ISSN-L =       "1017-1398",
  MRclass =      "subject classification; 30B70; 40A15; 65B99",
  bibdate =      "Tue Jul 8 19:14:28 MDT 2008",
  bibsource =    "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1017-1398&volume=41&issue=3;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.springerlink.com/openurl.asp?genre=article&issn=1017-1398&volume=41&issue=3&spage=297",
  acknowledgement = ack-nhfb,
  fjournal =     "Numerical Algorithms",
  journal-URL =  "http://link.springer.com/journal/11075",
  keywords =     "continued fraction; convergence acceleration; modified
                 approximant; tail",
}


bibtype: article
filename: elefunt.bib
label: Ozban:2006:NMA
author: Ahmet Yasar Ozban
title: New methods for approximating square roots
journal: j-APPL-MATH-COMP
volume: 175
number: 1
pages: 532--540
day: 1
month: apr
monthnumber: 4
year: 2006
CODEN: AMHCBQ
ISSN: 0096-3003 (print), 1873-5649 (electronic) OR 00963003 (print), 18735649 (electronic)
ISSNL: 0096-3003 OR 00963003
bibdate: Sat Jul 12 09:02:54 MDT 2008
bibtimestamp: 2008.07.12 09:02:54 MDT
fjournal: Applied Mathematics and Computation
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00963003
rawauthor: Ahmet Ya{\c{s}}ar {\"O}zban
rawtitle: New methods for approximating square roots
bibsource:
http://www.sciencedirect.com/science/journal/00963003; https://www.math.utah.edu/pub/tex/bib/applmathcomput2005.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib

entry:
@Article{Ozban:2006:NMA,
  author =       "Ahmet Ya{\c{s}}ar {\"O}zban",
  title =        "New methods for approximating square roots",
  journal =      j-APPL-MATH-COMP,
  volume =       "175",
  number =       "1",
  pages =        "532--540",
  day =          "1",
  month =        apr,
  year =         "2006",
  CODEN =        "AMHCBQ",
  ISSN =         "0096-3003 (print), 1873-5649 (electronic)",
  ISSN-L =       "0096-3003",
  bibdate =      "Sat Jul 12 09:02:54 MDT 2008",
  bibsource =    "http://www.sciencedirect.com/science/journal/00963003;
                 https://www.math.utah.edu/pub/tex/bib/applmathcomput2005.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "Applied Mathematics and Computation",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00963003",
}


bibtype: inproceedings
filename: elefunt.bib
label: Parks:2006:UTS
author: Michael Parks
title: Unifying Tests for Square Root
crossref: Anonymous:2006:PCR
pages: ??--??
year: 2006
bibdate: Tue Jun 27 10:28:05 2006
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2006.06.27 10:28:05 ???
keywords: elementary functions
acknowledgement: ack-nhfb
rawauthor: Michael Parks
rawtitle: Unifying Tests for Square Root
entry:
@InProceedings{Parks:2006:UTS,
  author =       "Michael Parks",
  title =        "Unifying Tests for Square Root",
  crossref =     "Anonymous:2006:PCR",
  pages =        "??--??",
  year =         "2006",
  bibdate =      "Tue Jun 27 10:28:05 2006",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
  keywords =     "elementary functions",
}


bibtype: article
filename: elefunt.bib
label: Qian:2006:HMP
author: Jianbo Qian and Cao An Wang
title: How much precision is needed to compare two sums of square roots of integers?
journal: j-INFO-PROC-LETT
volume: 100
number: 5
pages: 194--198
day: 16
month: dec
monthnumber: 12
year: 2006
CODEN: IFPLAT
ISSN: 0020-0190 (print), 1872-6119 (electronic) OR 00200190 (print), 18726119 (electronic)
ISSNL: 0020-0190 OR 00200190
bibdate: Thu Mar 31 15:52:31 MDT 2011
bibtimestamp: 2011.03.31 15:52:31 MDT
fjournal: Information Processing Letters
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00200190
rawauthor: Jianbo Qian and Cao An Wang
rawtitle: How much precision is needed to compare two sums of square roots of integers?
bibsource:
http://www.sciencedirect.com/science/journal/00200190; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib

entry:
@Article{Qian:2006:HMP,
  author =       "Jianbo Qian and Cao An Wang",
  title =        "How much precision is needed to compare two sums of
                 square roots of integers?",
  journal =      j-INFO-PROC-LETT,
  volume =       "100",
  number =       "5",
  pages =        "194--198",
  day =          "16",
  month =        dec,
  year =         "2006",
  CODEN =        "IFPLAT",
  ISSN =         "0020-0190 (print), 1872-6119 (electronic)",
  ISSN-L =       "0020-0190",
  bibdate =      "Thu Mar 31 15:52:31 MDT 2011",
  bibsource =    "http://www.sciencedirect.com/science/journal/00200190;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "Information Processing Letters",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00200190",
}


bibtype: article
filename: elefunt.bib
label: Shi:2006:NAS
author: Xiquan Shi and Fengshan Liu and Minghan Hu
title: A new asymptotic series for the Gamma function
journal: j-J-COMPUT-APPL-MATH
volume: 195
number: 1--2
pages: 134--154
day: 15
month: oct
monthnumber: 10
year: 2006
CODEN: JCAMDI
DOI: https://doi.org/10.1016/j.cam.2005.03.081
ISSN: 0377-0427 (print), 1879-1778 (electronic) OR 03770427 (print), 18791778 (electronic)
ISSNL: 0377-0427 OR 03770427
bibdate: Sat Feb 25 13:12:01 MST 2017
bibtimestamp: 2017.02.25 13:12:01 MST
URL: http://www.sciencedirect.com/science/article/pii/S0377042705004802
fjournal: Journal of Computational and Applied Mathematics
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/03770427
rawauthor: Xiquan Shi and Fengshan Liu and Minghan Hu
rawtitle: A new asymptotic series for the Gamma function
entry:
@Article{Shi:2006:NAS,
  author =       "Xiquan Shi and Fengshan Liu and Minghan Hu",
  title =        "A new asymptotic series for the Gamma function",
  journal =      j-J-COMPUT-APPL-MATH,
  volume =       "195",
  number =       "1--2",
  pages =        "134--154",
  day =          "15",
  month =        oct,
  year =         "2006",
  CODEN =        "JCAMDI",
  DOI =          "https://doi.org/10.1016/j.cam.2005.03.081",
  ISSN =         "0377-0427 (print), 1879-1778 (electronic)",
  ISSN-L =       "0377-0427",
  bibdate =      "Sat Feb 25 13:12:01 MST 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2005.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0377042705004802",
  acknowledgement = ack-nhfb,
  fjournal =     "Journal of Computational and Applied Mathematics",
  journal-URL =  "http://www.sciencedirect.com/science/journal/03770427",
}


bibtype: article
filename: elefunt.bib
label: Sidi:2006:CTC
author: Avram Sidi
title: A challenging test for convergence accelerators: Summation of a series with a special sign pattern
journal: App. Math. E-Notes
volume: 6
pages: 225--234
year: 2006
MRclass: 40A99 (11M41 40A05 65B10)
MRnumber: MR2231748 (2007h:40009)
bibdate: Thu Dec 01 10:33:54 2011
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2011.12.01 10:33:54 ???
keywords: convergence acceleration; Shanks transformation
acknowledgement: ack-nhfb
rawauthor: Avram Sidi
rawtitle: A challenging test for convergence accelerators: Summation of a series with a special sign pattern
entry:
@Article{Sidi:2006:CTC,
  author =       "Avram Sidi",
  title =        "A challenging test for convergence accelerators:
                 Summation of a series with a special sign pattern",
  journal =      "App. Math. E-Notes",
  volume =       "6",
  number =       "??",
  pages =        "225--234",
  month =        "????",
  year =         "2006",
  CODEN =        "????",
  ISSN =         "????",
  MRclass =      "40A99 (11M41 40A05 65B10)",
  MRnumber =     "MR2231748 (2007h:40009)",
  bibdate =      "Thu Dec 01 10:33:54 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
  keywords =     "convergence acceleration; Shanks transformation",
}


bibtype: inproceedings
filename: elefunt.bib
label: Thakkar:2006:PDP
author: Anuja J. Thakkar and Abdel Ejnioui
title: Pipelining of double precision floating point division and square root operations
crossref: Menezes:2006:PAS
pages: 488--493
year: 2006
DOI: https://doi.org/10.1145/1185448.1185555
bibdate: Sat Oct 9 13:04:49 2010
bibtimestamp: 2010.10.09 13:04:49 ???
acknowledgement: ack-nhfb
rawauthor: Anuja J. Thakkar and Abdel Ejnioui
rawtitle: Pipelining of double precision floating point division and square root operations
abstract:
Space applications rely increasingly on high data rate DSP algorithms. These algorithms use double precision floating point arithmetic operations. While most DSP applications can be compiled on DSP processors, high data rate DSP computations require novel implementation technologies to support their high throughputs. Only recently, gate densities in FPGAs have reached a level which makes them attractive platforms to implement compute-intensive DSP applications. In this context, this paper presents the sequential and pipelined designs of a double precision floating point divider and square root unit on FPGAs. Contrary to pipelined parallel implementations, the pipelining of these units is based on unrolling the iterations in low-radix digit recurrence algorithms. These units are mapped on generic FPGA reconfigurable fabric without taking advantage of any advanced architectural components available in high capacity FPGAs. The implementations of these designs show that their performances are comparable to, and sometimes higher than, the performances of non-iterative designs based of high radix numbers. The iterative divider and square root unit occupy less than 1% of an XC2V6000 FPGA chip while their pipelined counterparts can produce throughputs that reach the 100 MFLOPS mark by consuming a modest 8% of the chip area.

rawabstract:
Space applications rely increasingly on high data rate DSP algorithms. These algorithms use double precision floating point arithmetic operations. While most DSP applications can be compiled on DSP processors, high data rate DSP computations require novel implementation technologies to support their high throughputs. Only recently, gate densities in FPGAs have reached a level which makes them attractive platforms to implement compute-intensive DSP applications. In this context, this paper presents the sequential and pipelined designs of a double precision floating point divider and square root unit on FPGAs. Contrary to pipelined parallel implementations, the pipelining of these units is based on unrolling the iterations in low-radix digit recurrence algorithms. These units are mapped on generic FPGA reconfigurable fabric without taking advantage of any advanced architectural components available in high capacity FPGAs. The implementations of these designs show that their performances are comparable to, and sometimes higher than, the performances of non-iterative designs based of high radix numbers. The iterative divider and square root unit occupy less than 1\% of an XC2V6000 FPGA chip while their pipelined counterparts can produce throughputs that reach the 100 MFLOPS mark by consuming a modest 8\% of the chip area.

entry:
@InProceedings{Thakkar:2006:PDP,
  author =       "Anuja J. Thakkar and Abdel Ejnioui",
  title =        "Pipelining of double precision floating point division
                 and square root operations",
  crossref =     "Menezes:2006:PAS",
  pages =        "488--493",
  year =         "2006",
  DOI =          "https://doi.org/10.1145/1185448.1185555",
  bibdate =      "Sat Oct 9 13:04:49 2010",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  abstract =     "Space applications rely increasingly on high data rate
                 DSP algorithms. These algorithms use double precision
                 floating point arithmetic operations. While most DSP
                 applications can be compiled on DSP processors, high
                 data rate DSP computations require novel implementation
                 technologies to support their high throughputs. Only
                 recently, gate densities in FPGAs have reached a level
                 which makes them attractive platforms to implement
                 compute-intensive DSP applications. In this context,
                 this paper presents the sequential and pipelined
                 designs of a double precision floating point divider
                 and square root unit on FPGAs. Contrary to pipelined
                 parallel implementations, the pipelining of these units
                 is based on unrolling the iterations in low-radix digit
                 recurrence algorithms. These units are mapped on
                 generic FPGA reconfigurable fabric without taking
                 advantage of any advanced architectural components
                 available in high capacity FPGAs. The implementations
                 of these designs show that their performances are
                 comparable to, and sometimes higher than, the
                 performances of non-iterative designs based of high
                 radix numbers. The iterative divider and square root
                 unit occupy less than 1\% of an XC2V6000 FPGA chip
                 while their pipelined counterparts can produce
                 throughputs that reach the 100 MFLOPS mark by consuming
                 a modest 8\% of the chip area.",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: elefunt.bib
label: VanDeun:2006:ACI
author: Joris Van Deun and Ronald Cools
title: Algorithm 858: Computing infinite range integrals of an arbitrary product of Bessel functions
journal: j-TOMS
volume: 32
number: 4
pages: 580--596
month: dec
monthnumber: 12
year: 2006
CODEN: ACMSCU
DOI: https://doi.org/10.1145/1186785.1186790
ISSN: 0098-3500 (print), 1557-7295 (electronic) OR 00983500 (print), 15577295 (electronic)
ISSNL: 0098-3500 OR 00983500
bibdate: Sat Apr 14 09:48:57 MDT 2007
bibtimestamp: 2007.04.14 09:48:57 MDT
fjournal: ACM Transactions on Mathematical Software (TOMS)
acknowledgement: ack-nhfb
journalURL: https://dl.acm.org/loi/toms
rawauthor: Joris {Van Deun} and Ronald Cools
rawtitle: {Algorithm 858}: {Computing} infinite range integrals of an arbitrary product of {Bessel} functions
abstract:
We present an algorithm to compute integrals of the form $ int _0^infty x^m prod ^k_i = 1 J_nu _i(a_i x)d x $ with $ J_nu _i(x) $ the Bessel function of the first kind and (real) order $ nu _i $. The parameter $m$ is a real number such that $ sum _i nu _i + m > - 1 $ and the coefficients $ a_i $ are strictly positive real numbers. The main ingredients in this algorithm are the well-known asymptotic expansion for $ J_nu _i(x) $ and the observation that the infinite part of the integral can be approximated using the incomplete Gamma function $ Gamma (a, z) $. Accurate error estimates are included in the algorithm, which is implemented as a MATLAB program.

rawabstract:
We present an algorithm to compute integrals of the form $ \int_0^\infty x^m \prod^k_i = 1 J_{\nu_i}(a_i x)d x $ with $ J_{\nu_i}(x) $ the Bessel function of the first kind and (real) order $ \nu_i $. The parameter $m$ is a real number such that $ \sum_i \nu_i + m > - 1 $ and the coefficients $ a_i $ are strictly positive real numbers. The main ingredients in this algorithm are the well-known asymptotic expansion for $ J_{\nu_i}(x) $ and the observation that the infinite part of the integral can be approximated using the incomplete Gamma function $ \Gamma (a, z) $. Accurate error estimates are included in the algorithm, which is implemented as a MATLAB program.

entry:
@Article{VanDeun:2006:ACI,
  author =       "Joris {Van Deun} and Ronald Cools",
  title =        "{Algorithm 858}: {Computing} infinite range integrals
                 of an arbitrary product of {Bessel} functions",
  journal =      j-TOMS,
  volume =       "32",
  number =       "4",
  pages =        "580--596",
  month =        dec,
  year =         "2006",
  CODEN =        "ACMSCU",
  DOI =          "https://doi.org/10.1145/1186785.1186790",
  ISSN =         "0098-3500 (print), 1557-7295 (electronic)",
  ISSN-L =       "0098-3500",
  bibdate =      "Sat Apr 14 09:48:57 MDT 2007",
  bibsource =    "http://www.acm.org/pubs/contents/journals/toms/;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/toms.bib",
  abstract =     "We present an algorithm to compute integrals of the
                 form $ \int_0^\infty x^m \prod^k_i = 1 J_{\nu_i}(a_i
                 x)d x $ with $ J_{\nu_i}(x) $ the Bessel function of
                 the first kind and (real) order $ \nu_i $. The
                 parameter $m$ is a real number such that $ \sum_i \nu_i
                 + m > - 1 $ and the coefficients $ a_i $ are strictly
                 positive real numbers. The main ingredients in this
                 algorithm are the well-known asymptotic expansion for $
                 J_{\nu_i}(x) $ and the observation that the infinite
                 part of the integral can be approximated using the
                 incomplete Gamma function $ \Gamma (a, z) $. Accurate
                 error estimates are included in the algorithm, which is
                 implemented as a MATLAB program.",
  acknowledgement = ack-nhfb,
  fjournal =     "ACM Transactions on Mathematical Software (TOMS)",
  journal-URL =  "https://dl.acm.org/loi/toms",
}


bibtype: article
filename: elefunt.bib
label: VanDeun:2006:SRI
author: Joris Van Deun and Ronald Cools
title: A stable recurrence for the incomplete gamma function with imaginary second argument
journal: j-NUM-MATH
volume: 104
number: 4
pages: 445--456
month: oct
monthnumber: 10
year: 2006
CODEN: NUMMA7
DOI: https://doi.org/10.1007/s00211-006-0026-1
ISSN: 0029-599X (print), 0945-3245 (electronic) OR 0029599X (print), 09453245 (electronic)
ISSNL: 0029-599X OR 0029599X
bibdate: Tue Jul 8 10:28:23 MDT 2008
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2008.07.08 10:28:23 MDT
fjournal: Numerische Mathematik
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/211
rawauthor: Joris {Van Deun} and Ronald Cools
rawtitle: A stable recurrence for the incomplete gamma function with imaginary second argument
entry:
@Article{VanDeun:2006:SRI,
  author =       "Joris {Van Deun} and Ronald Cools",
  title =        "A stable recurrence for the incomplete gamma function
                 with imaginary second argument",
  journal =      j-NUM-MATH,
  volume =       "104",
  number =       "4",
  pages =        "445--456",
  month =        oct,
  year =         "2006",
  CODEN =        "NUMMA7",
  DOI =          "https://doi.org/10.1007/s00211-006-0026-1",
  ISSN =         "0029-599X (print), 0945-3245 (electronic)",
  ISSN-L =       "0029-599X",
  bibdate =      "Tue Jul 8 10:28:23 MDT 2008",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "Numerische Mathematik",
  journal-URL =  "http://link.springer.com/journal/211",
}


bibtype: inproceedings
filename: elefunt.bib
label: Wang:2006:EAR
author: Laixiong Wang and Yangping Chen and Shitan Huang
editor: IEEE
booktitle: Sixth World Congress on Intelligent Control and Automation, June 21--23, 2006, Dalian, China
publisher: pub-IEEE
address: pub-IEEE:adr
pages: 6800--6803
year: 2006
DOI: https://doi.org/10.1109/wcica.2006.1714401
bibdate: Mon Nov 10 06:50:26 2025
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2025.11.10 06:50:26 ???
acknowledgement: ack-nhfb
rawauthor: Laixiong Wang and Yangping Chen and Shitan Huang
rawbooktitle: {Sixth World Congress on Intelligent Control and Automation, June 21--23, 2006, Dalian, China}
raweditor: {IEEE}
title:
Efficient Argument Range Reduction for Implementation of Double-Precision Floating-Point Exponential Function

rawtitle:
Efficient Argument Range Reduction for Implementation of Double-Precision Floating-Point Exponential Function

entry:
@InProceedings{Wang:2006:EAR,
  author =       "Laixiong Wang and Yangping Chen and Shitan Huang",
  editor =       "{IEEE}",
  booktitle =    "{Sixth World Congress on Intelligent Control and
                 Automation, June 21--23, 2006, Dalian, China}",
  title =        "Efficient Argument Range Reduction for Implementation
                 of Double-Precision Floating-Point Exponential
                 Function",
  publisher =    pub-IEEE,
  address =      pub-IEEE:adr,
  pages =        "6800--6803",
  year =         "2006",
  DOI =          "https://doi.org/10.1109/wcica.2006.1714401",
  bibdate =      "Mon Nov 10 06:50:26 2025",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: elefunt.bib
label: Wires:2006:RRS
author: Kent E. Wires and Michael J. Schulte
title: Reciprocal and Reciprocal Square Root Units with Operand Modification and Multiplication
journal: j-J-VLSI-SIGNAL-PROC
volume: 42
number: 3
pages: 257--272
month: mar
monthnumber: 3
year: 2006
CODEN: JVSPED
DOI: https://doi.org/10.1007/s11265-006-4186-0
ISSN: 0922-5773 (print), 1573-109x (electronic) OR 09225773 (print), 1573109x (electronic)
ISSNL: 0922-5773 OR 09225773
bibdate: Mon Mar 05 08:26:23 2007
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2007.03.05 08:26:23 ???
URL: http://springerlink.metapress.com/content/t6027p6713727606/fulltext.pdf
fjournal: Journal of VLSI Signal Processing
acknowledgement: ack-nhfb
rawauthor: Kent E. Wires and Michael J. Schulte
rawtitle: Reciprocal and Reciprocal Square Root Units with Operand Modification and Multiplication
entry:
@Article{Wires:2006:RRS,
  author =       "Kent E. Wires and Michael J. Schulte",
  title =        "Reciprocal and Reciprocal Square Root Units with
                 Operand Modification and Multiplication",
  journal =      j-J-VLSI-SIGNAL-PROC,
  volume =       "42",
  number =       "3",
  pages =        "257--272",
  month =        mar,
  year =         "2006",
  CODEN =        "JVSPED",
  DOI =          "https://doi.org/10.1007/s11265-006-4186-0",
  ISSN =         "0922-5773 (print), 1573-109x (electronic)",
  ISSN-L =       "0922-5773",
  bibdate =      "Mon Mar 05 08:26:23 2007",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://springerlink.metapress.com/content/t6027p6713727606/fulltext.pdf",
  acknowledgement = ack-nhfb,
  fjournal =     "Journal of VLSI Signal Processing",
}


bibtype: inproceedings
filename: elefunt.bib
label: Barnett:2007:HPV
author: Ross Barnett and J. A. Youngman
title: High-Precision Values of the Gamma Function of real argument
publisher: pub-AMS
address: pub-AMS:adr
year: 2007
bibdate: Mon Jul 14 11:57:00 2008
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2008.07.14 11:57:00 ???
URL: http://homepages.mcs.vuw.ac.nz/~mathmeet/amsnzms2007/abstracts.pdf
acknowledgement: ack-nhfb
rawauthor: Ross Barnett and J. A. Youngman
rawtitle: High-Precision Values of the Gamma Function of real argument
booktitle:
1st Joint Meeting of the American Mathematical Society and the New Zealand Mathematical Society, Victoria University of Wellington, Wellington, New Zealand, December 12--15, 2007

abstract:
A method is described to calculate values of $ Gamma (nu ) $, $ 0 leq nu leq 1 $ to arbitrary precision by combining a Bessel function with a $_0 F_1$ function. Steed's algorithm is used to compute the regular Bessel function $ J_nu (x) $, for a suitable argument $x$ and real $ nu $, to arbitrary accuracy. Hence the gamma function is obtained. Example values are given to 200D. Verification is by the 80D-results of Fransen and Wrigge, by the use of the duplication formula, and by computing the closed form results of Borwein and Zucker. A caveat is offered concerning the coding of the Bessel functions in Numerical Recipes and in the GSL library.

rawabstract:
A method is described to calculate values of $ \Gamma (\nu) $, $ 0 \leq \nu \leq 1 $ to arbitrary precision by combining a Bessel function with a $_0 F_1$ function. Steed's algorithm is used to compute the regular Bessel function $ J_\nu (x) $, for a suitable argument $x$ and real $ \nu $, to arbitrary accuracy. Hence the gamma function is obtained. Example values are given to 200D. Verification is by the 80D-results of Frans{\'e}n and Wrigge, by the use of the duplication formula, and by computing the closed form results of Borwein and Zucker. A caveat is offered concerning the coding of the Bessel functions in Numerical Recipes and in the GSL library.

rawbooktitle:
{1st Joint Meeting of the American Mathematical Society and the New Zealand Mathematical Society, Victoria University of Wellington, Wellington, New Zealand, December 12--15, 2007}

entry:
@InProceedings{Barnett:2007:HPV,
  author =       "Ross Barnett and J. A. Youngman",
  booktitle =    "{1st Joint Meeting of the American Mathematical
                 Society and the New Zealand Mathematical Society,
                 Victoria University of Wellington, Wellington, New
                 Zealand, December 12--15, 2007}",
  title =        "High-Precision Values of the Gamma Function of real
                 argument",
  publisher =    pub-AMS,
  address =      pub-AMS:adr,
  pages =        "????",
  year =         "2007",
  ISBN =         "????",
  ISBN-13 =      "????",
  LCCN =         "????",
  bibdate =      "Mon Jul 14 11:57:00 2008",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://homepages.mcs.vuw.ac.nz/~mathmeet/amsnzms2007/abstracts.pdf",
  abstract =     "A method is described to calculate values of $ \Gamma
                 (\nu) $, $ 0 \leq \nu \leq 1 $ to arbitrary precision
                 by combining a Bessel function with a $_0 F_1$
                 function. Steed's algorithm is used to compute the
                 regular Bessel function $ J_\nu (x) $, for a suitable
                 argument $x$ and real $ \nu $, to arbitrary accuracy.
                 Hence the gamma function is obtained. Example values
                 are given to 200D. Verification is by the 80D-results
                 of Frans{\'e}n and Wrigge, by the use of the
                 duplication formula, and by computing the closed form
                 results of Borwein and Zucker. A caveat is offered
                 concerning the coding of the Bessel functions in
                 Numerical Recipes and in the GSL library.",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: elefunt.bib
label: Batterman:2007:SSF
author: Robert W. Batterman
title: On the Specialness of Special Functions (The Nonrandom Effusions of the Divine Mathematician)
journal: j-BRITISH-J-PHILOS-SCI
volume: 58
number: 2
pages: 263--286
month: jun
monthnumber: 6
year: 2007
CODEN: BJPIA5
DOI: https://doi.org/10.1093/bjps/axm007
ISSN: 0007-0882 (print), 1464-3537 (electronic) OR 00070882 (print), 14643537 (electronic)
ISSNL: 0007-0882 OR 00070882
bibdate: Thu Oct 7 14:03:55 MDT 2010
bibsource: http://bjps.oxfordjournals.org/content/58/2.toc; https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2010.10.07 14:03:55 MDT
URL: http://bjps.oxfordjournals.org/content/58/2/263.full.pdf+html
fjournal: British Journal for the Philosophy of Science
acknowledgement: ack-nhfb
journalURL: http://www.jstor.org/journals/00070882.html
onlinedate: May 18, 2007
rawauthor: Robert W. Batterman
rawtitle: On the Specialness of Special Functions (The Nonrandom Effusions of the Divine Mathematician)
entry:
@Article{Batterman:2007:SSF,
  author =       "Robert W. Batterman",
  title =        "On the Specialness of Special Functions (The Nonrandom
                 Effusions of the Divine Mathematician)",
  journal =      j-BRITISH-J-PHILOS-SCI,
  volume =       "58",
  number =       "2",
  pages =        "263--286",
  month =        jun,
  year =         "2007",
  CODEN =        "BJPIA5",
  DOI =          "https://doi.org/10.1093/bjps/axm007",
  ISSN =         "0007-0882 (print), 1464-3537 (electronic)",
  ISSN-L =       "0007-0882",
  bibdate =      "Thu Oct 7 14:03:55 MDT 2010",
  bibsource =    "http://bjps.oxfordjournals.org/content/58/2.toc;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://bjps.oxfordjournals.org/content/58/2/263.full.pdf+html",
  acknowledgement = ack-nhfb,
  fjournal =     "British Journal for the Philosophy of Science",
  journal-URL =  "http://www.jstor.org/journals/00070882.html",
  onlinedate =   "May 18, 2007",
}


bibtype: inproceedings
filename: elefunt.bib
label: Brisebarre:2007:EPA
author: Nicolas Brisebarre and Sylvain Chevillard
title: Efficient polynomial $ L^infty $-approximations
crossref: Kornerup:2007:PIS
pages: 169--176
year: 2007
DOI: https://doi.org/10.1109/ARITH.2007.17
bibdate: Tue Oct 9 17:16:03 MDT 2007
bibtimestamp: 2007.10.09 17:16:03 MDT
keywords: ARITH-18
acknowledgement: ack-nhfb
rawauthor: Nicolas Brisebarre and Sylvain Chevillard
rawtitle: Efficient polynomial {$ L^\infty $}-approximations
abstract:
We address the problem of computing a good floating-point-coefficient polynomial approximation to a function, with respect to the supremum norm. This is a key step in most processes of evaluation of a function. We present a fast and efficient method, based on lattice basis reduction, that often gives the best polynomial possible and most of the time returns a very good approximation.

rawabstract:
We address the problem of computing a good floating-point-coefficient polynomial approximation to a function, with respect to the supremum norm. This is a key step in most processes of evaluation of a function. We present a fast and efficient method, based on lattice basis reduction, that often gives the best polynomial possible and most of the time returns a very good approximation.

entry:
@InProceedings{Brisebarre:2007:EPA,
  author =       "Nicolas Brisebarre and Sylvain Chevillard",
  title =        "Efficient polynomial {$ L^\infty $}-approximations",
  crossref =     "Kornerup:2007:PIS",
  pages =        "169--176",
  year =         "2007",
  DOI =          "https://doi.org/10.1109/ARITH.2007.17",
  bibdate =      "Tue Oct 9 17:16:03 MDT 2007",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  abstract =     "We address the problem of computing a good
                 floating-point-coefficient polynomial approximation to
                 a function, with respect to the supremum norm. This is
                 a key step in most processes of evaluation of a
                 function. We present a fast and efficient method, based
                 on lattice basis reduction, that often gives the best
                 polynomial possible and most of the time returns a very
                 good approximation.",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH-18",
}


bibtype: inproceedings
filename: elefunt.bib
label: Burgess:2007:DAV
author: Neil Burgess and Chris N. Hinds
title: Design of the ARM VFP11 Divide and Square Root Synthesisable Macrocell
crossref: Kornerup:2007:PIS
pages: 87--96
year: 2007
DOI: https://doi.org/10.1109/ARITH.2007.15
bibdate: Tue Oct 9 16:32:41 MDT 2007
bibtimestamp: 2007.10.09 16:32:41 MDT
keywords: ARITH-18
acknowledgement: ack-nhfb
rawauthor: Neil Burgess and Chris N. Hinds
rawtitle: Design of the {ARM VFP11} Divide and Square Root Synthesisable Macrocell
abstract:
This paper presents the detailed design of the ARM VFP11 Divide and Square Root synthesisable macrocell. The macrocell was designed using the minimum-redundancy radix-4 SRT digit recurrence algorithm, and this paper describes a novel acceleration technique employed to achieve the required processor clock frequency of up to 750MHz in 90nm CMOS. Logical Effort theory is used to provide a delay analysis of the unit, which demonstrates the balanced nature of the two critical paths therein.

rawabstract:
This paper presents the detailed design of the ARM VFP11 Divide and Square Root synthesisable macrocell. The macrocell was designed using the minimum-redundancy radix-4 SRT digit recurrence algorithm, and this paper describes a novel acceleration technique employed to achieve the required processor clock frequency of up to 750MHz in 90nm CMOS. Logical Effort theory is used to provide a delay analysis of the unit, which demonstrates the balanced nature of the two critical paths therein.

entry:
@InProceedings{Burgess:2007:DAV,
  author =       "Neil Burgess and Chris N. Hinds",
  title =        "Design of the {ARM VFP11} Divide and Square Root
                 Synthesisable Macrocell",
  crossref =     "Kornerup:2007:PIS",
  pages =        "87--96",
  year =         "2007",
  DOI =          "https://doi.org/10.1109/ARITH.2007.15",
  bibdate =      "Tue Oct 9 16:32:41 MDT 2007",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  abstract =     "This paper presents the detailed design of the ARM
                 VFP11 Divide and Square Root synthesisable macrocell.
                 The macrocell was designed using the minimum-redundancy
                 radix-4 SRT digit recurrence algorithm, and this paper
                 describes a novel acceleration technique employed to
                 achieve the required processor clock frequency of up to
                 750MHz in 90nm CMOS. Logical Effort theory is used to
                 provide a delay analysis of the unit, which
                 demonstrates the balanced nature of the two critical
                 paths therein.",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH-18",
}


bibtype: article
filename: elefunt.bib
label: Cerone:2007:SFA
author: Pietro Cerone
title: Special functions: approximations and bounds
journal: Applicable Analysis and Discrete Mathematics
volume: 1
number: 1
pages: 72--91
year: 2007
DOI: https://doi.org/10.2298/AADM0701072C
ISSN: 1452-8630 (print), 2406-100X (electronic) OR 14528630 (print), 2406100X (electronic)
ISSNL: 1452-8630 OR 14528630
MRclass: 26D15 (26D20 33B15 33C05)
MRnumber: 2316589
MRreviewer: Pierpaolo Natalini
bibdate: Thu Jul 29 07:41:55 2021
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2021.07.29 07:41:55 ???
URL: http://pefmath.etf.rs/accepted/AADM-Vol1-No1-72-91.pdf
fjournal: Applicable Analysis and Discrete Mathematics
acknowledgement: ack-nhfb
ajournal: Appl. Anal. Discrete Math.
rawauthor: Pietro Cerone
rawtitle: Special functions: approximations and bounds
abstract:
The Steffensen inequality and bounds for the Cebysev functional are utilised to obtain bounds for some classical special functions. The technique relies on determining bounds on integrals of products of functions. The above techniques are used to obtain novel and useful bounds for the Bessel function of the first kind, the Beta function, and the Zeta function.

rawabstract:
The Steffensen inequality and bounds for the {\v{C}}eby{\v{s}}ev functional are utilised to obtain bounds for some classical special functions. The technique relies on determining bounds on integrals of products of functions. The above techniques are used to obtain novel and useful bounds for the Bessel function of the first kind, the Beta function, and the Zeta function.

entry:
@Article{Cerone:2007:SFA,
  author =       "Pietro Cerone",
  title =        "Special functions: approximations and bounds",
  journal =      "Applicable Analysis and Discrete Mathematics",
  volume =       "1",
  number =       "1",
  pages =        "72--91",
  year =         "2007",
  DOI =          "https://doi.org/10.2298/AADM0701072C",
  ISSN =         "1452-8630 (print), 2406-100X (electronic)",
  ISSN-L =       "1452-8630",
  MRclass =      "26D15 (26D20 33B15 33C05)",
  MRnumber =     "2316589",
  MRreviewer =   "Pierpaolo Natalini",
  bibdate =      "Thu Jul 29 07:41:55 2021",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://pefmath.etf.rs/accepted/AADM-Vol1-No1-72-91.pdf",
  abstract =     "The Steffensen inequality and bounds for the
                 {\v{C}}eby{\v{s}}ev functional are utilised to obtain
                 bounds for some classical special functions. The
                 technique relies on determining bounds on integrals of
                 products of functions. The above techniques are used to
                 obtain novel and useful bounds for the Bessel function
                 of the first kind, the Beta function, and the Zeta
                 function.",
  acknowledgement = ack-nhfb,
  ajournal =     "Appl. Anal. Discrete Math.",
  fjournal =     "Applicable Analysis and Discrete Mathematics",
}


bibtype: book
filename: elefunt.bib
label: Chakraborty:2007:VSF
author: Kalyan Chakraborty and Shigeru Kanemitsu and Haruo Tsukada
title: Vistas of special functions II
publisher: pub-WORLD-SCI
address: pub-WORLD-SCI:adr
pages: xii + 215
year: 2007
ISBN: 981-270-774-3 OR 9812707743
ISBN13: 978-981-270-774-1 OR 9789812707741
LCCN: QA351 .K35 2007
bibdate: Sat Oct 30 17:02:07 MDT 2010
bibtimestamp: 2010.10.30 17:02:07 MDT
subject: Functions, Special; Bernoulli polynomials
acknowledgement: ack-nhfb
rawauthor: Kalyan Chakraborty and Shigeru Kanemitsu and Haruo Tsukada
rawtitle: Vistas of special functions {II}
bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcw.bib; prodorbis.library.yale.edu:7090/voyager

abstract:
This is a unique book for studying special functions through zeta-functions. Many important formulas of special functions scattered throughout the literature are located in their proper positions and readers get enlightened access to them in this book. The areas covered include: Bernoulli polynomials, the gamma function (the beta and the digamma function), the zeta-functions (the Hurwitz, the Lerch, and the Epstein zeta-function), Bessel functions, an introduction to Fourier analysis, finite Fourier series, Dirichlet L-functions, the rudiments of complex functions and summation formulas. The Fourier series for the (first) periodic Bernoulli polynomial is effectively used, familiarizing the reader with the relationship between special functions and zeta-functions.

TOC:
The theory of Bernoulli and allied polynomials The theory of the gamma and related functions The theory of the Hurwitz--Lerch zeta-functions The theory of Bernoulli polynomials via zeta-functions The theory of the gamma and related functions via zeta-functions The theory of Bessel functions and the Epstein zeta-functions Fourier series and Fourier transforms Around Dirichlet's $L$-functions Appendix A: Complex functions Appendix B: Summation formulas and convergence theorems

rawabstract:
This is a unique book for studying special functions through zeta-functions. Many important formulas of special functions scattered throughout the literature are located in their proper positions and readers get enlightened access to them in this book. The areas covered include: Bernoulli polynomials, the gamma function (the beta and the digamma function), the zeta-functions (the Hurwitz, the Lerch, and the Epstein zeta-function), Bessel functions, an introduction to Fourier analysis, finite Fourier series, Dirichlet L-functions, the rudiments of complex functions and summation formulas. The Fourier series for the (first) periodic Bernoulli polynomial is effectively used, familiarizing the reader with the relationship between special functions and zeta-functions.

rawTOC:
The theory of Bernoulli and allied polynomials \\ The theory of the gamma and related functions \\ The theory of the Hurwitz--Lerch zeta-functions \\ The theory of Bernoulli polynomials via zeta-functions \\ The theory of the gamma and related functions via zeta-functions \\ The theory of Bessel functions and the Epstein zeta-functions \\ Fourier series and Fourier transforms \\ Around Dirichlet's $L$-functions \\ Appendix A: Complex functions \\ Appendix B: Summation formulas and convergence theorems

entry:
@Book{Chakraborty:2007:VSF,
  author =       "Kalyan Chakraborty and Shigeru Kanemitsu and Haruo
                 Tsukada",
  title =        "Vistas of special functions {II}",
  publisher =    pub-WORLD-SCI,
  address =      pub-WORLD-SCI:adr,
  pages =        "xii + 215",
  year =         "2007",
  ISBN =         "981-270-774-3",
  ISBN-13 =      "978-981-270-774-1",
  LCCN =         "QA351 .K35 2007",
  bibdate =      "Sat Oct 30 17:02:07 MDT 2010",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/mathcw.bib;
                 prodorbis.library.yale.edu:7090/voyager",
  abstract =     "This is a unique book for studying special functions
                 through zeta-functions. Many important formulas of
                 special functions scattered throughout the literature
                 are located in their proper positions and readers get
                 enlightened access to them in this book. The areas
                 covered include: Bernoulli polynomials, the gamma
                 function (the beta and the digamma function), the
                 zeta-functions (the Hurwitz, the Lerch, and the Epstein
                 zeta-function), Bessel functions, an introduction to
                 Fourier analysis, finite Fourier series, Dirichlet
                 L-functions, the rudiments of complex functions and
                 summation formulas. The Fourier series for the (first)
                 periodic Bernoulli polynomial is effectively used,
                 familiarizing the reader with the relationship between
                 special functions and zeta-functions.",
  acknowledgement = ack-nhfb,
  subject =      "Functions, Special; Bernoulli polynomials",
  tableofcontents = "The theory of Bernoulli and allied polynomials \\
                 The theory of the gamma and related functions \\
                 The theory of the Hurwitz--Lerch zeta-functions \\
                 The theory of Bernoulli polynomials via zeta-functions
                 \\
                 The theory of the gamma and related functions via
                 zeta-functions \\
                 The theory of Bessel functions and the Epstein
                 zeta-functions \\
                 Fourier series and Fourier transforms \\
                 Around Dirichlet's $L$-functions \\
                 Appendix A: Complex functions \\
                 Appendix B: Summation formulas and convergence
                 theorems",
}


bibtype: article
filename: elefunt.bib
label: Detrey:2007:PFP
author: Jeremie Detrey and Florent de Dinechin
title: Parameterized floating-point logarithm and exponential functions for FPGAs
journal: j-MICROPROC-MICROSYS
volume: 31
number: 8
pages: 537--545
day: 3
month: dec
monthnumber: 12
year: 2007
CODEN: MIMID5
DOI: https://doi.org/10.1016/j.micpro.2006.02.008
ISSN: 0141-9331 (print), 1872-9436 (electronic) OR 01419331 (print), 18729436 (electronic)
ISSNL: 0141-9331 OR 01419331
bibdate: Sat Nov 8 14:03:10 2025
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2025.11.08 14:03:10 ???
URL: http://www.ens-lyon.fr/LIP/Arenaire/
fjournal: Microprocessors and Microsystems
acknowledgement: ack-nhfb
rawauthor: J{\'e}r{\'e}mie Detrey and Florent de Dinechin
rawtitle: Parameterized floating-point logarithm and exponential functions for {FPGAs}
keywords:
Elementary functions; Exponential; Floating-point; FPGA; FPLibrary; Logarithm; Parameterized operators

entry:
@Article{Detrey:2007:PFP,
  author =       "J{\'e}r{\'e}mie Detrey and Florent de Dinechin",
  title =        "Parameterized floating-point logarithm and exponential
                 functions for {FPGAs}",
  journal =      j-MICROPROC-MICROSYS,
  volume =       "31",
  number =       "8",
  pages =        "537--545",
  day =          "3",
  month =        dec,
  year =         "2007",
  CODEN =        "MIMID5",
  DOI =          "https://doi.org/10.1016/j.micpro.2006.02.008",
  ISSN =         "0141-9331 (print), 1872-9436 (electronic)",
  ISSN-L =       "0141-9331",
  bibdate =      "Sat Nov 8 14:03:10 2025",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.ens-lyon.fr/LIP/Arenaire/",
  acknowledgement = ack-nhfb,
  fjournal =     "Microprocessors and Microsystems",
  keywords =     "Elementary functions; Exponential; Floating-point;
                 FPGA; FPLibrary; Logarithm; Parameterized operators",
}


bibtype: article
filename: elefunt.bib
label: Dyer:2007:AEF
author: Stephen Dyer and Justin Dyer
title: Approximations to Error Functions
journal: IEEE Instrumentation & Measurement Magazine
volume: 10
number: 6
pages: 45--48
month: dec
monthnumber: 12
year: 2007
DOI: https://doi.org/10.1109/mim.2007.4428581
bibdate: Sat Dec 16 16:26:38 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2017.12.16 16:26:38 ???
URL: https://ieeexplore.ieee.org/abstract/document/4428581/
acknowledgement: ack-nhfb
rawauthor: Stephen Dyer and Justin Dyer
rawtitle: Approximations to Error Functions
entry:
@Article{Dyer:2007:AEF,
  author =       "Stephen Dyer and Justin Dyer",
  title =        "Approximations to Error Functions",
  journal =      "IEEE Instrumentation \& Measurement Magazine",
  volume =       "10",
  number =       "6",
  pages =        "45--48",
  month =        dec,
  year =         "2007",
  DOI =          "https://doi.org/10.1109/mim.2007.4428581",
  bibdate =      "Sat Dec 16 16:26:38 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "https://ieeexplore.ieee.org/abstract/document/4428581/",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: elefunt.bib
label: Ercegovac:2007:CSR
author: Milos D. Ercegovac and Jean-Michel Muller
title: Complex Square Root with Operand Prescaling
journal: j-J-VLSI-SIGNAL-PROC
volume: 49
number: 1
pages: 19--30
month: oct
monthnumber: 10
year: 2007
CODEN: JVSPED
DOI: https://doi.org/10.1007/s11265-006-0029-2
ISSN: 0922-5773 (print), 1573-109x (electronic) OR 09225773 (print), 1573109x (electronic)
ISSNL: 0922-5773 OR 09225773
bibdate: Mon Nov 05 19:24:36 2007
bibtimestamp: 2007.11.05 19:24:36 ???
fjournal: Journal of VLSI Signal Processing
acknowledgement: ack-nhfb
rawauthor: Milo{\v{s}} D. Ercegovac and Jean-Michel Muller
rawtitle: Complex Square Root with Operand Prescaling
abstract:
We propose a radix-$r$ digit-recurrence algorithm for complex square-root. The operand is prescaled to allow the selection of square-root digits by rounding of the residual. This leads to a simple hardware implementation of digit selection. Moreover, the use of digit recurrence approach allows correct rounding of the result if needed. The algorithm, compatible with the complex division presented in Ercegovac and Muller (``Complex Division with Prescaling of the Operands,'' in Proc. Application-Specific Systems, Architectures, and Processors (ASAP'03), The Hague, The Netherlands, June 24---26, 2003), and its design are described. We also give rough estimates of its latency and cost with respect to implementation based on standard floating-point instructions as used in software routines for complex square root.

rawabstract:
We propose a radix-$r$ digit-recurrence algorithm for complex square-root. The operand is prescaled to allow the selection of square-root digits by rounding of the residual. This leads to a simple hardware implementation of digit selection. Moreover, the use of digit recurrence approach allows correct rounding of the result if needed. The algorithm, compatible with the complex division presented in Ercegovac and Muller (``Complex Division with Prescaling of the Operands,'' in Proc. Application-Specific Systems, Architectures, and Processors (ASAP'03), The Hague, The Netherlands, June 24---26, 2003), and its design are described. We also give rough estimates of its latency and cost with respect to implementation based on standard floating-point instructions as used in software routines for complex square root.

entry:
@Article{Ercegovac:2007:CSR,
  author =       "Milo{\v{s}} D. Ercegovac and Jean-Michel Muller",
  title =        "Complex Square Root with Operand Prescaling",
  journal =      j-J-VLSI-SIGNAL-PROC,
  volume =       "49",
  number =       "1",
  pages =        "19--30",
  month =        oct,
  year =         "2007",
  CODEN =        "JVSPED",
  DOI =          "https://doi.org/10.1007/s11265-006-0029-2",
  ISSN =         "0922-5773 (print), 1573-109x (electronic)",
  ISSN-L =       "0922-5773",
  bibdate =      "Mon Nov 05 19:24:36 2007",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  abstract =     "We propose a radix-$r$ digit-recurrence algorithm for
                 complex square-root. The operand is prescaled to allow
                 the selection of square-root digits by rounding of the
                 residual. This leads to a simple hardware
                 implementation of digit selection. Moreover, the use of
                 digit recurrence approach allows correct rounding of
                 the result if needed. The algorithm, compatible with
                 the complex division presented in Ercegovac and Muller
                 (``Complex Division with Prescaling of the Operands,''
                 in Proc. Application-Specific Systems, Architectures,
                 and Processors (ASAP'03), The Hague, The Netherlands,
                 June 24---26, 2003), and its design are described. We
                 also give rough estimates of its latency and cost with
                 respect to implementation based on standard
                 floating-point instructions as used in software
                 routines for complex square root.",
  acknowledgement = ack-nhfb,
  fjournal =     "Journal of VLSI Signal Processing",
}


bibtype: article
filename: elefunt.bib
label: Ferraro:2007:FAG
author: Giovanni Ferraro
title: The foundational aspects of Gauss's work on the hypergeometric, factorial and digamma functions
journal: j-ARCH-HIST-EXACT-SCI
volume: 61
number: 5
pages: 457--518
month: sep
monthnumber: 9
year: 2007
CODEN: AHESAN
DOI: https://doi.org/10.1007/s00407-007-0004-8
ISSN: 0003-9519 (print), 1432-0657 (electronic) OR 00039519 (print), 14320657 (electronic)
ISSNL: 0003-9519 OR 00039519
MRclass: 33-03 (01A50 33B15 33C05)
MRnumber: 2329096 (2009d:33002)
MRreviewer: M. E. Muldoon
bibdate: Fri Feb 4 21:50:42 MST 2011
bibtimestamp: 2011.02.04 21:50:42 MST
URL: http://www.springerlink.com/openurl.asp?genre=article&issn=0003-9519&volume=61&issue=5&spage=457
fjournal: Archive for History of Exact Sciences
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/407
rawauthor: Giovanni Ferraro
rawtitle: The foundational aspects of {Gauss}'s work on the hypergeometric, factorial and digamma functions
bibsource:
http://springerlink.metapress.com/openurl.asp?genre=issue&issn=0003-9519&volume=61&issue=5; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

entry:
@Article{Ferraro:2007:FAG,
  author =       "Giovanni Ferraro",
  title =        "The foundational aspects of {Gauss}'s work on the
                 hypergeometric, factorial and digamma functions",
  journal =      j-ARCH-HIST-EXACT-SCI,
  volume =       "61",
  number =       "5",
  pages =        "457--518",
  month =        sep,
  year =         "2007",
  CODEN =        "AHESAN",
  DOI =          "https://doi.org/10.1007/s00407-007-0004-8",
  ISSN =         "0003-9519 (print), 1432-0657 (electronic)",
  ISSN-L =       "0003-9519",
  MRclass =      "33-03 (01A50 33B15 33C05)",
  MRnumber =     "2329096 (2009d:33002)",
  MRreviewer =   "M. E. Muldoon",
  bibdate =      "Fri Feb 4 21:50:42 MST 2011",
  bibsource =    "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=0003-9519&volume=61&issue=5;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.springerlink.com/openurl.asp?genre=article&issn=0003-9519&volume=61&issue=5&spage=457",
  acknowledgement = ack-nhfb,
  fjournal =     "Archive for History of Exact Sciences",
  journal-URL =  "http://link.springer.com/journal/407",
  MRtitle =      "The foundational aspects of {Gauss}'s work on the
                 hypergeometric, factorial and digamma functions",
}


bibtype: article
filename: elefunt.bib
label: Fousse:2007:MMP
author: Laurent Fousse and Guillaume Hanrot and Vincent Lefevre and Patrick Pelissier and Paul Zimmermann
title: MPFR: a multiple-precision binary floating-point library with correct rounding
journal: j-TOMS
volume: 33
number: 2
pages: 1--15
month: jun
monthnumber: 6
year: 2007
CODEN: ACMSCU
DOI: https://doi.org/10.1145/1236463.1236468
ISSN: 0098-3500 (print), 1557-7295 (electronic) OR 00983500 (print), 15577295 (electronic)
ISSNL: 0098-3500 OR 00983500
MRclass: 65G99
MRnumber: MR2326955
bibdate: Thu Jul 26 17:36:59 MDT 2007
bibtimestamp: 2007.07.26 17:36:59 MDT
fjournal: ACM Transactions on Mathematical Software (TOMS)
acknowledgement: ack-nhfb
articleno: 13
journalURL: https://dl.acm.org/loi/toms
rawtitle: {MPFR}: a multiple-precision binary floating-point library with correct rounding
bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib

abstract:
This article presents a multiple-precision binary floating-point library, written in the ISO C language, and based on the GNU MP library. Its particularity is to extend to arbitrary-precision, ideas from the IEEE 754 standard, by providing correct rounding and exceptions. We demonstrate how these strong semantics are achieved---with no significant slowdown with respect to other arbitrary-precision tools---and discuss a few applications where such a library can be useful.

rawabstract:
This article presents a multiple-precision binary floating-point library, written in the ISO C language, and based on the GNU MP library. Its particularity is to extend to arbitrary-precision, ideas from the IEEE 754 standard, by providing correct rounding and exceptions. We demonstrate how these strong semantics are achieved---with no significant slowdown with respect to other arbitrary-precision tools---and discuss a few applications where such a library can be useful.

rawauthor:
Laurent Fousse and Guillaume Hanrot and Vincent Lef{\`e}vre and Patrick P{\'e}lissier and Paul Zimmermann

entry:
@Article{Fousse:2007:MMP,
  author =       "Laurent Fousse and Guillaume Hanrot and Vincent
                 Lef{\`e}vre and Patrick P{\'e}lissier and Paul
                 Zimmermann",
  title =        "{MPFR}: a multiple-precision binary floating-point
                 library with correct rounding",
  journal =      j-TOMS,
  volume =       "33",
  number =       "2",
  pages =        "1--15",
  month =        jun,
  year =         "2007",
  CODEN =        "ACMSCU",
  DOI =          "https://doi.org/10.1145/1236463.1236468",
  ISSN =         "0098-3500 (print), 1557-7295 (electronic)",
  ISSN-L =       "0098-3500",
  MRclass =      "65G99",
  MRnumber =     "MR2326955",
  bibdate =      "Thu Jul 26 17:36:59 MDT 2007",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/toms.bib",
  abstract =     "This article presents a multiple-precision binary
                 floating-point library, written in the ISO C language,
                 and based on the GNU MP library. Its particularity is
                 to extend to arbitrary-precision, ideas from the IEEE
                 754 standard, by providing correct rounding and
                 exceptions. We demonstrate how these strong semantics
                 are achieved---with no significant slowdown with
                 respect to other arbitrary-precision tools---and
                 discuss a few applications where such a library can be
                 useful.",
  acknowledgement = ack-nhfb,
  articleno =    "13",
  fjournal =     "ACM Transactions on Mathematical Software (TOMS)",
  journal-URL =  "https://dl.acm.org/loi/toms",
}


bibtype: book
filename: elefunt.bib
label: Gil:2007:NMS
author: Amparo Gil and Javier Segura and N. M. Temme
title: Numerical Methods for Special Functions
publisher: pub-SIAM
address: pub-SIAM:adr
pages: xvi + 415
year: 2007
DOI: https://doi.org/10.1137/1.9780898717822
ISBN: 0-89871-634-9 OR 0898716349
ISBN13: 978-0-89871-634-4 OR 9780898716344
LCCN: QA351 .G455 2007
bibdate: Fri Sep 14 10:24:22 MDT 2007
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager
bibtimestamp: 2007.09.14 10:24:22 MDT
acknowledgement: ack-nhfb
rawauthor: Amparo Gil and Javier Segura and N. M. Temme
rawtitle: Numerical Methods for Special Functions
subject:
functions, special; data processing; numerical analysis; asymptotic expansions; approximation theory

TOC:
Preface xiii 1 Introduction / 1 I Basic Methods / 13 2 Convergent and Divergent Series / 15 2.1 Introduction / 15 2.1.1 Power series: First steps / 15 2.1.2 Further practical aspects / 17 2.2 Differential equations and Frobenius series solutions / 18 2.2.1 Singular points / 19 2.2.2 The solution near a regular point / 20 2.2.3 Power series expansions around a regular singular point / 22 2.2.4 The Liouville transformation / 25 2.3 Hypergeometric series / 26 2.3.1 The Gauss hypergeometric function / 28 2.3.2 Other power series for the Gauss hypergeometric function / 30 2.3.3 Removable singularities / 33 2.4 Asymptotic expansions / 34 2.4.1 Watson's lemma / 36 2.4.2 Estimating the remainders of asymptotic expansions / 38 2.4.3 Exponentially improved asymptotic expansions / 39 2.4.4 Alternatives of asymptotic expansions / 40 3 Chebyshev Expansions / 51 3.1 Introduction / 51 3.2 Basic results on interpolation / 52 3.2.1 The Runge phenomenon and the Chebyshev nodes / 54 3.3 Chebyshev polynomials: Basic properties / 56 3.3.1 Properties of the Chebyshev polynomials $T_n(x)$ / 56 3.3.2 Chebyshev polynomials of the second, third, and fourth kinds / 60 vii 3.4 Chebyshev interpolation / 62 3.4.1 Computing the Chebyshev interpolation polynomial / 64 3.5 Expansions in terms of Chebyshev polynomials / 66 3.5.1 Convergence properties of Chebyshev expansions / 68 3.6 Computing the coefficients of a Chebyshev expansion / 69 3.6.1 Clenshaw's method for solutions of linear differential equations with polynomial coefficients / 70 3.7 Evaluation of a Chebyshev sum / 75 3.7.1 Clenshaw's method for the evaluation of a Chebyshev sum / 75 3.8 Economization of power series / 80 3.9 Example: Computation of Airy functions of real variable / 80 3.10 Chebyshev expansions with coefficients in terms of special functions / 83 4 Linear Recurrence Relations and Associated Continued Fractions / 87 4.1 Introduction / 87 4.2 Condition of three-term recurrence relations / 88 4.2.1 Minimal solutions / 89 4.3 Perron's theorem / 92 4.3.1 Scaled recurrence relations / 94 4.4 Minimal solutions of TTRRs and continued fractions / 95 4.5 Some notable recurrence relations / 96 4.5.1 The confluent hypergeometric family / 96 4.5.2 The Gauss hypergeometric family / 102 4.6 Computing the minimal solution of a TTRR / 105 4.6.1 Miller's algorithm when a function value is known / 105 4.6.2 Miller's algorithm with a normalizing sum / 107 4.6.3 ``Anti-Miller'' algorithm / 110 4.7 Inhomogeneous linear difference equations / 112 4.7.1 Inhomogeneous first order difference equations. Examples / 112 4.7.2 Inhomogeneous second order difference equations / 115 4.7.3 Olver's method / 116 4.8 Anomalous behavior of some second order homogeneous and first order inhomogeneous recurrences / 118 4.8.1 A canonical example: Modified Bessel function / 118 4.8.2 Other examples: Hypergeometric recursions / 120 4.8.3 A first order inhomogeneous equation / 121 4.8.4 A warning / 122 5 Quadrature Methods / 123 5.1 Introduction / 123 5.2 Newton--Cotes quadrature: The trapezoidal and Simpson's rule / 124 5.2.1 The compound trapezoidal rule / 126 5.2.2 The recurrent trapezoidal rule / 129 5.2.3 Euler's summation formula and the trapezoidal rule / 130 5.3 Gauss quadrature / 132 5.3.1 Basics of the theory of orthogonal polynomials and Gauss quadrature / 133 5.3.2 The Golub--Welsch algorithm / 141 5.3.3 Example: The Airy function in the complex plane / 145 5.3.4 Further practical aspects of Gauss quadrature / 146 5.4 The trapezoidal rule on $mathbb R$ / 147 5.4.1 Contour integral formulas for the truncation errors / 148 5.4.2 Transforming the variable of integration / 153 5.5 Contour integrals and the saddle point method / 157 5.5.1 The saddle point method / 158 5.5.2 Other integration contours / 163 5.5.3 Integrating along the saddle point contours and examples / 165 II Further Tools and Methods / 171 6 Numerical Aspects of Continued Fractions / 173 6.1 Introduction / 173 6.2 Definitions and notation / 173 6.3 Equivalence transformations and contractions / 175 6.4 Special forms of continued fractions / 178 6.4.1 Stieltjes fractions / 178 6.4.2 Jacobi fractions / 179 6.4.3 Relation with Pade approximants / 179 6.5 Convergence of continued fractions / 179 6.6 Numerical evaluation of continued fractions / 181 6.6.1 Steed's algorithm / 181 6.6.2 The modified Lentz algorithm / 183 6.7 Special functions and continued fractions / 185 6.7.1 Incomplete gamma function / 186 6.7.2 Gauss hypergeometric functions / 187 7 Computation of the Zeros of Special Functions / 191 7.1 Introduction / 191 7.2 Some classical methods / 193 7.2.1 The bisection method / 193 7.2.2 The fixed point method and the Newton--Raphson method / 193 7.2.3 Complex zeros / 197 7.3 Local strategies: Asymptotic and other approximations / 197 7.3.1 Asymptotic approximations for large zeros / 199 7.3.2 Other approximations / 202 7.4 Global strategies I: Matrix methods / 205 7.4.1 The eigenvalue problem for orthogonal polynomials / 206 7.4.2 The eigenvalue problem for minimal solutions of TTRRs / 207 7.5 Global strategies II: Global fixed point methods / 213 7.5.1 Zeros of Bessel functions / 213 7.5.2 The general case / 219 7.6 Asymptotic methods: Further examples / 224 7.6.1 Airy functions / 224 7.6.2 Scorer functions / 227 7.6.3 The error functions / 229 7.6.4 The parabolic cylinder function / 233 7.6.5 Bessel functions / 233 7.6.6 Orthogonal polynomials / 234 8 Uniform Asymptotic Expansions / 237 8.1 Asymptotic expansions for the incomplete gamma functions / 237 8.2 Uniform asymptotic expansions / 239 8.3 Uniform asymptotic expansions for the incomplete gamma functions / 240 8.3.1 The uniform expansion / 242 8.3.2 Expansions for the coefficients / 244 8.3.3 Numerical algorithm for small values of $eta $ / 245 8.3.4 A simpler uniform expansion / 247 8.4 Airy-type expansions for Bessel functions / 249 8.4.1 The Airy-type asymptotic expansions / 250 8.4.2 Representations of $a_s()$, $b_s()$, $c_s()$, $d_s()$ / 253 8.4.3 Properties of the functions $A_nu $, $B_nu $, $C_nu $, $D_nu $ / 254 8.4.4 Expansions for $a_s()$, $b_s()$, $c_s()$, $d_s()$ / 256 8.4.5 Evaluation of the functions $A_nu ()$, $B_nu ()$ by iteration / 258 8.5 Airy-type asymptotic expansions obtained from integrals / 263 8.5.1 Airy-type asymptotic expansions / 264 8.5.2 How to compute the coefficients $alpha _n$, $beta _n$ / 267 8.5.3 Application to parabolic cylinder functions / 270 9 Other Methods / 275 9.1 Introduction / 275 9.2 Pade approximations / 276 9.2.1 Pade approximants and continued fractions / 278 9.2.2 How to compute the Pade approximants / 278 9.2.3 Pade approximants to the exponential function / 280 9.2.4 Analytic forms of Pade approximations / 283 9.3 Sequence transformations / 286 9.3.1 The principles of sequence transformations / 286 9.3.2 Examples of sequence transformations / 287 9.3.3 The transformation of power series / 288 9.3.4 Numerical examples / 288 9.4 Best rational approximations / 290 9.5 Numerical solution of ordinary differential equations: Taylor expansion method / 291 9.5.1 Taylor-series method: Initial value problems / 292 9.5.2 Taylor-series method: Boundary value problem / 293 9.6 Other quadrature methods / 294 9.6.1 Romberg quadrature / 294 9.6.2 Fejer and Clenshaw--Curtis quadratures / 296 9.6.3 Other Gaussian quadratures / 298 9.6.4 Oscillatory integrals / 301 III Related Topics and Examples / 307 10 Inversion of Cumulative Distribution Functions / 309 10.1 Introduction / 309 10.2 Asymptotic inversion of the complementary error function / 309 10.3 Asymptotic inversion of incomplete gamma functions / 312 10.3.1 The asymptotic inversion method / 312 10.3.2 Determination of the coefficients i / 314 10.3.3 Expansions of the coefficients i / 316 10.3.4 Numerical examples / 316 10.4 Generalizations / 317 10.5 Asymptotic inversion of the incomplete beta function / 318 10.5.1 The nearly symmetric case / 319 10.5.2 The general error function case / 322 10.5.3 The incomplete gamma function case / 324 10.5.4 Numerical aspects / 326 10.6 High order Newton-like methods / 327 11 Further Examples / 331 11.1 Introduction / 331 11.2 The Euler summation formula / 331 11.3 Approximations of Stirling numbers / 336 11.3.1 Definitions / 337 11.3.2 Asymptotics for Stirling numbers of the second kind / 338 11.3.3 Stirling numbers of the first kind / 343 11.4 Symmetric elliptic integrals / 344 11.4.1 The standard forms in terms of symmetric integrals / 345 11.4.2 An algorithm / 346 11.4.3 Other elliptic integrals / 347 11.5 Numerical inversion of Laplace transforms / 347 11.5.1 Complex Gauss quadrature / 348 11.5.2 Deforming the contour / 349 11.5.3 Using Pade approximations / 352 IV Software / 353 12 Associated Algorithms / 355 12.1 Introduction / 355 12.1.1 Errors and stability: Basic terminology / 356 12.1.2 Design and testing of software for computing functions: General philosophy / 357 12.1.3 Scaling the functions / 358 12.2 Airy and Scorer functions of complex arguments / 359 12.2.1 Purpose / 359 12.2.2 Algorithms / 359 12.3 Associated Legendre functions of integer and half-integer degrees / 363 12.3.1 Purpose / 363 12.3.2 Algorithms / 364 12.4 Bessel functions / 369 12.4.1 Modified Bessel functions of integer and half-integer orders / 370 12.4.2 Modified Bessel functions of purely imaginary orders / 372 12.5 Parabolic cylinder functions / 377 12.5.1 Purpose / 377 12.5.2 Algorithm / 378 12.6 Zeros of Bessel functions / 385 12.6.1 Purpose / 385 12.6.2 Algorithm / 385 List of Algorithms / 387 Bibliography / 389 Index / 405

shortTOC:
Preface 1. Introduction I. Basic methods 2. Convergent and divergent series 3. Chebyshev expansions 4. Linear recurrence relations and associated continued fractions 5. Quadrature methods II. Further tools and methods 6. Numerical aspects of continued fractions 7. Computation of the zeros of special functions 8. Uniform asymptotic expansions 9. Other methods III. Related topics and examples 10. Inversion of cumulative distribution functions 11. Further examples 12. Associated algorithms List of algorithms Bibliography Index

rawshortTOC:
Preface \\ 1. Introduction \\ I. Basic methods \\ 2. Convergent and divergent series \\ 3. Chebyshev expansions \\ 4. Linear recurrence relations and associated continued fractions \\ 5. Quadrature methods \\ II. Further tools and methods \\ 6. Numerical aspects of continued fractions \\ 7. Computation of the zeros of special functions \\ 8. Uniform asymptotic expansions \\ 9. Other methods \\ III. Related topics and examples \\ 10. Inversion of cumulative distribution functions \\ 11. Further examples \\ 12. Associated algorithms \\ List of algorithms \\ Bibliography \\ Index

rawTOC:
Preface xiii 1 Introduction / 1 \\ I Basic Methods / 13 \\ 2 Convergent and Divergent Series / 15 \\ 2.1 Introduction / 15 \\ 2.1.1 Power series: First steps / 15 \\ 2.1.2 Further practical aspects / 17 \\ 2.2 Differential equations and Frobenius series solutions / 18 \\ 2.2.1 Singular points / 19 \\ 2.2.2 The solution near a regular point / 20 \\ 2.2.3 Power series expansions around a regular singular point / 22 \\ 2.2.4 The Liouville transformation / 25 \\ 2.3 Hypergeometric series / 26 \\ 2.3.1 The Gauss hypergeometric function / 28 \\ 2.3.2 Other power series for the Gauss hypergeometric function / 30 \\ 2.3.3 Removable singularities / 33 \\ 2.4 Asymptotic expansions / 34 \\ 2.4.1 Watson's lemma / 36 \\ 2.4.2 Estimating the remainders of asymptotic expansions / 38 \\ 2.4.3 Exponentially improved asymptotic expansions / 39 \\ 2.4.4 Alternatives of asymptotic expansions / 40 \\ 3 Chebyshev Expansions / 51 \\ 3.1 Introduction / 51 \\ 3.2 Basic results on interpolation / 52 \\ 3.2.1 The Runge phenomenon and the Chebyshev nodes / 54 \\ 3.3 Chebyshev polynomials: Basic properties / 56 \\ 3.3.1 Properties of the Chebyshev polynomials $T_n(x)$ / 56 \\ 3.3.2 Chebyshev polynomials of the second, third, and fourth kinds / 60 \\ vii 3.4 Chebyshev interpolation / 62 \\ 3.4.1 Computing the Chebyshev interpolation polynomial / 64 \\ 3.5 Expansions in terms of Chebyshev polynomials / 66 \\ 3.5.1 Convergence properties of Chebyshev expansions / 68 \\ 3.6 Computing the coefficients of a Chebyshev expansion / 69 \\ 3.6.1 Clenshaw's method for solutions of linear differential equations with polynomial coefficients / 70 \\ 3.7 Evaluation of a Chebyshev sum / 75 \\ 3.7.1 Clenshaw's method for the evaluation of a Chebyshev sum / 75 \\ 3.8 Economization of power series / 80 \\ 3.9 Example: Computation of Airy functions of real variable / 80 \\ 3.10 Chebyshev expansions with coefficients in terms of special functions / 83 \\ 4 Linear Recurrence Relations and Associated Continued Fractions / 87 \\ 4.1 Introduction / 87 \\ 4.2 Condition of three-term recurrence relations / 88 \\ 4.2.1 Minimal solutions / 89 \\ 4.3 Perron's theorem / 92 \\ 4.3.1 Scaled recurrence relations / 94 \\ 4.4 Minimal solutions of TTRRs and continued fractions / 95 \\ 4.5 Some notable recurrence relations / 96 \\ 4.5.1 The confluent hypergeometric family / 96 \\ 4.5.2 The Gauss hypergeometric family / 102 \\ 4.6 Computing the minimal solution of a TTRR / 105 \\ 4.6.1 Miller's algorithm when a function value is known / 105 \\ 4.6.2 Miller's algorithm with a normalizing sum / 107 \\ 4.6.3 ``Anti-Miller'' algorithm / 110 \\ 4.7 Inhomogeneous linear difference equations / 112 \\ 4.7.1 Inhomogeneous first order difference equations. Examples / 112 \\ 4.7.2 Inhomogeneous second order difference equations / 115 \\ 4.7.3 Olver's method / 116 \\ 4.8 Anomalous behavior of some second order homogeneous and first order inhomogeneous recurrences / 118 \\ 4.8.1 A canonical example: Modified Bessel function / 118 \\ 4.8.2 Other examples: Hypergeometric recursions / 120 \\ 4.8.3 A first order inhomogeneous equation / 121 \\ 4.8.4 A warning / 122 \\ 5 Quadrature Methods / 123 \\ 5.1 Introduction / 123 \\ 5.2 Newton--Cotes quadrature: The trapezoidal and Simpson's rule / 124 \\ 5.2.1 The compound trapezoidal rule / 126 \\ 5.2.2 The recurrent trapezoidal rule / 129 \\ 5.2.3 Euler's summation formula and the trapezoidal rule / 130 \\ 5.3 Gauss quadrature / 132 \\ 5.3.1 Basics of the theory of orthogonal polynomials and Gauss quadrature / 133 \\ 5.3.2 The Golub--Welsch algorithm / 141 \\ 5.3.3 Example: The Airy function in the complex plane / 145 \\ 5.3.4 Further practical aspects of Gauss quadrature / 146 \\ 5.4 The trapezoidal rule on $\mathbb{R}$ / 147 \\ 5.4.1 Contour integral formulas for the truncation errors / 148 \\ 5.4.2 Transforming the variable of integration / 153 \\ 5.5 Contour integrals and the saddle point method / 157 \\ 5.5.1 The saddle point method / 158 \\ 5.5.2 Other integration contours / 163 \\ 5.5.3 Integrating along the saddle point contours and examples / 165 \\ II Further Tools and Methods / 171 \\ 6 Numerical Aspects of Continued Fractions / 173 \\ 6.1 Introduction / 173 \\ 6.2 Definitions and notation / 173 \\ 6.3 Equivalence transformations and contractions / 175 \\ 6.4 Special forms of continued fractions / 178 \\ 6.4.1 Stieltjes fractions / 178 \\ 6.4.2 Jacobi fractions / 179 \\ 6.4.3 Relation with Pad{\'e} approximants / 179 \\ 6.5 Convergence of continued fractions / 179 \\ 6.6 Numerical evaluation of continued fractions / 181 \\ 6.6.1 Steed's algorithm / 181 \\ 6.6.2 The modified Lentz algorithm / 183 \\ 6.7 Special functions and continued fractions / 185 \\ 6.7.1 Incomplete gamma function / 186 \\ 6.7.2 Gauss hypergeometric functions / 187 \\ 7 Computation of the Zeros of Special Functions / 191 \\ 7.1 Introduction / 191 \\ 7.2 Some classical methods / 193 \\ 7.2.1 The bisection method / 193 \\ 7.2.2 The fixed point method and the Newton--Raphson method / 193 \\ 7.2.3 Complex zeros / 197 \\ 7.3 Local strategies: Asymptotic and other approximations / 197 \\ 7.3.1 Asymptotic approximations for large zeros / 199 \\ 7.3.2 Other approximations / 202 \\ 7.4 Global strategies I: Matrix methods / 205 \\ 7.4.1 The eigenvalue problem for orthogonal polynomials / 206 \\ 7.4.2 The eigenvalue problem for minimal solutions of TTRRs / 207 \\ 7.5 Global strategies II: Global fixed point methods / 213 \\ 7.5.1 Zeros of Bessel functions / 213 \\ 7.5.2 The general case / 219 \\ 7.6 Asymptotic methods: Further examples / 224 \\ 7.6.1 Airy functions / 224 \\ 7.6.2 Scorer functions / 227 \\ 7.6.3 The error functions / 229 \\ 7.6.4 The parabolic cylinder function / 233 \\ 7.6.5 Bessel functions / 233 \\ 7.6.6 Orthogonal polynomials / 234 \\ 8 Uniform Asymptotic Expansions / 237 \\ 8.1 Asymptotic expansions for the incomplete gamma functions / 237 \\ 8.2 Uniform asymptotic expansions / 239 \\ 8.3 Uniform asymptotic expansions for the incomplete gamma functions / 240 \\ 8.3.1 The uniform expansion / 242 \\ 8.3.2 Expansions for the coefficients / 244 \\ 8.3.3 Numerical algorithm for small values of $\eta$ / 245 \\ 8.3.4 A simpler uniform expansion / 247 \\ 8.4 Airy-type expansions for Bessel functions / 249 \\ 8.4.1 The Airy-type asymptotic expansions / 250 \\ 8.4.2 Representations of $a_s()$, $b_s()$, $c_s()$, $d_s()$ / 253 \\ 8.4.3 Properties of the functions $A_\nu$, $B_\nu$, $C_\nu$, $D_\nu$ / 254 \\ 8.4.4 Expansions for $a_s()$, $b_s()$, $c_s()$, $d_s()$ / 256 \\ 8.4.5 Evaluation of the functions $A_\nu()$, $B_\nu()$ by iteration / 258 \\ 8.5 Airy-type asymptotic expansions obtained from integrals / 263 \\ 8.5.1 Airy-type asymptotic expansions / 264 \\ 8.5.2 How to compute the coefficients $\alpha_n$, $\beta_n$ / 267 \\ 8.5.3 Application to parabolic cylinder functions / 270 \\ 9 Other Methods / 275 \\ 9.1 Introduction / 275 \\ 9.2 Pad{\'e} approximations / 276 \\ 9.2.1 Pad{\'e} approximants and continued fractions / 278 \\ 9.2.2 How to compute the Pad{\'e} approximants / 278 \\ 9.2.3 Pad{\'e} approximants to the exponential function / 280 \\ 9.2.4 Analytic forms of Pad{\'e} approximations / 283 \\ 9.3 Sequence transformations / 286 \\ 9.3.1 The principles of sequence transformations / 286 \\ 9.3.2 Examples of sequence transformations / 287 \\ 9.3.3 The transformation of power series / 288 \\ 9.3.4 Numerical examples / 288 \\ 9.4 Best rational approximations / 290 \\ 9.5 Numerical solution of ordinary differential equations: Taylor expansion method / 291 \\ 9.5.1 Taylor-series method: Initial value problems / 292 \\ 9.5.2 Taylor-series method: Boundary value problem / 293 \\ 9.6 Other quadrature methods / 294 \\ 9.6.1 Romberg quadrature / 294 \\ 9.6.2 Fej{\'e}r and Clenshaw--Curtis quadratures / 296 \\ 9.6.3 Other Gaussian quadratures / 298 \\ 9.6.4 Oscillatory integrals / 301 \\ III Related Topics and Examples / 307 \\ 10 Inversion of Cumulative Distribution Functions / 309 \\ 10.1 Introduction / 309 \\ 10.2 Asymptotic inversion of the complementary error function / 309 \\ 10.3 Asymptotic inversion of incomplete gamma functions / 312 \\ 10.3.1 The asymptotic inversion method / 312 \\ 10.3.2 Determination of the coefficients i / 314 \\ 10.3.3 Expansions of the coefficients i / 316 \\ 10.3.4 Numerical examples / 316 \\ 10.4 Generalizations / 317 \\ 10.5 Asymptotic inversion of the incomplete beta function / 318 \\ 10.5.1 The nearly symmetric case / 319 \\ 10.5.2 The general error function case / 322 \\ 10.5.3 The incomplete gamma function case / 324 \\ 10.5.4 Numerical aspects / 326 \\ 10.6 High order Newton-like methods / 327 \\ 11 Further Examples / 331 \\ 11.1 Introduction / 331 \\ 11.2 The Euler summation formula / 331 \\ 11.3 Approximations of Stirling numbers / 336 \\ 11.3.1 Definitions / 337 \\ 11.3.2 Asymptotics for Stirling numbers of the second kind / 338 \\ 11.3.3 Stirling numbers of the first kind / 343 \\ 11.4 Symmetric elliptic integrals / 344 \\ 11.4.1 The standard forms in terms of symmetric integrals / 345 \\ 11.4.2 An algorithm / 346 \\ 11.4.3 Other elliptic integrals / 347 \\ 11.5 Numerical inversion of Laplace transforms / 347 \\ 11.5.1 Complex Gauss quadrature / 348 \\ 11.5.2 Deforming the contour / 349 \\ 11.5.3 Using Pad{\'e} approximations / 352 \\ IV Software / 353 \\ 12 Associated Algorithms / 355 \\ 12.1 Introduction / 355 \\ 12.1.1 Errors and stability: Basic terminology / 356 \\ 12.1.2 Design and testing of software for computing functions: General philosophy / 357 \\ 12.1.3 Scaling the functions / 358 \\ 12.2 Airy and Scorer functions of complex arguments / 359 \\ 12.2.1 Purpose / 359 \\ 12.2.2 Algorithms / 359 \\ 12.3 Associated Legendre functions of integer and half-integer degrees / 363 \\ 12.3.1 Purpose / 363 \\ 12.3.2 Algorithms / 364 \\ 12.4 Bessel functions / 369 \\ 12.4.1 Modified Bessel functions of integer and half-integer orders / 370 \\ 12.4.2 Modified Bessel functions of purely imaginary orders / 372 \\ 12.5 Parabolic cylinder functions / 377 \\ 12.5.1 Purpose / 377 \\ 12.5.2 Algorithm / 378 \\ 12.6 Zeros of Bessel functions / 385 \\ 12.6.1 Purpose / 385 \\ 12.6.2 Algorithm / 385 \\ List of Algorithms / 387 \\ Bibliography / 389 \\ Index / 405

entry:
@Book{Gil:2007:NMS,
  author =       "Amparo Gil and Javier Segura and N. M. Temme",
  title =        "Numerical Methods for Special Functions",
  publisher =    pub-SIAM,
  address =      pub-SIAM:adr,
  pages =        "xvi + 415",
  year =         "2007",
  DOI =          "https://doi.org/10.1137/1.9780898717822",
  ISBN =         "0-89871-634-9",
  ISBN-13 =      "978-0-89871-634-4",
  LCCN =         "QA351 .G455 2007",
  bibdate =      "Fri Sep 14 10:24:22 MDT 2007",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 z3950.loc.gov:7090/Voyager",
  price =        "US\$99.00",
  acknowledgement = ack-nhfb,
  shorttableofcontents = "Preface \\
                 1. Introduction \\
                 I. Basic methods \\
                 2. Convergent and divergent series \\
                 3. Chebyshev expansions \\
                 4. Linear recurrence relations and associated continued
                 fractions \\
                 5. Quadrature methods \\
                 II. Further tools and methods \\
                 6. Numerical aspects of continued fractions \\
                 7. Computation of the zeros of special functions \\
                 8. Uniform asymptotic expansions \\
                 9. Other methods \\
                 III. Related topics and examples \\
                 10. Inversion of cumulative distribution functions \\
                 11. Further examples \\
                 12. Associated algorithms \\
                 List of algorithms \\
                 Bibliography \\
                 Index",
  subject =      "functions, special; data processing; numerical
                 analysis; asymptotic expansions; approximation theory",
  tableofcontents = "Preface xiii 1 Introduction / 1 \\
                 I Basic Methods / 13 \\
                 2 Convergent and Divergent Series / 15 \\
                 2.1 Introduction / 15 \\
                 2.1.1 Power series: First steps / 15 \\
                 2.1.2 Further practical aspects / 17 \\
                 2.2 Differential equations and Frobenius series
                 solutions / 18 \\
                 2.2.1 Singular points / 19 \\
                 2.2.2 The solution near a regular point / 20 \\
                 2.2.3 Power series expansions around a regular singular
                 point / 22 \\
                 2.2.4 The Liouville transformation / 25 \\
                 2.3 Hypergeometric series / 26 \\
                 2.3.1 The Gauss hypergeometric function / 28 \\
                 2.3.2 Other power series for the Gauss hypergeometric
                 function / 30 \\
                 2.3.3 Removable singularities / 33 \\
                 2.4 Asymptotic expansions / 34 \\
                 2.4.1 Watson's lemma / 36 \\
                 2.4.2 Estimating the remainders of asymptotic
                 expansions / 38 \\
                 2.4.3 Exponentially improved asymptotic expansions / 39
                 \\
                 2.4.4 Alternatives of asymptotic expansions / 40 \\
                 3 Chebyshev Expansions / 51 \\
                 3.1 Introduction / 51 \\
                 3.2 Basic results on interpolation / 52 \\
                 3.2.1 The Runge phenomenon and the Chebyshev nodes / 54
                 \\
                 3.3 Chebyshev polynomials: Basic properties / 56 \\
                 3.3.1 Properties of the Chebyshev polynomials $T_n(x)$
                 / 56 \\
                 3.3.2 Chebyshev polynomials of the second, third, and
                 fourth kinds / 60 \\
                 vii 3.4 Chebyshev interpolation / 62 \\
                 3.4.1 Computing the Chebyshev interpolation polynomial
                 / 64 \\
                 3.5 Expansions in terms of Chebyshev polynomials / 66
                 \\
                 3.5.1 Convergence properties of Chebyshev expansions /
                 68 \\
                 3.6 Computing the coefficients of a Chebyshev expansion
                 / 69 \\
                 3.6.1 Clenshaw's method for solutions of linear
                 differential equations with polynomial coefficients /
                 70 \\
                 3.7 Evaluation of a Chebyshev sum / 75 \\
                 3.7.1 Clenshaw's method for the evaluation of a
                 Chebyshev sum / 75 \\
                 3.8 Economization of power series / 80 \\
                 3.9 Example: Computation of Airy functions of real
                 variable / 80 \\
                 3.10 Chebyshev expansions with coefficients in terms of
                 special functions / 83 \\
                 4 Linear Recurrence Relations and Associated Continued
                 Fractions / 87 \\
                 4.1 Introduction / 87 \\
                 4.2 Condition of three-term recurrence relations / 88
                 \\
                 4.2.1 Minimal solutions / 89 \\
                 4.3 Perron's theorem / 92 \\
                 4.3.1 Scaled recurrence relations / 94 \\
                 4.4 Minimal solutions of TTRRs and continued fractions
                 / 95 \\
                 4.5 Some notable recurrence relations / 96 \\
                 4.5.1 The confluent hypergeometric family / 96 \\
                 4.5.2 The Gauss hypergeometric family / 102 \\
                 4.6 Computing the minimal solution of a TTRR / 105 \\
                 4.6.1 Miller's algorithm when a function value is known
                 / 105 \\
                 4.6.2 Miller's algorithm with a normalizing sum / 107
                 \\
                 4.6.3 ``Anti-Miller'' algorithm / 110 \\
                 4.7 Inhomogeneous linear difference equations / 112 \\
                 4.7.1 Inhomogeneous first order difference equations.
                 Examples / 112 \\
                 4.7.2 Inhomogeneous second order difference equations /
                 115 \\
                 4.7.3 Olver's method / 116 \\
                 4.8 Anomalous behavior of some second order homogeneous
                 and first order inhomogeneous recurrences / 118 \\
                 4.8.1 A canonical example: Modified Bessel function /
                 118 \\
                 4.8.2 Other examples: Hypergeometric recursions / 120
                 \\
                 4.8.3 A first order inhomogeneous equation / 121 \\
                 4.8.4 A warning / 122 \\
                 5 Quadrature Methods / 123 \\
                 5.1 Introduction / 123 \\
                 5.2 Newton--Cotes quadrature: The trapezoidal and
                 Simpson's rule / 124 \\
                 5.2.1 The compound trapezoidal rule / 126 \\
                 5.2.2 The recurrent trapezoidal rule / 129 \\
                 5.2.3 Euler's summation formula and the trapezoidal
                 rule / 130 \\
                 5.3 Gauss quadrature / 132 \\
                 5.3.1 Basics of the theory of orthogonal polynomials
                 and Gauss quadrature / 133 \\
                 5.3.2 The Golub--Welsch algorithm / 141 \\
                 5.3.3 Example: The Airy function in the complex plane /
                 145 \\
                 5.3.4 Further practical aspects of Gauss quadrature /
                 146 \\
                 5.4 The trapezoidal rule on $\mathbb{R}$ / 147 \\
                 5.4.1 Contour integral formulas for the truncation
                 errors / 148 \\
                 5.4.2 Transforming the variable of integration / 153
                 \\
                 5.5 Contour integrals and the saddle point method / 157
                 \\
                 5.5.1 The saddle point method / 158 \\
                 5.5.2 Other integration contours / 163 \\
                 5.5.3 Integrating along the saddle point contours and
                 examples / 165 \\
                 II Further Tools and Methods / 171 \\
                 6 Numerical Aspects of Continued Fractions / 173 \\
                 6.1 Introduction / 173 \\
                 6.2 Definitions and notation / 173 \\
                 6.3 Equivalence transformations and contractions / 175
                 \\
                 6.4 Special forms of continued fractions / 178 \\
                 6.4.1 Stieltjes fractions / 178 \\
                 6.4.2 Jacobi fractions / 179 \\
                 6.4.3 Relation with Pad{\'e} approximants / 179 \\
                 6.5 Convergence of continued fractions / 179 \\
                 6.6 Numerical evaluation of continued fractions / 181
                 \\
                 6.6.1 Steed's algorithm / 181 \\
                 6.6.2 The modified Lentz algorithm / 183 \\
                 6.7 Special functions and continued fractions / 185 \\
                 6.7.1 Incomplete gamma function / 186 \\
                 6.7.2 Gauss hypergeometric functions / 187 \\
                 7 Computation of the Zeros of Special Functions / 191
                 \\
                 7.1 Introduction / 191 \\
                 7.2 Some classical methods / 193 \\
                 7.2.1 The bisection method / 193 \\
                 7.2.2 The fixed point method and the Newton--Raphson
                 method / 193 \\
                 7.2.3 Complex zeros / 197 \\
                 7.3 Local strategies: Asymptotic and other
                 approximations / 197 \\
                 7.3.1 Asymptotic approximations for large zeros / 199
                 \\
                 7.3.2 Other approximations / 202 \\
                 7.4 Global strategies I: Matrix methods / 205 \\
                 7.4.1 The eigenvalue problem for orthogonal polynomials
                 / 206 \\
                 7.4.2 The eigenvalue problem for minimal solutions of
                 TTRRs / 207 \\
                 7.5 Global strategies II: Global fixed point methods /
                 213 \\
                 7.5.1 Zeros of Bessel functions / 213 \\
                 7.5.2 The general case / 219 \\
                 7.6 Asymptotic methods: Further examples / 224 \\
                 7.6.1 Airy functions / 224 \\
                 7.6.2 Scorer functions / 227 \\
                 7.6.3 The error functions / 229 \\
                 7.6.4 The parabolic cylinder function / 233 \\
                 7.6.5 Bessel functions / 233 \\
                 7.6.6 Orthogonal polynomials / 234 \\
                 8 Uniform Asymptotic Expansions / 237 \\
                 8.1 Asymptotic expansions for the incomplete gamma
                 functions / 237 \\
                 8.2 Uniform asymptotic expansions / 239 \\
                 8.3 Uniform asymptotic expansions for the incomplete
                 gamma functions / 240 \\
                 8.3.1 The uniform expansion / 242 \\
                 8.3.2 Expansions for the coefficients / 244 \\
                 8.3.3 Numerical algorithm for small values of $\eta$ /
                 245 \\
                 8.3.4 A simpler uniform expansion / 247 \\
                 8.4 Airy-type expansions for Bessel functions / 249 \\
                 8.4.1 The Airy-type asymptotic expansions / 250 \\
                 8.4.2 Representations of $a_s()$, $b_s()$, $c_s()$,
                 $d_s()$ / 253 \\
                 8.4.3 Properties of the functions $A_\nu$, $B_\nu$,
                 $C_\nu$, $D_\nu$ / 254 \\
                 8.4.4 Expansions for $a_s()$, $b_s()$, $c_s()$, $d_s()$
                 / 256 \\
                 8.4.5 Evaluation of the functions $A_\nu()$, $B_\nu()$
                 by iteration / 258 \\
                 8.5 Airy-type asymptotic expansions obtained from
                 integrals / 263 \\
                 8.5.1 Airy-type asymptotic expansions / 264 \\
                 8.5.2 How to compute the coefficients $\alpha_n$,
                 $\beta_n$ / 267 \\
                 8.5.3 Application to parabolic cylinder functions / 270
                 \\
                 9 Other Methods / 275 \\
                 9.1 Introduction / 275 \\
                 9.2 Pad{\'e} approximations / 276 \\
                 9.2.1 Pad{\'e} approximants and continued fractions /
                 278 \\
                 9.2.2 How to compute the Pad{\'e} approximants / 278
                 \\
                 9.2.3 Pad{\'e} approximants to the exponential function
                 / 280 \\
                 9.2.4 Analytic forms of Pad{\'e} approximations / 283
                 \\
                 9.3 Sequence transformations / 286 \\
                 9.3.1 The principles of sequence transformations / 286
                 \\
                 9.3.2 Examples of sequence transformations / 287 \\
                 9.3.3 The transformation of power series / 288 \\
                 9.3.4 Numerical examples / 288 \\
                 9.4 Best rational approximations / 290 \\
                 9.5 Numerical solution of ordinary differential
                 equations: Taylor expansion method / 291 \\
                 9.5.1 Taylor-series method: Initial value problems /
                 292 \\
                 9.5.2 Taylor-series method: Boundary value problem /
                 293 \\
                 9.6 Other quadrature methods / 294 \\
                 9.6.1 Romberg quadrature / 294 \\
                 9.6.2 Fej{\'e}r and Clenshaw--Curtis quadratures / 296
                 \\
                 9.6.3 Other Gaussian quadratures / 298 \\
                 9.6.4 Oscillatory integrals / 301 \\
                 III Related Topics and Examples / 307 \\
                 10 Inversion of Cumulative Distribution Functions / 309
                 \\
                 10.1 Introduction / 309 \\
                 10.2 Asymptotic inversion of the complementary error
                 function / 309 \\
                 10.3 Asymptotic inversion of incomplete gamma functions
                 / 312 \\
                 10.3.1 The asymptotic inversion method / 312 \\
                 10.3.2 Determination of the coefficients i / 314 \\
                 10.3.3 Expansions of the coefficients i / 316 \\
                 10.3.4 Numerical examples / 316 \\
                 10.4 Generalizations / 317 \\
                 10.5 Asymptotic inversion of the incomplete beta
                 function / 318 \\
                 10.5.1 The nearly symmetric case / 319 \\
                 10.5.2 The general error function case / 322 \\
                 10.5.3 The incomplete gamma function case / 324 \\
                 10.5.4 Numerical aspects / 326 \\
                 10.6 High order Newton-like methods / 327 \\
                 11 Further Examples / 331 \\
                 11.1 Introduction / 331 \\
                 11.2 The Euler summation formula / 331 \\
                 11.3 Approximations of Stirling numbers / 336 \\
                 11.3.1 Definitions / 337 \\
                 11.3.2 Asymptotics for Stirling numbers of the second
                 kind / 338 \\
                 11.3.3 Stirling numbers of the first kind / 343 \\
                 11.4 Symmetric elliptic integrals / 344 \\
                 11.4.1 The standard forms in terms of symmetric
                 integrals / 345 \\
                 11.4.2 An algorithm / 346 \\
                 11.4.3 Other elliptic integrals / 347 \\
                 11.5 Numerical inversion of Laplace transforms / 347
                 \\
                 11.5.1 Complex Gauss quadrature / 348 \\
                 11.5.2 Deforming the contour / 349 \\
                 11.5.3 Using Pad{\'e} approximations / 352 \\
                 IV Software / 353 \\
                 12 Associated Algorithms / 355 \\
                 12.1 Introduction / 355 \\
                 12.1.1 Errors and stability: Basic terminology / 356
                 \\
                 12.1.2 Design and testing of software for computing
                 functions: General philosophy / 357 \\
                 12.1.3 Scaling the functions / 358 \\
                 12.2 Airy and Scorer functions of complex arguments /
                 359 \\
                 12.2.1 Purpose / 359 \\
                 12.2.2 Algorithms / 359 \\
                 12.3 Associated Legendre functions of integer and
                 half-integer degrees / 363 \\
                 12.3.1 Purpose / 363 \\
                 12.3.2 Algorithms / 364 \\
                 12.4 Bessel functions / 369 \\
                 12.4.1 Modified Bessel functions of integer and
                 half-integer orders / 370 \\
                 12.4.2 Modified Bessel functions of purely imaginary
                 orders / 372 \\
                 12.5 Parabolic cylinder functions / 377 \\
                 12.5.1 Purpose / 377 \\
                 12.5.2 Algorithm / 378 \\
                 12.6 Zeros of Bessel functions / 385 \\
                 12.6.1 Purpose / 385 \\
                 12.6.2 Algorithm / 385 \\
                 List of Algorithms / 387 \\
                 Bibliography / 389 \\
                 Index / 405",
}


bibtype: article
filename: elefunt.bib
label: Gil:2007:NSS
author: Amparo Gil and Javier Segura and Nico M. Temme
title: Numerically satisfactory solutions of hypergeometric recursions
journal: j-MATH-COMPUT
volume: 76
number: 259
pages: 1449--1468
month: jul
monthnumber: 7
year: 2007
CODEN: MCMPAF
ISSN: 0025-5718 (print), 1088-6842 (electronic) OR 00255718 (print), 10886842 (electronic)
ISSNL: 0025-5718 OR 00255718
bibdate: Tue Jul 8 06:24:22 MDT 2008
bibtimestamp: 2008.07.08 06:24:22 MDT
fjournal: Mathematics of Computation
acknowledgement: ack-nhfb
journalURL: http://www.ams.org/mcom/
rawauthor: Amparo Gil and Javier Segura and Nico M. Temme
rawtitle: Numerically satisfactory solutions of hypergeometric recursions
bibsource:
http://www.ams.org/mcom/2007-76-259; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp2000.bib

URL:
http://www.ams.org/mcom/2007-76-259/S0025-5718-07-01918-7/home.html; http://www.ams.org/mcom/2007-76-259/S0025-5718-07-01918-7/S0025-5718-07-01918-7.dvi; http://www.ams.org/mcom/2007-76-259/S0025-5718-07-01918-7/S0025-5718-07-01918-7.pdf; http://www.ams.org/mcom/2007-76-259/S0025-5718-07-01918-7/S0025-5718-07-01918-7.ps

entry:
@Article{Gil:2007:NSS,
  author =       "Amparo Gil and Javier Segura and Nico M. Temme",
  title =        "Numerically satisfactory solutions of hypergeometric
                 recursions",
  journal =      j-MATH-COMPUT,
  volume =       "76",
  number =       "259",
  pages =        "1449--1468",
  month =        jul,
  year =         "2007",
  CODEN =        "MCMPAF",
  ISSN =         "0025-5718 (print), 1088-6842 (electronic)",
  ISSN-L =       "0025-5718",
  bibdate =      "Tue Jul 8 06:24:22 MDT 2008",
  bibsource =    "http://www.ams.org/mcom/2007-76-259;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/mathcomp2000.bib",
  URL =          "http://www.ams.org/mcom/2007-76-259/S0025-5718-07-01918-7/home.html;
                 http://www.ams.org/mcom/2007-76-259/S0025-5718-07-01918-7/S0025-5718-07-01918-7.dvi;
                 http://www.ams.org/mcom/2007-76-259/S0025-5718-07-01918-7/S0025-5718-07-01918-7.pdf;
                 http://www.ams.org/mcom/2007-76-259/S0025-5718-07-01918-7/S0025-5718-07-01918-7.ps",
  acknowledgement = ack-nhfb,
  fjournal =     "Mathematics of Computation",
  journal-URL =  "http://www.ams.org/mcom/",
}


bibtype: article
filename: elefunt.bib
label: Glaser:2007:FAC
author: Andreas Glaser and Xiangtao Liu and Vladimir Rokhlin
title: A Fast Algorithm for the Calculation of the Roots of Special Functions
journal: j-SIAM-J-SCI-COMP
volume: 29
number: 4
pages: 1420--1438
year: 2007
CODEN: SJOCE3
DOI: https://doi.org/10.1137/06067016X
ISSN: 1064-8275 (print), 1095-7197 (electronic) OR 10648275 (print), 10957197 (electronic)
ISSNL: 1064-8275 OR 10648275
bibdate: Wed May 19 10:43:53 MDT 2010
bibsource: http://epubs.siam.org/sam-bin/dbq/toc/SISC/29/4; https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2010.05.19 10:43:53 MDT
fjournal: SIAM Journal on Scientific Computing
acknowledgement: ack-nhfb
journalURL: http://epubs.siam.org/sisc
rawauthor: Andreas Glaser and Xiangtao Liu and Vladimir Rokhlin
rawtitle: A Fast Algorithm for the Calculation of the Roots of Special Functions
abstract:
We describe a procedure for the determination of the roots of functions satisfying second-order ordinary differential equations, including the classical special functions. The scheme is based on a combination of the Prufer transform with the classical Taylor series method for the solution of ordinary differential equations and requires $ O(1) $ operations for the determination of each root. When the functions in question are classical orthogonal polynomials (Legendre, Hermite, etc.), the techniques presented here also provide tools for the evaluation of the weights for the associated Gaussian quadratures. The performance of the scheme for several classical special functions (prolate spheroidal wave functions, Bessel functions, and Legendre, Hermite, and Laguerre polynomials) is illustrated with numerical examples.

rawabstract:
We describe a procedure for the determination of the roots of functions satisfying second-order ordinary differential equations, including the classical special functions. The scheme is based on a combination of the Pr{\"u}fer transform with the classical Taylor series method for the solution of ordinary differential equations and requires $ O(1) $ operations for the determination of each root. When the functions in question are classical orthogonal polynomials (Legendre, Hermite, etc.), the techniques presented here also provide tools for the evaluation of the weights for the associated Gaussian quadratures. The performance of the scheme for several classical special functions (prolate spheroidal wave functions, Bessel functions, and Legendre, Hermite, and Laguerre polynomials) is illustrated with numerical examples.

entry:
@Article{Glaser:2007:FAC,
  author =       "Andreas Glaser and Xiangtao Liu and Vladimir Rokhlin",
  title =        "A Fast Algorithm for the Calculation of the Roots of
                 Special Functions",
  journal =      j-SIAM-J-SCI-COMP,
  volume =       "29",
  number =       "4",
  pages =        "1420--1438",
  month =        "????",
  year =         "2007",
  CODEN =        "SJOCE3",
  DOI =          "https://doi.org/10.1137/06067016X",
  ISSN =         "1064-8275 (print), 1095-7197 (electronic)",
  ISSN-L =       "1064-8275",
  bibdate =      "Wed May 19 10:43:53 MDT 2010",
  bibsource =    "http://epubs.siam.org/sam-bin/dbq/toc/SISC/29/4;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  abstract =     "We describe a procedure for the determination of the
                 roots of functions satisfying second-order ordinary
                 differential equations, including the classical special
                 functions. The scheme is based on a combination of the
                 Pr{\"u}fer transform with the classical Taylor series
                 method for the solution of ordinary differential
                 equations and requires $ O(1) $ operations for the
                 determination of each root. When the functions in
                 question are classical orthogonal polynomials
                 (Legendre, Hermite, etc.), the techniques presented
                 here also provide tools for the evaluation of the
                 weights for the associated Gaussian quadratures. The
                 performance of the scheme for several classical special
                 functions (prolate spheroidal wave functions, Bessel
                 functions, and Legendre, Hermite, and Laguerre
                 polynomials) is illustrated with numerical examples.",
  acknowledgement = ack-nhfb,
  fjournal =     "SIAM Journal on Scientific Computing",
  journal-URL =  "http://epubs.siam.org/sisc",
}


bibtype: book
filename: elefunt.bib
label: Gradshteyn:2007:TIS
author: I. S. Gradshteyn and I. M. Ryzhik and Alan Jeffrey and Daniel Zwillinger
title: Table of Integrals, Series and Products
publisher: pub-ACADEMIC
address: pub-ACADEMIC:adr
edition: Seventh
pages: xlv + 1171
year: 2007
ISBN: 0-12-373637-4 (hardcover) OR 0123736374 (hardcover)
ISBN13: 978-0-12-373637-6 (hardcover) OR 9780123736376 (hardcover)
LCCN: QA55 .G6613 2007
bibdate: Thu Feb 18 12:04:10 MST 2010
bibtimestamp: 2010.02.18 12:04:10 MST
remark: Previous edition 2000. Includes CD-ROM.
subject: Mathematics; Tables
acknowledgement: ack-nhfb
rawauthor: I. S. Gradshteyn and I. M. Ryzhik and Alan Jeffrey and Daniel Zwillinger
rawtitle: Table of Integrals, Series and Products
bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/master.bib; https://www.math.utah.edu/pub/tex/bib/mathcw.bib; prodorbis.library.yale.edu:7090/voyager

TOC:
0 Introduction 1 Elementary Functions 2 Indefinite Integrals of Elementary Functions 3 Definite Integrals of Elementary Functions 4.Combinations involving trigonometric and hyperbolic functions and power 5 Indefinite Integrals of Special Functions 6 Definite Integrals of Special Functions 7.Associated Legendre Functions 8 Special Functions 9 Hypergeometric Functions 10 Vector Field Theory 11 Algebraic Inequalities 12 Integral Inequalities 13 Matrices and related results 14 Determinants 15 Norms 16 Ordinary differential equations 17 Fourier, Laplace, and Mellin Transforms 18 The z-transform

rawTOC:
0 Introduction \\ 1 Elementary Functions \\ 2 Indefinite Integrals of Elementary Functions \\ 3 Definite Integrals of Elementary Functions \\ 4.Combinations involving trigonometric and hyperbolic functions and power \\ 5 Indefinite Integrals of Special Functions \\ 6 Definite Integrals of Special Functions \\ 7.Associated Legendre Functions \\ 8 Special Functions \\ 9 Hypergeometric Functions \\ 10 Vector Field Theory \\ 11 Algebraic Inequalities \\ 12 Integral Inequalities \\ 13 Matrices and related results \\ 14 Determinants \\ 15 Norms \\ 16 Ordinary differential equations \\ 17 Fourier, Laplace, and Mellin Transforms \\ 18 The z-transform

entry:
@Book{Gradshteyn:2007:TIS,
  author =       "I. S. Gradshteyn and I. M. Ryzhik and Alan Jeffrey and
                 Daniel Zwillinger",
  title =        "Table of Integrals, Series and Products",
  publisher =    pub-ACADEMIC,
  address =      pub-ACADEMIC:adr,
  edition =      "Seventh",
  pages =        "xlv + 1171",
  year =         "2007",
  ISBN =         "0-12-373637-4 (hardcover)",
  ISBN-13 =      "978-0-12-373637-6 (hardcover)",
  LCCN =         "QA55 .G6613 2007",
  bibdate =      "Thu Feb 18 12:04:10 MST 2010",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/master.bib;
                 https://www.math.utah.edu/pub/tex/bib/mathcw.bib;
                 prodorbis.library.yale.edu:7090/voyager",
  acknowledgement = ack-nhfb,
  remark =       "Previous edition 2000. Includes CD-ROM.",
  subject =      "Mathematics; Tables",
  tableofcontents = "0 Introduction \\
                 1 Elementary Functions \\
                 2 Indefinite Integrals of Elementary Functions \\
                 3 Definite Integrals of Elementary Functions \\
                 4.Combinations involving trigonometric and hyperbolic
                 functions and power \\
                 5 Indefinite Integrals of Special Functions \\
                 6 Definite Integrals of Special Functions \\
                 7.Associated Legendre Functions \\
                 8 Special Functions \\
                 9 Hypergeometric Functions \\
                 10 Vector Field Theory \\
                 11 Algebraic Inequalities \\
                 12 Integral Inequalities \\
                 13 Matrices and related results \\
                 14 Determinants \\
                 15 Norms \\
                 16 Ordinary differential equations \\
                 17 Fourier, Laplace, and Mellin Transforms \\
                 18 The z-transform",
  xxauthor =     "I. S. (Izrail Solomonovich) Gradshteyn and I. M.
                 (Iosif Moiseevich) Ryzhik and Alan Jeffrey and Daniel
                 Zwillinger",
}


bibtype: article
filename: elefunt.bib
label: Guseinov:2007:UTE
author: I. I. Guseinov and B. A. Mamedov
title: Unified treatment for the evaluation of generalized complete and incomplete gamma functions
journal: j-J-COMPUT-APPL-MATH
volume: 202
number: 2
pages: 435--439
day: 15
month: may
monthnumber: 5
year: 2007
CODEN: JCAMDI
ISSN: 0377-0427 (print), 1879-1778 (electronic) OR 03770427 (print), 18791778 (electronic)
ISSNL: 0377-0427 OR 03770427
bibdate: Sat Feb 25 13:13:14 MST 2017
bibtimestamp: 2017.02.25 13:13:14 MST
URL: http://www.sciencedirect.com/science/article/pii/S0377042706001506
fjournal: Journal of Computational and Applied Mathematics
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/03770427
rawauthor: I. I. Guseinov and B. A. Mamedov
rawtitle: Unified treatment for the evaluation of generalized complete and incomplete gamma functions
entry:
@Article{Guseinov:2007:UTE,
  author =       "I. I. Guseinov and B. A. Mamedov",
  title =        "Unified treatment for the evaluation of generalized
                 complete and incomplete gamma functions",
  journal =      j-J-COMPUT-APPL-MATH,
  volume =       "202",
  number =       "2",
  pages =        "435--439",
  day =          "15",
  month =        may,
  year =         "2007",
  CODEN =        "JCAMDI",
  ISSN =         "0377-0427 (print), 1879-1778 (electronic)",
  ISSN-L =       "0377-0427",
  bibdate =      "Sat Feb 25 13:13:14 MST 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2005.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0377042706001506",
  acknowledgement = ack-nhfb,
  fjournal =     "Journal of Computational and Applied Mathematics",
  journal-URL =  "http://www.sciencedirect.com/science/journal/03770427",
}


bibtype: article
filename: elefunt.bib
label: Hernandez:2007:MPO
author: M. A. Hernandez and N. Romero
title: Methods with prefixed order for approximating square roots with global and general convergence
journal: j-APPL-MATH-COMP
volume: 194
number: 2
pages: 346--353
day: 15
month: dec
monthnumber: 12
year: 2007
CODEN: AMHCBQ
ISSN: 0096-3003 (print), 1873-5649 (electronic) OR 00963003 (print), 18735649 (electronic)
ISSNL: 0096-3003 OR 00963003
bibdate: Sat Jul 12 09:03:09 MDT 2008
bibtimestamp: 2008.07.12 09:03:09 MDT
fjournal: Applied Mathematics and Computation
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00963003
rawauthor: M. A. Hern{\'a}ndez and N. Romero
rawtitle: Methods with prefixed order for approximating square roots with global and general convergence
bibsource:
http://www.sciencedirect.com/science/journal/00963003; https://www.math.utah.edu/pub/tex/bib/applmathcomput2005.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib

entry:
@Article{Hernandez:2007:MPO,
  author =       "M. A. Hern{\'a}ndez and N. Romero",
  title =        "Methods with prefixed order for approximating square
                 roots with global and general convergence",
  journal =      j-APPL-MATH-COMP,
  volume =       "194",
  number =       "2",
  pages =        "346--353",
  day =          "15",
  month =        dec,
  year =         "2007",
  CODEN =        "AMHCBQ",
  ISSN =         "0096-3003 (print), 1873-5649 (electronic)",
  ISSN-L =       "0096-3003",
  bibdate =      "Sat Jul 12 09:03:09 MDT 2008",
  bibsource =    "http://www.sciencedirect.com/science/journal/00963003;
                 https://www.math.utah.edu/pub/tex/bib/applmathcomput2005.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "Applied Mathematics and Computation",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00963003",
}


bibtype: article
filename: elefunt.bib
label: Kalmykov:2007:AOEa
author: M. Y. Kalmykov and B. F. L. Ward and Y. Yost
journal: j-J-HIGH-ENERGY-PHYS
volume: 2007
number: 02
pages: 040--??
year: 2007
CODEN: JHEPAB
ISSN: 1126-6708 OR 11266708
ISSNL: 1029-8479 OR 10298479
MRclass: 33C05 (33B30)
MRnumber: MR2318011 (2009g:33004)
bibdate: Thu Dec 01 09:16:04 2011
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2011.12.01 09:16:04 ???
fjournal: Journal of High Energy Physics
acknowledgement: ack-nhfb
ajournal: JHEP
rawauthor: M. Y. Kalmykov and B. F. L. Ward and Y. Yost
title:
All order $ epsilon $-expansion of Gauss hypergeometric functions with integer and half / integer values of parameters

rawtitle:
All order $ \epsilon $-expansion of {Gauss} hypergeometric functions with integer and half\slash integer values of parameters

entry:
@Article{Kalmykov:2007:AOEa,
  author =       "M. Y. Kalmykov and B. F. L. Ward and Y. Yost",
  title =        "All order $ \epsilon $-expansion of {Gauss}
                 hypergeometric functions with integer and half\slash
                 integer values of parameters",
  journal =      j-J-HIGH-ENERGY-PHYS,
  volume =       "2007",
  number =       "02",
  pages =        "040--??",
  month =        "????",
  year =         "2007",
  CODEN =        "JHEPAB",
  ISSN =         "1126-6708",
  ISSN-L =       "1029-8479",
  MRclass =      "33C05 (33B30)",
  MRnumber =     "MR2318011 (2009g:33004)",
  bibdate =      "Thu Dec 01 09:16:04 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
  ajournal =     "JHEP",
  fjournal =     "Journal of High Energy Physics",
  pagecount =    "21",
}


bibtype: article
filename: elefunt.bib
label: Kalmykov:2007:AOEb
author: M. Y. Kalmykov and B. F. L. Ward and Y. Yost
journal: j-J-HIGH-ENERGY-PHYS
volume: 2007
number: 11
pages: 009
year: 2007
CODEN: JHEPAB
ISSN: 1126-6708 OR 11266708
ISSNL: 1029-8479 OR 10298479
MRclass: 33C20 (33B30 41A58)
MRnumber: MR2362140 (2008m:33016)
bibdate: Thu Dec 01 09:16:04 2011
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2011.12.01 09:16:04 ???
fjournal: Journal of High Energy Physics
acknowledgement: ack-nhfb
ajournal: JHEP
rawauthor: M. Y. Kalmykov and B. F. L. Ward and Y. Yost
title:
On the all-order $ epsilon $-expansion of generalized hypergeometric functions with integer values of parameters

rawtitle:
On the all-order $ \epsilon $-expansion of generalized hypergeometric functions with integer values of parameters

entry:
@Article{Kalmykov:2007:AOEb,
  author =       "M. Y. Kalmykov and B. F. L. Ward and Y. Yost",
  title =        "On the all-order $ \epsilon $-expansion of generalized
                 hypergeometric functions with integer values of
                 parameters",
  journal =      j-J-HIGH-ENERGY-PHYS,
  volume =       "2007",
  number =       "11",
  pages =        "009",
  month =        "????",
  year =         "2007",
  CODEN =        "JHEPAB",
  ISSN =         "1126-6708",
  ISSN-L =       "1029-8479",
  MRclass =      "33C20 (33B30 41A58)",
  MRnumber =     "MR2362140 (2008m:33016)",
  bibdate =      "Thu Dec 01 09:16:04 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
  ajournal =     "JHEP",
  fjournal =     "Journal of High Energy Physics",
  pagecount =    "13",
}


bibtype: article
filename: elefunt.bib
label: Karagiannidis:2007:IAG
author: George Karagiannidis and Athanasios Lioumpas
title: An Improved Approximation for the Gaussian $Q$-Function
journal: j-IEEE-COMMUN-LET
volume: 11
number: 8
pages: 644--646
month: aug
monthnumber: 8
year: 2007
CODEN: ICLEF6
DOI: https://doi.org/10.1109/lcomm.2007.070470
ISSN: 1089-7798 (print), 1558-2558 (electronic) OR 10897798 (print), 15582558 (electronic)
ISSNL: 1089-7798 OR 10897798
bibdate: Sat Dec 16 16:49:58 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2017.12.16 16:49:58 ???
note: See corrections and comments cite Dyer:2008:CCI.
fjournal: IEEE Communications Letters
acknowledgement: ack-nhfb
journalURL: https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=4234
rawauthor: George Karagiannidis and Athanasios Lioumpas
rawnote: See corrections and comments \cite{Dyer:2008:CCI}.
rawtitle: An Improved Approximation for the {Gaussian} {$Q$}-Function
entry:
@Article{Karagiannidis:2007:IAG,
  author =       "George Karagiannidis and Athanasios Lioumpas",
  title =        "An Improved Approximation for the {Gaussian}
                 {$Q$}-Function",
  journal =      j-IEEE-COMMUN-LET,
  volume =       "11",
  number =       "8",
  pages =        "644--646",
  month =        aug,
  year =         "2007",
  CODEN =        "ICLEF6",
  DOI =          "https://doi.org/10.1109/lcomm.2007.070470",
  ISSN =         "1089-7798 (print), 1558-2558 (electronic)",
  ISSN-L =       "1089-7798",
  bibdate =      "Sat Dec 16 16:49:58 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  note =         "See corrections and comments \cite{Dyer:2008:CCI}.",
  acknowledgement = ack-nhfb,
  fjournal =     "IEEE Communications Letters",
  journal-URL =  "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=4234",
}


bibtype: book
filename: elefunt.bib
label: King:2007:DNC
author: Louis Vessot King
title: On the Direct Numerical Calculation of Elliptic Functions and Integrals
publisher: Mellon Press
pages: 56
year: 2007
ISBN: 1-4067-4226-0 OR 1406742260
ISBN13: 978-1-4067-4226-8 OR 9781406742268
bibdate: Wed Feb 03 08:53:04 2010
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcw.bib
bibtimestamp: 2010.02.03 08:53:04 ???
URL: https://www.google.com/books/edition/On_the_Direct_Numerical_Calculation_of_E/CMM4AAAAMAAJ
acknowledgement: ack-nhfb
rawauthor: Louis Vessot King
rawtitle: On the Direct Numerical Calculation of Elliptic Functions and Integrals
note:
Dedicated to the memory of James Harkness, Peter Redpath Professor of Pure Mathematics, McGill University 1903--1923.

remark:
The AGM method for Jacobian elliptic functions was discovered by this book's author at McGill University in 1913, first published in cite King:1921:SNF, and then in a 1924 monograph, of which this is a reprint.

TOC:
Preface / v Introduction / 1 II. Historical Note on Landen's Transformation and the Various Scales of Moduli and Amplitudes / 2 III. On the Scale of Arithmetico-Geometrical Means / 6 IV. Landen's Scale of Increasing Amplitudes: $tan (phi _n + 1 - phi _n) = (b_n / a_n) tan phi _n$ / 7 (i) Calculation of $F(phi , k)$, $E(phi , k)$, $K$, and $E$ / 7 (ii) Calculation of $sn (u,k)$, $cn (u,k)$, $dn (u,k)$ in terms of the argument $u$ / 9 V. The Hyperbolic Scale of Increasing Amplitudes: $tanh (phi _n + 1 - phi _n) = (b_n / a_n) tanh (phi _n)$ / 10 (i) Calculation of $F(phi , k')$, $E(phi , k')$, etc. / 10 (ii) Calculation of $sn (u,k')$, $cn (u,k')$, $dn (u,k')$ / 11 (iii) Calculation of $sn (i u,k)$, $cn (i u,k)$, $dn (i u,k)$ / 12 VI: Gauss' Scale of Increasing Amplitudes VII: Landen's Scale of Decreasing Amplitudes: $sin (2psi _n + 1 - psi _n) = (b_n / a_n) sin psi _n$ / 14 (i) Calculation of $F(psi , k')$, $E(psi , k')$, etc. / 14 (ii) Calculation of $sn (u,k')$, $cn (u,k')$, $dn (u,k')$ / 16 VIII. The Hyperbolic Scale of Decreasing Amplitudes: $sinh (2psi _n + 1 - psi _n) = (b_n / a_n) sinh psi _n$ / 17 (i) Calculation of $F(psi , k)$, $E(psi , k)$, etc. / 17 (ii) Calculation of $sn (u,k)$, $cn (u,k)$, $dn (u,k)$ / 18 IX. On the Numerical Computation of the Third Elliptic Integral / 19 Case I. $n$ negative, between $0$ and $-k^2$ (Hyperbolic case) / 19 Case II. $n$ negative, between $-1$ and $-infty $ (Hyperbolic case) / 20 Case III. $n$ negative, between $-k^2$ and $-1$ (Circular case) / 21 First Method. (i) Use of circular recurrence formulae / 22 (ii) Use of hyperbolic recurrence formulae / 23 Second Method. (i) Use of circular recurrence formulae / 25 (ii) Use of hyperbolic recurrence formulae / 25 Case IV. $n$ positive, between $0$ and $infty $ (Circular case) / 26 First Method. (i) Use of circular recurrence formulae / 26 (ii) Use of hyperbolic recurrence formulae / 27 Second Method. (i) Use of circular recurrence formulae / 28 (ii) Use of hyperbolic recurrence formulae / 29 X. Note on the Calculation of the Third Elliptic Integral in Terms of the Complementary A.G.M. Scale / 30 Summary of Formulae / 31 Appendix, Examples 1--35 / 35

rawnote:
Dedicated to the memory of James Harkness, Peter Redpath Professor of Pure Mathematics, McGill University 1903--1923.

rawTOC:
Preface / v \\ Introduction / 1 \\ II. Historical Note on Landen's Transformation and the Various Scales of Moduli and Amplitudes / 2 \\ III. On the Scale of Arithmetico-Geometrical Means / 6 \\ IV. Landen's Scale of Increasing Amplitudes: $\tan(\phi_{n + 1} - \phi_n) = (b_n / a_n) \tan \phi_n$ / 7 \\ (i) Calculation of $F(\phi, k)$, $E(\phi, k)$, $K$, and $E$ / 7 \\ (ii) Calculation of $\sn(u,k)$, $\cn(u,k)$, $\dn(u,k)$ in terms of the argument $u$ / 9 \\ V. The Hyperbolic Scale of Increasing Amplitudes: $\tanh(\phi_{n + 1} - \phi_n) = (b_n / a_n) \tanh(\phi_n)$ / 10 \\ (i) Calculation of $F(\phi, k')$, $E(\phi, k')$, etc. / 10 \\ (ii) Calculation of $\sn(u,k')$, $\cn(u,k')$, $\dn(u,k')$ / 11 \\ (iii) Calculation of $\sn(i u,k)$, $\cn(i u,k)$, $\dn(i u,k)$ / 12 \\ VI: Gauss' Scale of Increasing Amplitudes \\ VII: Landen's Scale of Decreasing Amplitudes: $\sin(2\psi_{n + 1} - \psi_n) = (b_n / a_n) \sin \psi_n$ / 14 \\ (i) Calculation of $F(\psi, k')$, $E(\psi, k')$, etc. / 14 \\ (ii) Calculation of $\sn(u,k')$, $\cn(u,k')$, $\dn(u,k')$ / 16 \\ VIII. The Hyperbolic Scale of Decreasing Amplitudes: $\sinh(2\psi_{n + 1} - \psi_n) = (b_n / a_n) \sinh \psi_n$ / 17 \\ (i) Calculation of $F(\psi, k)$, $E(\psi, k)$, etc. / 17 \\ (ii) Calculation of $\sn(u,k)$, $\cn(u,k)$, $\dn(u,k)$ / 18 \\ IX. On the Numerical Computation of the Third Elliptic Integral / 19 \\ Case I. $n$ negative, between $0$ and $-k^2$ (Hyperbolic case) / 19 \\ Case II. $n$ negative, between $-1$ and $-\infty$ (Hyperbolic case) / 20 \\ Case III. $n$ negative, between $-k^2$ and $-1$ (Circular case) / 21 \\ First Method. (i) Use of circular recurrence formulae / 22 \\ (ii) Use of hyperbolic recurrence formulae / 23 \\ Second Method. (i) Use of circular recurrence formulae / 25 \\ (ii) Use of hyperbolic recurrence formulae / 25 \\ Case IV. $n$ positive, between $0$ and $\infty$ (Circular case) / 26 \\ First Method. (i) Use of circular recurrence formulae / 26 \\ (ii) Use of hyperbolic recurrence formulae / 27 \\ Second Method. (i) Use of circular recurrence formulae / 28 \\ (ii) Use of hyperbolic recurrence formulae / 29 \\ X. Note on the Calculation of the Third Elliptic Integral in Terms of the Complementary A.G.M. Scale / 30 \\ Summary of Formulae / 31 \\ Appendix, Examples 1--35 / 35

entry:
@Book{King:2007:DNC,
  author =       "Louis Vessot King",
  title =        "On the Direct Numerical Calculation of Elliptic
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  publisher =    "Mellon Press",
  address =      "????",
  pages =        "56",
  year =         "2007",
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  bibdate =      "Wed Feb 03 08:53:04 2010",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/mathcw.bib",
  note =         "Dedicated to the memory of James Harkness, Peter
                 Redpath Professor of Pure Mathematics, McGill
                 University 1903--1923.",
  URL =          "https://www.google.com/books/edition/On_the_Direct_Numerical_Calculation_of_E/CMM4AAAAMAAJ",
  acknowledgement = ack-nhfb,
  remark =       "The AGM method for Jacobian elliptic functions was
                 discovered by this book's author at McGill University
                 in 1913, first published in \cite{King:1921:SNF}, and
                 then in a 1924 monograph, of which this is a reprint.",
  tableofcontents = "Preface / v \\
                 Introduction / 1 \\
                 II. Historical Note on Landen's Transformation and the
                 Various Scales of Moduli and Amplitudes / 2 \\
                 III. On the Scale of Arithmetico-Geometrical Means / 6
                 \\
                 IV. Landen's Scale of Increasing Amplitudes:
                 $\tan(\phi_{n + 1} - \phi_n) = (b_n / a_n) \tan \phi_n$
                 / 7 \\
                 (i) Calculation of $F(\phi, k)$, $E(\phi, k)$, $K$, and
                 $E$ / 7 \\
                 (ii) Calculation of $\sn(u,k)$, $\cn(u,k)$, $\dn(u,k)$
                 in terms of the argument $u$ / 9 \\
                 V. The Hyperbolic Scale of Increasing Amplitudes:
                 $\tanh(\phi_{n + 1} - \phi_n) = (b_n / a_n)
                 \tanh(\phi_n)$ / 10 \\
                 (i) Calculation of $F(\phi, k')$, $E(\phi, k')$, etc. /
                 10 \\
                 (ii) Calculation of $\sn(u,k')$, $\cn(u,k')$,
                 $\dn(u,k')$ / 11 \\
                 (iii) Calculation of $\sn(i u,k)$, $\cn(i u,k)$, $\dn(i
                 u,k)$ / 12 \\
                 VI: Gauss' Scale of Increasing Amplitudes \\
                 VII: Landen's Scale of Decreasing Amplitudes:
                 $\sin(2\psi_{n + 1} - \psi_n) = (b_n / a_n) \sin
                 \psi_n$ / 14 \\
                 (i) Calculation of $F(\psi, k')$, $E(\psi, k')$, etc. /
                 14 \\
                 (ii) Calculation of $\sn(u,k')$, $\cn(u,k')$,
                 $\dn(u,k')$ / 16 \\
                 VIII. The Hyperbolic Scale of Decreasing Amplitudes:
                 $\sinh(2\psi_{n + 1} - \psi_n) = (b_n / a_n) \sinh
                 \psi_n$ / 17 \\
                 (i) Calculation of $F(\psi, k)$, $E(\psi, k)$, etc. /
                 17 \\
                 (ii) Calculation of $\sn(u,k)$, $\cn(u,k)$, $\dn(u,k)$
                 / 18 \\
                 IX. On the Numerical Computation of the Third Elliptic
                 Integral / 19 \\
                 Case I. $n$ negative, between $0$ and $-k^2$
                 (Hyperbolic case) / 19 \\
                 Case II. $n$ negative, between $-1$ and $-\infty$
                 (Hyperbolic case) / 20 \\
                 Case III. $n$ negative, between $-k^2$ and $-1$
                 (Circular case) / 21 \\
                 First Method. (i) Use of circular recurrence formulae /
                 22 \\
                 (ii) Use of hyperbolic recurrence formulae / 23 \\
                 Second Method. (i) Use of circular recurrence formulae
                 / 25 \\
                 (ii) Use of hyperbolic recurrence formulae / 25 \\
                 Case IV. $n$ positive, between $0$ and $\infty$
                 (Circular case) / 26 \\
                 First Method. (i) Use of circular recurrence formulae /
                 26 \\
                 (ii) Use of hyperbolic recurrence formulae / 27 \\
                 Second Method. (i) Use of circular recurrence formulae
                 / 28 \\
                 (ii) Use of hyperbolic recurrence formulae / 29 \\
                 X. Note on the Calculation of the Third Elliptic
                 Integral in Terms of the Complementary A.G.M. Scale /
                 30 \\
                 Summary of Formulae / 31 \\
                 Appendix, Examples 1--35 / 35",
}


bibtype: article
filename: elefunt.bib
label: Kodama:2007:RA
author: Masao Kodama
title: Remark on Algorithm 644
journal: j-TOMS
volume: 33
number: 4
pages: 28:1--28:3
month: aug
monthnumber: 8
year: 2007
CODEN: ACMSCU
DOI: https://doi.org/10.1145/1268776.1268783
ISSN: 0098-3500 (print), 1557-7295 (electronic) OR 00983500 (print), 15577295 (electronic)
ISSNL: 0098-3500 OR 00983500
bibdate: Mon Dec 17 18:09:13 MST 2007
bibtimestamp: 2007.12.17 18:09:13 MST
note: See cite Amos:1986:APP,Amos:1990:RPP,Amos:1995:RAP.
fjournal: ACM Transactions on Mathematical Software (TOMS)
acknowledgement: ack-nhfb
articleno: 28
journalURL: https://dl.acm.org/loi/toms
rawauthor: Masao Kodama
rawnote: See \cite{Amos:1986:APP,Amos:1990:RPP,Amos:1995:RAP}.
rawtitle: Remark on {Algorithm 644}
abstract:
This remark details correction for errors in the functions which compute the modified Bessel function of the second kind and the log of the gamma function. In both cases these errors cause a loss of precision for a small range of values of the $ nu $ argument. These routines are used in the calculation of a number of other functions within the package whose accuracy is thus similarly affected.

rawabstract:
This remark details correction for errors in the functions which compute the modified Bessel function of the second kind and the log of the gamma function. In both cases these errors cause a loss of precision for a small range of values of the $ \nu $ argument. These routines are used in the calculation of a number of other functions within the package whose accuracy is thus similarly affected.

entry:
@Article{Kodama:2007:RA,
  author =       "Masao Kodama",
  title =        "Remark on {Algorithm 644}",
  journal =      j-TOMS,
  volume =       "33",
  number =       "4",
  pages =        "28:1--28:3",
  month =        aug,
  year =         "2007",
  CODEN =        "ACMSCU",
  DOI =          "https://doi.org/10.1145/1268776.1268783",
  ISSN =         "0098-3500 (print), 1557-7295 (electronic)",
  ISSN-L =       "0098-3500",
  bibdate =      "Mon Dec 17 18:09:13 MST 2007",
  bibsource =    "http://www.acm.org/pubs/contents/journals/toms/;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/toms.bib",
  note =         "See
                 \cite{Amos:1986:APP,Amos:1990:RPP,Amos:1995:RAP}.",
  abstract =     "This remark details correction for errors in the
                 functions which compute the modified Bessel function of
                 the second kind and the log of the gamma function. In
                 both cases these errors cause a loss of precision for a
                 small range of values of the $ \nu $ argument. These
                 routines are used in the calculation of a number of
                 other functions within the package whose accuracy is
                 thus similarly affected.",
  acknowledgement = ack-nhfb,
  articleno =    "28",
  fjournal =     "ACM Transactions on Mathematical Software (TOMS)",
  journal-URL =  "https://dl.acm.org/loi/toms",
}


bibtype: article
filename: elefunt.bib
label: Kuijlaars:2007:TIH
author: A. B. J. Kuijlaars and H. Stahl and W. Van Assche and F. Wielonsky
title: Type II Hermite--Pade approximation to the exponential function
journal: j-J-COMPUT-APPL-MATH
volume: 207
number: 2
pages: 227--244
day: 15
month: oct
monthnumber: 10
year: 2007
CODEN: JCAMDI
ISSN: 0377-0427 (print), 1879-1778 (electronic) OR 03770427 (print), 18791778 (electronic)
ISSNL: 0377-0427 OR 03770427
bibdate: Sat Feb 25 13:13:18 MST 2017
bibtimestamp: 2017.02.25 13:13:18 MST
URL: http://www.sciencedirect.com/science/article/pii/S0377042706005978
fjournal: Journal of Computational and Applied Mathematics
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/03770427
rawauthor: A. B. J. Kuijlaars and H. Stahl and W. {Van Assche} and F. Wielonsky
rawtitle: {Type II} {Hermite--Pad{\'e}} approximation to the exponential function
entry:
@Article{Kuijlaars:2007:TIH,
  author =       "A. B. J. Kuijlaars and H. Stahl and W. {Van Assche}
                 and F. Wielonsky",
  title =        "{Type II} {Hermite--Pad{\'e}} approximation to the
                 exponential function",
  journal =      j-J-COMPUT-APPL-MATH,
  volume =       "207",
  number =       "2",
  pages =        "227--244",
  day =          "15",
  month =        oct,
  year =         "2007",
  CODEN =        "JCAMDI",
  ISSN =         "0377-0427 (print), 1879-1778 (electronic)",
  ISSN-L =       "0377-0427",
  bibdate =      "Sat Feb 25 13:13:18 MST 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2005.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0377042706005978",
  acknowledgement = ack-nhfb,
  fjournal =     "Journal of Computational and Applied Mathematics",
  journal-URL =  "http://www.sciencedirect.com/science/journal/03770427",
}


bibtype: article
filename: elefunt.bib
label: Kuliamin:2007:STI
author: V. V. Kuliamin
title: Standardization and testing of implementations of mathematical functions in floating point numbers
journal: j-PROG-COMP-SOFT
volume: 33
number: 3
pages: 154--173
year: 2007
CODEN: PCSODA
DOI: https://doi.org/10.1134/S036176880703005X
ISSN: 0361-7688 (print), 1608-3261 (electronic) OR 03617688 (print), 16083261 (electronic)
ISSNL: 0361-7688 OR 03617688
bibdate: Fri Aug 08 09:01:30 2008
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2008.08.08 09:01:30 ???
fjournal: Programming and Computer Software; translation of Programmirovaniye (Moscow, USSR) Plenum
keywords: floating-point function testing and verification
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/11086
rawauthor: V. V. Kuliamin
rawtitle: Standardization and testing of implementations of mathematical functions in floating point numbers
abstract:
Requirements definition and test suites development for implementations of mathematical functions in floating point arithmetic in the framework of the IEEE 754 standard are considered. A method based on this standard is proposed for defining requirements for such functions. This method can be used for the standardization of implementations of such functions; this kind of standardization extends IEEE 754. A method for designing test suites for the verification of those requirements is presented. The proposed methods are based on specific properties of the representation of floating point numbers and on some features of the functions under examination.

rawabstract:
Requirements definition and test suites development for implementations of mathematical functions in floating point arithmetic in the framework of the IEEE 754 standard are considered. A method based on this standard is proposed for defining requirements for such functions. This method can be used for the standardization of implementations of such functions; this kind of standardization extends IEEE 754. A method for designing test suites for the verification of those requirements is presented. The proposed methods are based on specific properties of the representation of floating point numbers and on some features of the functions under examination.

entry:
@Article{Kuliamin:2007:STI,
  author =       "V. V. Kuliamin",
  title =        "Standardization and testing of implementations of
                 mathematical functions in floating point numbers",
  journal =      j-PROG-COMP-SOFT,
  volume =       "33",
  number =       "3",
  pages =        "154--173",
  year =         "2007",
  CODEN =        "PCSODA",
  DOI =          "https://doi.org/10.1134/S036176880703005X",
  ISSN =         "0361-7688 (print), 1608-3261 (electronic)",
  ISSN-L =       "0361-7688",
  bibdate =      "Fri Aug 08 09:01:30 2008",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  abstract =     "Requirements definition and test suites development
                 for implementations of mathematical functions in
                 floating point arithmetic in the framework of the IEEE
                 754 standard are considered. A method based on this
                 standard is proposed for defining requirements for such
                 functions. This method can be used for the
                 standardization of implementations of such functions;
                 this kind of standardization extends IEEE 754. A method
                 for designing test suites for the verification of those
                 requirements is presented. The proposed methods are
                 based on specific properties of the representation of
                 floating point numbers and on some features of the
                 functions under examination.",
  acknowledgement = ack-nhfb,
  fjournal =     "Programming and Computer Software; translation of
                 Programmirovaniye (Moscow, USSR) Plenum",
  journal-URL =  "http://link.springer.com/journal/11086",
  keywords =     "floating-point function testing and verification",
}


bibtype: techreport
filename: elefunt.bib
label: Lefevre:2007:SNP
author: Vincent Lefevre and Jean-Michel Muller
title: Some notes on the possible under / overflow of the most common elementary functions
type: Report
institution: LIP, Ecole Normale Superieure de Lyon
address: Lyon, France
pages: 7
year: 2007
bibdate: Fri May 25 16:18:32 2007
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2007.05.25 16:18:32 ???
URL: http://prunel.ccsd.cnrs.fr/ensl-00149414
keywords: elementary functions; floating-point arithmetic; overflow; underflow
acknowledgement: ack-nhfb
rawauthor: Vincent Lef{\'e}vre and Jean-Michel Muller
rawtitle: Some notes on the possible under\slash overflow of the most common elementary functions
abstract:
The purpose of this short note is not to describe when underflow or overflow must be signalled (it is quite clear that the rules are the same as for the basic arithmetic operations). We just want to show that for some of the most common functions and floating-point formats, in many cases, we can know in advance that the results will always lie in the range of the numbers that are representable by normal floating-point numbers, so that in these cases there is no need to worry about underflow or overflow. Note that when it is not the case, an implementation is still possible using a run-time test.

rawabstract:
The purpose of this short note is not to describe when underflow or overflow must be signalled (it is quite clear that the rules are the same as for the basic arithmetic operations). We just want to show that for some of the most common functions and floating-point formats, in many cases, we can know in advance that the results will always lie in the range of the numbers that are representable by normal floating-point numbers, so that in these cases there is no need to worry about underflow or overflow. Note that when it is not the case, an implementation is still possible using a run-time test.

entry:
@TechReport{Lefevre:2007:SNP,
  author =       "Vincent Lef{\'e}vre and Jean-Michel Muller",
  title =        "Some notes on the possible under\slash overflow of the
                 most common elementary functions",
  type =         "Report",
  institution =  "LIP, {\'E}cole Normale Sup{\'e}rieure de Lyon",
  address =      "Lyon, France",
  pages =        "7",
  year =         "2007",
  bibdate =      "Fri May 25 16:18:32 2007",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://prunel.ccsd.cnrs.fr/ensl-00149414",
  abstract =     "The purpose of this short note is not to describe when
                 underflow or overflow must be signalled (it is quite
                 clear that the rules are the same as for the basic
                 arithmetic operations). We just want to show that for
                 some of the most common functions and floating-point
                 formats, in many cases, we can know in advance that the
                 results will always lie in the range of the numbers
                 that are representable by normal floating-point
                 numbers, so that in these cases there is no need to
                 worry about underflow or overflow. Note that when it is
                 not the case, an implementation is still possible using
                 a run-time test.",
  acknowledgement = ack-nhfb,
  keywords =     "elementary functions; floating-point arithmetic;
                 overflow; underflow",
}


bibtype: article
filename: elefunt.bib
label: Neher:2007:CSF
author: Markus Neher
title: Complex standard functions and their implementation in the CoStLy library
journal: j-TOMS
volume: 33
number: 1
pages: 2:1--2:27
month: mar
monthnumber: 3
year: 2007
CODEN: ACMSCU
DOI: https://doi.org/10.1145/1206040.1206042
ISSN: 0098-3500 (print), 1557-7295 (electronic) OR 00983500 (print), 15577295 (electronic)
ISSNL: 0098-3500 OR 00983500
bibdate: Sat Apr 14 09:48:58 MDT 2007
bibtimestamp: 2007.04.14 09:48:58 MDT
fjournal: ACM Transactions on Mathematical Software (TOMS)
acknowledgement: ack-nhfb
articleno: 2
journalURL: https://dl.acm.org/loi/toms
rawauthor: Markus Neher
rawtitle: Complex standard functions and their implementation in the {CoStLy} library
abstract:
The practical calculation of range bounds for some complex standard functions is addressed in this article. The functions under consideration are root and power functions, the exponential, trigonometric and hyperbolic functions, and their inverse functions. For such a function $f$ and a given rectangular complex interval $z$, some interval $w$ is computed that contains all function values of $f$ in $z$. This is done by expressing the real and the imaginary part of $f$ as compositions of real standard functions and then estimating the ranges of these compositions. In many cases, the inclusions are optimal, such that $w$ is the smallest rectangular interval containing the range of $f$. The algorithms presented in this article have been implemented in a C++ class library called CoStLy (Complex Standard Functions License), which is distributed under the conditions of the GNU General Public License.

rawabstract:
The practical calculation of range bounds for some complex standard functions is addressed in this article. The functions under consideration are root and power functions, the exponential, trigonometric and hyperbolic functions, and their inverse functions. For such a function $f$ and a given rectangular complex interval $z$, some interval $w$ is computed that contains all function values of $f$ in $z$. This is done by expressing the real and the imaginary part of $f$ as compositions of real standard functions and then estimating the ranges of these compositions. In many cases, the inclusions are optimal, such that $w$ is the smallest rectangular interval containing the range of $f$. The algorithms presented in this article have been implemented in a C++ class library called CoStLy (Complex Standard Functions License), which is distributed under the conditions of the GNU General Public License.

entry:
@Article{Neher:2007:CSF,
  author =       "Markus Neher",
  title =        "Complex standard functions and their implementation in
                 the {CoStLy} library",
  journal =      j-TOMS,
  volume =       "33",
  number =       "1",
  pages =        "2:1--2:27",
  month =        mar,
  year =         "2007",
  CODEN =        "ACMSCU",
  DOI =          "https://doi.org/10.1145/1206040.1206042",
  ISSN =         "0098-3500 (print), 1557-7295 (electronic)",
  ISSN-L =       "0098-3500",
  bibdate =      "Sat Apr 14 09:48:58 MDT 2007",
  bibsource =    "http://www.acm.org/pubs/contents/journals/toms/;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/toms.bib",
  abstract =     "The practical calculation of range bounds for some
                 complex standard functions is addressed in this
                 article. The functions under consideration are root and
                 power functions, the exponential, trigonometric and
                 hyperbolic functions, and their inverse functions. For
                 such a function $f$ and a given rectangular complex
                 interval $z$, some interval $w$ is computed that
                 contains all function values of $f$ in $z$. This is
                 done by expressing the real and the imaginary part of
                 $f$ as compositions of real standard functions and then
                 estimating the ranges of these compositions. In many
                 cases, the inclusions are optimal, such that $w$ is the
                 smallest rectangular interval containing the range of
                 $f$.

                 The algorithms presented in this article have been
                 implemented in a C++ class library called CoStLy
                 (Complex Standard Functions License), which is
                 distributed under the conditions of the GNU General
                 Public License.",
  acknowledgement = ack-nhfb,
  articleno =    "2",
  fjournal =     "ACM Transactions on Mathematical Software (TOMS)",
  journal-URL =  "https://dl.acm.org/loi/toms",
}


bibtype: book
filename: elefunt.bib
label: Press:2007:NRA
author: William H. Press and Saul A. Teukolsky and William T. Vetterling and Brian P. Flannery
title: Numerical Recipes --- The Art of Scientific Computing
publisher: pub-CAMBRIDGE
address: pub-CAMBRIDGE:adr
edition: Third
pages: xxi + 1235
year: 2007
LCCN: QA297 .N866 2007
bibdate: Wed Dec 15 10:40:52 1993
bibtimestamp: 1993.12.15 10:40:52 ???
URL: http://www.cambridge.org/numericalrecipes
subject: numerical analysis; computer programs; science; mathematics; C++ (computer program language)
acknowledgement: ack-nhfb
rawauthor: William H. Press and Saul A. Teukolsky and William T. Vetterling and Brian P. Flannery
rawtitle: Numerical Recipes --- The Art of Scientific Computing
ISBN:
0-521-88068-8 (hardcover), 0-521-88407-1 (with source code CD ROM), 0-521-70685-8 (source code CD ROM) OR 0521880688 (hardcover), 0521884071 (with source code CD ROM), 0521706858 (source code CD ROM)

ISBN13:
978-0-521-88068-8 (hardcover), 978-0-521-88407-5 (with source code CD ROM), 978-0-521-70685-8 (source code CD ROM) OR 9780521880688 (hardcover), 9780521884075 (with source code CD ROM), 9780521706858 (source code CD ROM)

bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/master.bib; https://www.math.utah.edu/pub/tex/bib/numana2000.bib

TOC:
1. Preliminaries 2. Solution of linear algebraic equations 3. Interpolation and extrapolation 4. Integration of functions 5. Evaluation of functions 6. Special functions 7. Random numbers 8. Sorting and selection 9. Root finding and nonlinear sets of equations 10. Minimization or maximization of functions 11. Eigensystems 12. Fast Fourier Transform 13. Fourier and spectral applications 14. Statistical description of data 15. Modeling of data 16. Classification and inference 17. Integration of ordinary differential equations 18. Two-point boundary value problems 19. Integral equations and inverse theory 20. Partial differential equations 21. Computational geometry 22. Less-numerical algorithms

rawTOC:
1. Preliminaries \\ 2. Solution of linear algebraic equations \\ 3. Interpolation and extrapolation \\ 4. Integration of functions \\ 5. Evaluation of functions \\ 6. Special functions \\ 7. Random numbers \\ 8. Sorting and selection \\ 9. Root finding and nonlinear sets of equations \\ 10. Minimization or maximization of functions \\ 11. Eigensystems \\ 12. Fast Fourier Transform \\ 13. Fourier and spectral applications \\ 14. Statistical description of data \\ 15. Modeling of data \\ 16. Classification and inference \\ 17. Integration of ordinary differential equations \\ 18. Two-point boundary value problems \\ 19. Integral equations and inverse theory \\ 20. Partial differential equations \\ 21. Computational geometry \\ 22. Less-numerical algorithms

entry:
@Book{Press:2007:NRA,
  author =       "William H. Press and Saul A. Teukolsky and William T.
                 Vetterling and Brian P. Flannery",
  title =        "Numerical Recipes --- The Art of Scientific
                 Computing",
  publisher =    pub-CAMBRIDGE,
  address =      pub-CAMBRIDGE:adr,
  edition =      "Third",
  pages =        "xxi + 1235",
  year =         "2007",
  ISBN =         "0-521-88068-8 (hardcover), 0-521-88407-1 (with source
                 code CD ROM), 0-521-70685-8 (source code CD ROM)",
  ISBN-13 =      "978-0-521-88068-8 (hardcover), 978-0-521-88407-5 (with
                 source code CD ROM), 978-0-521-70685-8 (source code CD
                 ROM)",
  LCCN =         "QA297 .N866 2007",
  bibdate =      "Wed Dec 15 10:40:52 1993",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/master.bib;
                 https://www.math.utah.edu/pub/tex/bib/numana2000.bib",
  URL =          "http://www.cambridge.org/numericalrecipes",
  acknowledgement = ack-nhfb,
  subject =      "numerical analysis; computer programs; science;
                 mathematics; C++ (computer program language)",
  tableofcontents = "1. Preliminaries \\
                 2. Solution of linear algebraic equations \\
                 3. Interpolation and extrapolation \\
                 4. Integration of functions \\
                 5. Evaluation of functions \\
                 6. Special functions \\
                 7. Random numbers \\
                 8. Sorting and selection \\
                 9. Root finding and nonlinear sets of equations \\
                 10. Minimization or maximization of functions \\
                 11. Eigensystems \\
                 12. Fast Fourier Transform \\
                 13. Fourier and spectral applications \\
                 14. Statistical description of data \\
                 15. Modeling of data \\
                 16. Classification and inference \\
                 17. Integration of ordinary differential equations \\
                 18. Two-point boundary value problems \\
                 19. Integral equations and inverse theory \\
                 20. Partial differential equations \\
                 21. Computational geometry \\
                 22. Less-numerical algorithms",
}


bibtype: article
filename: elefunt.bib
label: Ren:2007:CFA
author: C. Ren and A. R. MacKenzie
journal: j-ATMOS-SCI-LETT
volume: 8
number: 3
pages: 70--73
year: 2007
DOI: https://doi.org/10.1002/asl.154
ISSN: 1530-261X OR 1530261X
ISSNL: 1530-261X OR 1530261X
bibdate: Sat Dec 16 17:25:42 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2017.12.16 17:25:42 ???
URL: http://onlinelibrary.wiley.com/doi/10.1002/asl.154/full
fjournal: Atmospheric Science Letters
acknowledgement: ack-nhfb
rawauthor: C. Ren and A. R. MacKenzie
title:
Closed-form approximations to the error and complementary error functions and their applications in atmospheric science

journalURL:
http://www.sciencedirect.com/science/journal/1530261X; http://rmets.onlinelibrary.wiley.com/hub/journal/10.1002/(ISSN)1530-261X/

rawtitle:
Closed-form approximations to the error and complementary error functions and their applications in atmospheric science

entry:
@Article{Ren:2007:CFA,
  author =       "C. Ren and A. R. MacKenzie",
  title =        "Closed-form approximations to the error and
                 complementary error functions and their applications in
                 atmospheric science",
  journal =      j-ATMOS-SCI-LETT,
  volume =       "8",
  number =       "3",
  pages =        "70--73",
  month =        "????",
  year =         "2007",
  DOI =          "https://doi.org/10.1002/asl.154",
  ISSN =         "1530-261X",
  ISSN-L =       "1530-261X",
  bibdate =      "Sat Dec 16 17:25:42 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://onlinelibrary.wiley.com/doi/10.1002/asl.154/full",
  acknowledgement = ack-nhfb,
  fjournal =     "Atmospheric Science Letters",
  journal-URL =  "http://www.sciencedirect.com/science/journal/1530261X;
                 http://rmets.onlinelibrary.wiley.com/hub/journal/10.1002/(ISSN)1530-261X/",
}


bibtype: article
filename: elefunt.bib
label: Rokhlin:2007:AFC
author: Vladimir Rokhlin and Hong Xiao
journal: j-APPL-COMPUT-HARMON-ANAL
volume: 22
number: 1
pages: 105--123
month: jan
monthnumber: 1
year: 2007
DOI: https://doi.org/10.1016/j.acha.2006.05.004
ISSN: 1063-5203 (print), 1096-603x (electronic) OR 10635203 (print), 1096603x (electronic)
ISSNL: 1063-5203 OR 10635203
bibdate: Sun Oct 31 10:00:51 2010
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2010.10.31 10:00:51 ???
keywords: approximation; asymptotic; band-limit; prolate spheroidal wave functions
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/10635203
rawauthor: Vladimir Rokhlin and Hong Xiao
title:
Approximate formulae for certain prolate spheroidal wave functions valid for large values of both order and band-limit

abstract:
We construct asymptotic formulae for the approximation of certain prolate spheroidal wave functions and of the corresponding eigenvalues. We investigate two regimes: when the ratio $ c / m $ decays, and when both $c$ and $m$ grow, but the ratio $ c / m $ stays bounded. Both the regions of validity and the accuracies of the obtained expansions are illustrated with numerical examples.

fjournal:
Applied and Computational Harmonic Analysis. Time-Frequency and Time-Scale Analysis, Wavelets, Numerical Algorithms, and Applications

rawabstract:
We construct asymptotic formulae for the approximation of certain prolate spheroidal wave functions and of the corresponding eigenvalues. We investigate two regimes: when the ratio $ c / m $ decays, and when both $c$ and $m$ grow, but the ratio $ c / m $ stays bounded. Both the regions of validity and the accuracies of the obtained expansions are illustrated with numerical examples.

rawtitle:
Approximate formulae for certain prolate spheroidal wave functions valid for large values of both order and band-limit

entry:
@Article{Rokhlin:2007:AFC,
  author =       "Vladimir Rokhlin and Hong Xiao",
  title =        "Approximate formulae for certain prolate spheroidal
                 wave functions valid for large values of both order and
                 band-limit",
  journal =      j-APPL-COMPUT-HARMON-ANAL,
  volume =       "22",
  number =       "1",
  pages =        "105--123",
  month =        jan,
  year =         "2007",
  DOI =          "https://doi.org/10.1016/j.acha.2006.05.004",
  ISSN =         "1063-5203 (print), 1096-603x (electronic)",
  ISSN-L =       "1063-5203",
  bibdate =      "Sun Oct 31 10:00:51 2010",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  abstract =     "We construct asymptotic formulae for the approximation
                 of certain prolate spheroidal wave functions and of the
                 corresponding eigenvalues. We investigate two regimes:
                 when the ratio $ c / m $ decays, and when both $c$ and
                 $m$ grow, but the ratio $ c / m $ stays bounded. Both
                 the regions of validity and the accuracies of the
                 obtained expansions are illustrated with numerical
                 examples.",
  acknowledgement = ack-nhfb,
  fjournal =     "Applied and Computational Harmonic Analysis.
                 Time-Frequency and Time-Scale Analysis, Wavelets,
                 Numerical Algorithms, and Applications",
  journal-URL =  "http://www.sciencedirect.com/science/journal/10635203",
  keywords =     "approximation; asymptotic; band-limit; prolate
                 spheroidal wave functions",
}


bibtype: article
filename: elefunt.bib
label: Schmelzer:2007:CGF
author: Thomas Schmelzer and Lloyd N. Trefethen
title: Computing the Gamma Function Using Contour Integrals and Rational Approximations
journal: j-SIAM-J-NUMER-ANAL
volume: 45
number: 2
pages: 558--571
year: 2007
CODEN: SJNAAM
DOI: https://doi.org/10.1137/050646342
ISSN: 0036-1429 (print), 1095-7170 (electronic) OR 00361429 (print), 10957170 (electronic)
ISSNL: 0036-1429 OR 00361429
bibdate: Mon Nov 24 18:03:07 MST 2008
bibtimestamp: 2008.11.24 18:03:07 MST
fjournal: SIAM Journal on Numerical Analysis
acknowledgement: ack-nhfb
journalURL: http://epubs.siam.org/sinum
rawauthor: Thomas Schmelzer and Lloyd N. Trefethen
rawtitle: Computing the Gamma Function Using Contour Integrals and Rational Approximations
bibsource:
http://siamdl.aip.org/dbt/dbt.jsp?KEY=SJNAAM&Volume=45&Issue=2; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

entry:
@Article{Schmelzer:2007:CGF,
  author =       "Thomas Schmelzer and Lloyd N. Trefethen",
  title =        "Computing the Gamma Function Using Contour Integrals
                 and Rational Approximations",
  journal =      j-SIAM-J-NUMER-ANAL,
  volume =       "45",
  number =       "2",
  pages =        "558--571",
  month =        "????",
  year =         "2007",
  CODEN =        "SJNAAM",
  DOI =          "https://doi.org/10.1137/050646342",
  ISSN =         "0036-1429 (print), 1095-7170 (electronic)",
  ISSN-L =       "0036-1429",
  bibdate =      "Mon Nov 24 18:03:07 MST 2008",
  bibsource =    "http://siamdl.aip.org/dbt/dbt.jsp?KEY=SJNAAM&Volume=45&Issue=2;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "SIAM Journal on Numerical Analysis",
  journal-URL =  "http://epubs.siam.org/sinum",
}


bibtype: book
filename: elefunt.bib
label: Slavjanov:2007:SFU
author: Sergej J. Slavjanov and Wolfgang Lay
title: Special Functions: a Unified Theory Based on Singularities
publisher: pub-OXFORD
address: pub-OXFORD:adr
pages: xvi + 293
year: 2007
ISBN: 0-19-850573-6 OR 0198505736
ISBN13: 978-0-19-850573-0 OR 9780198505730
bibdate: Tue Dec 5 11:27:46 MST 2023
bibsource: fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2023.12.05 11:27:46 MST
series: Oxford mathematical monographs; Oxford science publications
subject: Functions, Special; Fonctions speciales; Functions, Special
acknowledgement: ack-nhfb
rawauthor: Sergej J. Slavjanov and Wolfgang Lay
rawtitle: Special Functions: a Unified Theory Based on Singularities
TOC:
1: Linear second-order ODEs with polynomial coefficients Regular singularities and Fuchsian equations Regular and Fuchsian singularities Fuchsian equations and their transformations Characteristic exponents Frobenius solutions Irregular singularities and confluent equations The $s$-rank of a singularity Confluent and reduced confluent equations The $s$-homotopic transformation Asymptotic solutions at irregular singularities Canonical forms A generalization of Fuchs's theorem Confluence and reduction processes Strong and weak confluence. A confluence theorem A confluence principle Reduction of an equation Classes and types of equations Standard forms of equations Invariants of $s$-homotopic transformations Types of solutions Eigenfunctions of singular Sturm--Liouville problems Central and lateral connection problems Stokes lines at singularities. Stokes matrices Generalized Riemann scheme Applications Central two-point connection problems (CTCPs) Two regular singularities as relevant endpoints One regular singularity and one irregular singularity as the endpoints A proof Two irregular singularities 2: The hypergeometric class of equations Classification scheme General presentation Hypergeometric equation Confluent equations Reduced confluent equations Difference equations General consideration Difference equations for hypergeometric functions Confluent hypergeometric functions ldots 

shortTOC:
1: Linear second-order ODEs with polynomial coefficients 2: The hypergeometric class of equations 3: The Heun class of equations 4: Application to physical sciences 5: The Painleve class of equations Appendix A: The gamma function and related functions Appendix B: CTCPs Heun equations in general form Appendix C: Multipole Matrix elements Appendix D: SFTools Database of the special functions

rawshortTOC:
1: Linear second-order ODEs with polynomial coefficients \\ 2: The hypergeometric class of equations \\ 3: The Heun class of equations \\ 4: Application to physical sciences \\ 5: The Painlev{\'e} class of equations \\ Appendix A: The gamma function and related functions \\ Appendix B: CTCPs Heun equations in general form \\ Appendix C: Multipole Matrix elements \\ Appendix D: SFTools \\ Database of the special functions

rawTOC:
1: Linear second-order ODEs with polynomial coefficients \\ Regular singularities and Fuchsian equations \\ Regular and Fuchsian singularities \\ Fuchsian equations and their transformations \\ Characteristic exponents \\ Frobenius solutions \\ Irregular singularities and confluent equations \\ The $s$-rank of a singularity \\ Confluent and reduced confluent equations \\ The $s$-homotopic transformation \\ Asymptotic solutions at irregular singularities \\ Canonical forms \\ A generalization of Fuchs's theorem \\ Confluence and reduction processes \\ Strong and weak confluence. A confluence theorem \\ A confluence principle \\ Reduction of an equation \\ Classes and types of equations \\ Standard forms of equations \\ Invariants of $s$-homotopic transformations \\ Types of solutions \\ Eigenfunctions of singular Sturm--Liouville problems \\ Central and lateral connection problems \\ Stokes lines at singularities. Stokes matrices \\ Generalized Riemann scheme \\ Applications \\ Central two-point connection problems (CTCPs) \\ Two regular singularities as relevant endpoints \\ One regular singularity and one irregular singularity as the endpoints \\ A proof \\ Two irregular singularities \\ 2: The hypergeometric class of equations \\ Classification scheme \\ General presentation \\ Hypergeometric equation \\ Confluent equations \\ Reduced confluent equations \\ Difference equations \\ General consideration \\ Difference equations for hypergeometric functions \\ Confluent hypergeometric functions \\ \ldots{}

entry:
@Book{Slavjanov:2007:SFU,
  author =       "Sergej J. Slavjanov and Wolfgang Lay",
  title =        "Special Functions: a Unified Theory Based on
                 Singularities",
  publisher =    pub-OXFORD,
  address =      pub-OXFORD:adr,
  pages =        "xvi + 293",
  year =         "2007",
  ISBN =         "0-19-850573-6",
  ISBN-13 =      "978-0-19-850573-0",
  LCCN =         "????",
  bibdate =      "Tue Dec 5 11:27:46 MST 2023",
  bibsource =    "fsz3950.oclc.org:210/WorldCat;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  series =       "Oxford mathematical monographs; Oxford science
                 publications",
  acknowledgement = ack-nhfb,
  shorttableofcontents = "1: Linear second-order ODEs with polynomial
                 coefficients \\
                 2: The hypergeometric class of equations \\
                 3: The Heun class of equations \\
                 4: Application to physical sciences \\
                 5: The Painlev{\'e} class of equations \\
                 Appendix A: The gamma function and related functions
                 \\
                 Appendix B: CTCPs Heun equations in general form \\
                 Appendix C: Multipole Matrix elements \\
                 Appendix D: SFTools \\
                 Database of the special functions",
  subject =      "Functions, Special; Fonctions sp{\'e}ciales;
                 Functions, Special",
  tableofcontents = "1: Linear second-order ODEs with polynomial
                 coefficients \\
                 Regular singularities and Fuchsian equations \\
                 Regular and Fuchsian singularities \\
                 Fuchsian equations and their transformations \\
                 Characteristic exponents \\
                 Frobenius solutions \\
                 Irregular singularities and confluent equations \\
                 The $s$-rank of a singularity \\
                 Confluent and reduced confluent equations \\
                 The $s$-homotopic transformation \\
                 Asymptotic solutions at irregular singularities \\
                 Canonical forms \\
                 A generalization of Fuchs's theorem \\
                 Confluence and reduction processes \\
                 Strong and weak confluence. A confluence theorem \\
                 A confluence principle \\
                 Reduction of an equation \\
                 Classes and types of equations \\
                 Standard forms of equations \\
                 Invariants of $s$-homotopic transformations \\
                 Types of solutions \\
                 Eigenfunctions of singular Sturm--Liouville problems
                 \\
                 Central and lateral connection problems \\
                 Stokes lines at singularities. Stokes matrices \\
                 Generalized Riemann scheme \\
                 Applications \\
                 Central two-point connection problems (CTCPs) \\
                 Two regular singularities as relevant endpoints \\
                 One regular singularity and one irregular singularity
                 as the endpoints \\
                 A proof \\
                 Two irregular singularities \\
                 2: The hypergeometric class of equations \\
                 Classification scheme \\
                 General presentation \\
                 Hypergeometric equation \\
                 Confluent equations \\
                 Reduced confluent equations \\
                 Difference equations \\
                 General consideration \\
                 Difference equations for hypergeometric functions \\
                 Confluent hypergeometric functions \\
                 \ldots{}",
}


bibtype: article
filename: elefunt.bib
label: Srinivasan:2007:GFE
author: Gopala Krishna Srinivasan
title: The Gamma Function: An Eclectic Tour
journal: j-AMER-MATH-MONTHLY
volume: 114
number: 4
pages: 297--315
month: apr
monthnumber: 4
year: 2007
CODEN: AMMYAE
ISSN: 0002-9890 (print), 1930-0972 (electronic) OR 00029890 (print), 19300972 (electronic)
ISSNL: 0002-9890 OR 00029890
bibdate: Mon Jan 30 12:00:28 MST 2012
bibtimestamp: 2012.01.30 12:00:28 MST
URL: http://www.jstor.org/stable/27642193
fjournal: American Mathematical Monthly
acknowledgement: ack-nhfb
journalURL: https://www.jstor.org/journals/00029890.htm
rawauthor: Gopala Krishna Srinivasan
rawtitle: The Gamma Function: An Eclectic Tour
bibsource:
http://www.jstor.org/journals/00029890.html; http://www.jstor.org/stable/i27642189; https://www.math.utah.edu/pub/tex/bib/amermathmonthly2000.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

entry:
@Article{Srinivasan:2007:GFE,
  author =       "Gopala Krishna Srinivasan",
  title =        "The Gamma Function: An Eclectic Tour",
  journal =      j-AMER-MATH-MONTHLY,
  volume =       "114",
  number =       "4",
  pages =        "297--315",
  month =        apr,
  year =         "2007",
  CODEN =        "AMMYAE",
  ISSN =         "0002-9890 (print), 1930-0972 (electronic)",
  ISSN-L =       "0002-9890",
  bibdate =      "Mon Jan 30 12:00:28 MST 2012",
  bibsource =    "http://www.jstor.org/journals/00029890.html;
                 http://www.jstor.org/stable/i27642189;
                 https://www.math.utah.edu/pub/tex/bib/amermathmonthly2000.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.jstor.org/stable/27642193",
  acknowledgement = ack-nhfb,
  fjournal =     "American Mathematical Monthly",
  journal-URL =  "https://www.jstor.org/journals/00029890.htm",
}


bibtype: article
filename: elefunt.bib
label: Temme:2007:NAS
author: Nico M. Temme
title: Numerical aspects of special functions
journal: j-ACTA-NUMERICA
volume: 16
pages: 379--478
year: 2007
CODEN: ANUMFU
DOI: https://doi.org/10.1017/S0962492906330012
ISBN: 0-521-87743-1 OR 0521877431
ISBN13: 978-0-521-87743-5 OR 9780521877435
ISSN: 0962-4929 (print), 1474-0508 (electronic) OR 09624929 (print), 14740508 (electronic)
ISSNL: 0962-4929 OR 09624929
MRclass: 33F05 (65D20)
MRnumber: 2417932 (2009g:33027)
MRreviewer: Amparo Gil
bibdate: Sat Sep 24 11:37:18 MDT 2011
bibtimestamp: 2011.09.24 11:37:18 MDT
fjournal: Acta Numerica
acknowledgement: ack-nhfb
ajournal: Acta Numer.
journalURL: http://journals.cambridge.org/action/displayJournal?jid=ANU
onlinedate: 24 April 2007
rawauthor: Nico M. Temme
rawtitle: Numerical aspects of special functions
bibsource:
https://www.math.utah.edu/pub/tex/bib/actanumerica.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

abstract:
This paper describes methods that are important for the numerical evaluation of certain functions that frequently occur in applied mathematics, physics and mathematical statistics. This includes what we consider to be the basic methods, such as recurrence relations, series expansions (both convergent and asymptotic), and numerical quadrature. Several other methods are available and some of these will be discussed in less detail. Examples will be given on the use of special functions in certain problems from mathematical physics and mathematical statistics (integrals and series with special functions).

rawabstract:
This paper describes methods that are important for the numerical evaluation of certain functions that frequently occur in applied mathematics, physics and mathematical statistics. This includes what we consider to be the basic methods, such as recurrence relations, series expansions (both convergent and asymptotic), and numerical quadrature. Several other methods are available and some of these will be discussed in less detail. Examples will be given on the use of special functions in certain problems from mathematical physics and mathematical statistics (integrals and series with special functions).

entry:
@Article{Temme:2007:NAS,
  author =       "Nico M. Temme",
  title =        "Numerical aspects of special functions",
  journal =      j-ACTA-NUMERICA,
  volume =       "16",
  pages =        "379--478",
  year =         "2007",
  CODEN =        "ANUMFU",
  DOI =          "https://doi.org/10.1017/S0962492906330012",
  ISBN =         "0-521-87743-1",
  ISBN-13 =      "978-0-521-87743-5",
  ISSN =         "0962-4929 (print), 1474-0508 (electronic)",
  ISSN-L =       "0962-4929",
  MRclass =      "33F05 (65D20)",
  MRnumber =     "2417932 (2009g:33027)",
  MRreviewer =   "Amparo Gil",
  bibdate =      "Sat Sep 24 11:37:18 MDT 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/actanumerica.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  abstract =     "This paper describes methods that are important for
                 the numerical evaluation of certain functions that
                 frequently occur in applied mathematics, physics and
                 mathematical statistics. This includes what we consider
                 to be the basic methods, such as recurrence relations,
                 series expansions (both convergent and asymptotic), and
                 numerical quadrature. Several other methods are
                 available and some of these will be discussed in less
                 detail. Examples will be given on the use of special
                 functions in certain problems from mathematical physics
                 and mathematical statistics (integrals and series with
                 special functions).",
  acknowledgement = ack-nhfb,
  ajournal =     "Acta Numer.",
  fjournal =     "Acta Numerica",
  journal-URL =  "http://journals.cambridge.org/action/displayJournal?jid=ANU",
  onlinedate =   "24 April 2007",
}


bibtype: incollection
filename: elefunt.bib
label: Weniger:2007:AAT
author: Ernst Joachim Weniger
title: Asymptotic approximations to truncation errors of series representations for special functions
crossref: Iske:2007:AAP
pages: 331--348
year: 2007
MRclass: 33F05
MRnumber: MR2335174 (2008h:33051)
bibdate: Thu Dec 01 09:38:02 2011
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2011.12.01 09:38:02 ???
remark: Available as math.CA/0511074.
acknowledgement: ack-nhfb
rawauthor: Ernst Joachim Weniger
rawtitle: Asymptotic approximations to truncation errors of series representations for special functions
keywords:
Bernoulli numbers; Euler--MacLaurin formula; exponential integer $E_1(z)$; Gaussian hypergeometric series $_2F_1(a / b / c / z)$; Riemann zeta function

entry:
@InCollection{Weniger:2007:AAT,
  author =       "Ernst Joachim Weniger",
  title =        "Asymptotic approximations to truncation errors of
                 series representations for special functions",
  crossref =     "Iske:2007:AAP",
  pages =        "331--348",
  year =         "2007",
  MRclass =      "33F05",
  MRnumber =     "MR2335174 (2008h:33051)",
  bibdate =      "Thu Dec 01 09:38:02 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
  keywords =     "Bernoulli numbers; Euler--MacLaurin formula;
                 exponential integer $E_1(z)$; Gaussian hypergeometric
                 series $_2F_1(a / b / c / z)$; Riemann zeta function",
  remark =       "Available as math.CA/0511074.",
}


bibtype: book
filename: elefunt.bib
label: Agarwal:2008:OPD
author: Ravi P. Agarwal and Donal O'Regan
publisher: pub-SV
address: pub-SV:adr
pages: xiv + 410
year: 2008
ISBN: 0-387-79145-0 (paperback) OR 0387791450 (paperback)
ISBN13: 978-0-387-79145-6 (paperback) OR 9780387791456 (paperback)
bibdate: Sat Oct 30 17:22:04 MDT 2010
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib; prodorbis.library.yale.edu:7090/voyager
bibtimestamp: 2010.10.30 17:22:04 MDT
series: Universitext
subject: differential equations; differential equations, partial; Fourier analysis; boundary value problems
acknowledgement: ack-nhfb
rawauthor: Ravi P. Agarwal and Donal O'Regan
title:
Ordinary and Partial Differential Equations: with Special Functions, Fourier Series, and Boundary Value Problems

TOC:
Preface vii l. Solvable Differential Equations / 1 2. Second-Order Differential Equations / 8 3. Preliminaries to Series Solutions / 15 4. Solution at an Ordinary Point / 23 5. Solution at a Singular Point / 31 6. Solution at a Singular Point (Cont'd.) / 37 7. Legendre Polynomials and Functions / 47 8. Chebyshev, Hermite and Laguerre Polynomials / 57 9. Bessel Functions / 64 10. Hypergeometrie Functions / 75 11. Piecewise Continuous and Periodic Functions / 83 12. Orthogonal Functions and Polynomials / 90 13. Orthogonal Functions and Polynomials (Cont'd.) / 95 14. Boundary Value Problems / 104 15. Boundary Value Problems (Cont'd.) / 109 16. Green's Functions / 119 17. Regular Perturbations / 129 18. Singular Perturbations / 138 19. Sturm--Liouville Problems / 145 20. Eigenfunction Expansions / 157 21. Eigenfunction Expansions (Cont'd.) / 163 22. Convergence of the Fourier Series / 171 23. Convergence of the Fourier Series (Cont'd.) / 176 24. Fourier Series Solutions of Ordinary Differential Equations / 187 25. Partial Differential Equations / 194 26. First-Order Partial Differential Equations / 202 27. Solvable Partial Differential Equations / 210 28. The Canonical Forms / 219 29. The Method of Separation of Variables / 227 30. The One-Dimensional Heat Equation / 234 31. The One-Dimensional Heat Equation (Cont'd.) / 241 32. The One-Dimensional Wave Equation / 249 33. The One-Dimensional Wave Equation (Cont'd.) / 256 34. Laplace Equation in Two Dimensions / 266 35. Laplace Equation in Polar Coordinates / 275 36. Two-Dimensional Heat Equation / 284 37. Two-Dimensional Wave Equation / 292 38. Laplace Equation in Three Dimensions / 300 39. Laplace Equation in Three Dimensions (Cont'd.) / 306 40. Nonhomogeneous Equations / 316 41. Fourier Integral and Transforms / 323 42. Fourier Integral and Transforms (Cont'd.) / 330 43. Fourier Transform Method for Partial DEs / 338 44. Fourier Transform Method for Partial DEs (Cont'd.) / 344 45. Laplace Transforms / 354 46. Laplace Transforms (Cont'd.) / 361 47. Laplace Transform Method for Ordinary DEs / 374 48. Laplace Transform Method for Partial DEs / 384 49. Well-Posed Problems / 394 50. Verification of Solutions / 399 References for Further Reading / 405 Index / 407

rawtitle:
Ordinary and Partial Differential Equations: with Special Functions, {Fourier} Series, and Boundary Value Problems

rawTOC:
Preface vii l. Solvable Differential Equations / 1 \\ 2. Second-Order Differential Equations / 8 \\ 3. Preliminaries to Series Solutions / 15 \\ 4. Solution at an Ordinary Point / 23 \\ 5. Solution at a Singular Point / 31 \\ 6. Solution at a Singular Point (Cont'd.) / 37 \\ 7. Legendre Polynomials and Functions / 47 \\ 8. Chebyshev, Hermite and Laguerre Polynomials / 57 \\ 9. Bessel Functions / 64 \\ 10. Hypergeometrie Functions / 75 \\ 11. Piecewise Continuous and Periodic Functions / 83 \\ 12. Orthogonal Functions and Polynomials / 90 \\ 13. Orthogonal Functions and Polynomials (Cont'd.) / 95 \\ 14. Boundary Value Problems / 104 \\ 15. Boundary Value Problems (Cont'd.) / 109 \\ 16. Green's Functions / 119 \\ 17. Regular Perturbations / 129 \\ 18. Singular Perturbations / 138 \\ 19. Sturm--Liouville Problems / 145 \\ 20. Eigenfunction Expansions / 157 \\ 21. Eigenfunction Expansions (Cont'd.) / 163 \\ 22. Convergence of the Fourier Series / 171 \\ 23. Convergence of the Fourier Series (Cont'd.) / 176 \\ 24. Fourier Series Solutions of Ordinary Differential Equations / 187 \\ 25. Partial Differential Equations / 194 \\ 26. First-Order Partial Differential Equations / 202 \\ 27. Solvable Partial Differential Equations / 210 \\ 28. The Canonical Forms / 219 \\ 29. The Method of Separation of Variables / 227 \\ 30. The One-Dimensional Heat Equation / 234 \\ 31. The One-Dimensional Heat Equation (Cont'd.) / 241 \\ 32. The One-Dimensional Wave Equation / 249 \\ 33. The One-Dimensional Wave Equation (Cont'd.) / 256 \\ 34. Laplace Equation in Two Dimensions / 266 \\ 35. Laplace Equation in Polar Coordinates / 275 \\ 36. Two-Dimensional Heat Equation / 284 \\ 37. Two-Dimensional Wave Equation / 292 \\ 38. Laplace Equation in Three Dimensions / 300 \\ 39. Laplace Equation in Three Dimensions (Cont'd.) / 306 \\ 40. Nonhomogeneous Equations / 316 \\ 41. Fourier Integral and Transforms / 323 \\ 42. Fourier Integral and Transforms (Cont'd.) / 330 \\ 43. Fourier Transform Method for Partial DEs / 338 \\ 44. Fourier Transform Method for Partial DEs (Cont'd.) / 344 \\ 45. Laplace Transforms / 354 \\ 46. Laplace Transforms (Cont'd.) / 361 \\ 47. Laplace Transform Method for Ordinary DEs / 374 \\ 48. Laplace Transform Method for Partial DEs / 384 \\ 49. Well-Posed Problems / 394 \\ 50. Verification of Solutions / 399 \\ References for Further Reading / 405 \\ Index / 407

entry:
@Book{Agarwal:2008:OPD,
  author =       "Ravi P. Agarwal and Donal O'Regan",
  title =        "Ordinary and Partial Differential Equations: with
                 Special Functions, {Fourier} Series, and Boundary Value
                 Problems",
  publisher =    pub-SV,
  address =      pub-SV:adr,
  pages =        "xiv + 410",
  year =         "2008",
  ISBN =         "0-387-79145-0 (paperback)",
  ISBN-13 =      "978-0-387-79145-6 (paperback)",
  LCCN =         "????",
  bibdate =      "Sat Oct 30 17:22:04 MDT 2010",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 prodorbis.library.yale.edu:7090/voyager",
  series =       "Universitext",
  acknowledgement = ack-nhfb,
  subject =      "differential equations; differential equations,
                 partial; Fourier analysis; boundary value problems",
  tableofcontents = "Preface vii l. Solvable Differential Equations / 1
                 \\
                 2. Second-Order Differential Equations / 8 \\
                 3. Preliminaries to Series Solutions / 15 \\
                 4. Solution at an Ordinary Point / 23 \\
                 5. Solution at a Singular Point / 31 \\
                 6. Solution at a Singular Point (Cont'd.) / 37 \\
                 7. Legendre Polynomials and Functions / 47 \\
                 8. Chebyshev, Hermite and Laguerre Polynomials / 57 \\
                 9. Bessel Functions / 64 \\
                 10. Hypergeometrie Functions / 75 \\
                 11. Piecewise Continuous and Periodic Functions / 83
                 \\
                 12. Orthogonal Functions and Polynomials / 90 \\
                 13. Orthogonal Functions and Polynomials (Cont'd.) / 95
                 \\
                 14. Boundary Value Problems / 104 \\
                 15. Boundary Value Problems (Cont'd.) / 109 \\
                 16. Green's Functions / 119 \\
                 17. Regular Perturbations / 129 \\
                 18. Singular Perturbations / 138 \\
                 19. Sturm--Liouville Problems / 145 \\
                 20. Eigenfunction Expansions / 157 \\
                 21. Eigenfunction Expansions (Cont'd.) / 163 \\
                 22. Convergence of the Fourier Series / 171 \\
                 23. Convergence of the Fourier Series (Cont'd.) / 176
                 \\
                 24. Fourier Series Solutions of Ordinary Differential
                 Equations / 187 \\
                 25. Partial Differential Equations / 194 \\
                 26. First-Order Partial Differential Equations / 202
                 \\
                 27. Solvable Partial Differential Equations / 210 \\
                 28. The Canonical Forms / 219 \\
                 29. The Method of Separation of Variables / 227 \\
                 30. The One-Dimensional Heat Equation / 234 \\
                 31. The One-Dimensional Heat Equation (Cont'd.) / 241
                 \\
                 32. The One-Dimensional Wave Equation / 249 \\
                 33. The One-Dimensional Wave Equation (Cont'd.) / 256
                 \\
                 34. Laplace Equation in Two Dimensions / 266 \\
                 35. Laplace Equation in Polar Coordinates / 275 \\
                 36. Two-Dimensional Heat Equation / 284 \\
                 37. Two-Dimensional Wave Equation / 292 \\
                 38. Laplace Equation in Three Dimensions / 300 \\
                 39. Laplace Equation in Three Dimensions (Cont'd.) /
                 306 \\
                 40. Nonhomogeneous Equations / 316 \\
                 41. Fourier Integral and Transforms / 323 \\
                 42. Fourier Integral and Transforms (Cont'd.) / 330 \\
                 43. Fourier Transform Method for Partial DEs / 338 \\
                 44. Fourier Transform Method for Partial DEs (Cont'd.)
                 / 344 \\
                 45. Laplace Transforms / 354 \\
                 46. Laplace Transforms (Cont'd.) / 361 \\
                 47. Laplace Transform Method for Ordinary DEs / 374 \\
                 48. Laplace Transform Method for Partial DEs / 384 \\
                 49. Well-Posed Problems / 394 \\
                 50. Verification of Solutions / 399 \\
                 References for Further Reading / 405 \\
                 Index / 407",
}


bibtype: article
filename: elefunt.bib
label: Alzer:2008:GFI
author: Horst Alzer
title: Gamma function inequalities
journal: j-NUMER-ALGORITHMS
volume: 49
number: 1--4
pages: 53--84
month: dec
monthnumber: 12
year: 2008
CODEN: NUALEG
ISSN: 1017-1398 (print), 1572-9265 (electronic) OR 10171398 (print), 15729265 (electronic)
ISSNL: 1017-1398 OR 10171398
bibdate: Mon May 17 14:08:26 MDT 2010
bibtimestamp: 2010.05.17 14:08:26 MDT
URL: http://www.springerlink.com/openurl.asp?genre=article&issn=1017-1398&volume=49&issue=1&spage=53
fjournal: Numerical Algorithms
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/11075
rawauthor: Horst Alzer
rawtitle: Gamma function inequalities
bibsource:
http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1017-1398&volume=49&issue=1; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

entry:
@Article{Alzer:2008:GFI,
  author =       "Horst Alzer",
  title =        "Gamma function inequalities",
  journal =      j-NUMER-ALGORITHMS,
  volume =       "49",
  number =       "1--4",
  pages =        "53--84",
  month =        dec,
  year =         "2008",
  CODEN =        "NUALEG",
  ISSN =         "1017-1398 (print), 1572-9265 (electronic)",
  ISSN-L =       "1017-1398",
  bibdate =      "Mon May 17 14:08:26 MDT 2010",
  bibsource =    "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1017-1398&volume=49&issue=1;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.springerlink.com/openurl.asp?genre=article&issn=1017-1398&volume=49&issue=1&spage=53",
  acknowledgement = ack-nhfb,
  fjournal =     "Numerical Algorithms",
  journal-URL =  "http://link.springer.com/journal/11075",
}


bibtype: article
filename: elefunt.bib
label: Ancarani:2008:DOC
author: L. U. Ancarani and G. Gasaneo
journal: j-J-MATH-PHYS
volume: 49
number: 6
pages: 063508
month: jun
monthnumber: 6
year: 2008
CODEN: JMAPAQ
DOI: https://doi.org/10.1063/1.2939395
ISSNL: 0022-2488 OR 00222488
bibdate: Wed Oct 26 09:06:03 MDT 2011
bibtimestamp: 2011.10.26 09:06:03 MDT
URL: http://jmp.aip.org/resource/1/jmapaq/v49/i6/p063508_s1
fjournal: Journal of Mathematical Physics
acknowledgement: ack-nhfb
journalURL: http://jmp.aip.org/
onlinedate: 20 June 2008
rawauthor: L. U. Ancarani and G. Gasaneo
title:
Derivatives of any order of the confluent hypergeometric function $_1 F_1 (a, b, z)$ with respect to the parameter $a$ or $b$

bibsource:
http://www.aip.org/ojs/jmp.html; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jmathphys2005.bib

rawtitle:
Derivatives of any order of the confluent hypergeometric function {$_1 F_1 (a, b, z)$} with respect to the parameter $a$ or $b$

entry:
@Article{Ancarani:2008:DOC,
  author =       "L. U. Ancarani and G. Gasaneo",
  title =        "Derivatives of any order of the confluent
                 hypergeometric function {$_1 F_1 (a, b, z)$} with
                 respect to the parameter $a$ or $b$",
  journal =      j-J-MATH-PHYS,
  volume =       "49",
  number =       "6",
  pages =        "063508",
  month =        jun,
  year =         "2008",
  CODEN =        "JMAPAQ",
  DOI =          "https://doi.org/10.1063/1.2939395",
  ISSN =         "0022-2488 (print), 1089-7658 (electronic), 1527-2427",
  ISSN-L =       "0022-2488",
  bibdate =      "Wed Oct 26 09:06:03 MDT 2011",
  bibsource =    "http://www.aip.org/ojs/jmp.html;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jmathphys2005.bib",
  URL =          "http://jmp.aip.org/resource/1/jmapaq/v49/i6/p063508_s1",
  acknowledgement = ack-nhfb,
  fjournal =     "Journal of Mathematical Physics",
  journal-URL =  "http://jmp.aip.org/",
  onlinedate =   "20 June 2008",
  pagecount =    "16",
}


bibtype: article
filename: elefunt.bib
label: Borwein:2008:EBF
author: David Borwein and Jonathan M. Borwein and O-Yeat Chan
title: The evaluation of Bessel functions via exp--arc integrals
journal: j-J-MATH-ANAL-APPL
volume: 341
number: 1
pages: 478--500
month: may
monthnumber: 5
year: 2008
CODEN: JMANAK
DOI: https://doi.org/10.1016/j.jmaa.2007.10.003
ISSN: 0022-247X (print), 1096-0813 (electronic) OR 0022247X (print), 10960813 (electronic)
ISSNL: 0022-247X OR 0022247X
MRclass: 33C10 (33F05 65D20)
MRnumber: 2394100
MRreviewer: Richard B. Paris
bibdate: Thu Aug 11 10:27:38 2016
bibtimestamp: 2016.08.11 10:27:38 ???
fjournal: Journal of Mathematical Analysis and Applications
keywords: Bessel function; Exponential-hyperbolic expansions; Uniform series expansion
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/0022247X
rawauthor: David Borwein and Jonathan M. Borwein and O-Yeat Chan
rawtitle: The evaluation of {Bessel} functions via exp--arc integrals
ORCID: Borwein, Jonathan/0000-0002-1263-0646
bibsource:
https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

URL:
http://adsabs.harvard.edu/abs/2008JMAA..341..478B; http://docserver.carma.newcastle.edu.au/1231/; http://www.sciencedirect.com/science/article/pii/S0022247X07012346

abstract:
A standard method for computing values of Bessel functions has been to use the well-known ascending series for small argument, and to use an asymptotic series for large argument; with the choice of the series changing at some appropriate argument magnitude, depending on the number of digits required. In a recent paper, D. Borwein, J. Borwein, and R. Crandall [D. Borwein, J. M. Borwein, R. Crandall, Effective Laguerre asymptotics, preprint at http://locutus.cs.dal.ca:8088/archive/00000334/] derived a series for an ``exp-arc'' integral which gave rise to an absolutely convergent series for the J and I Bessel functions with integral order. Such series can be rapidly evaluated via recursion and elementary operations, and provide a viable alternative to the conventional ascending-asymptotic switching. In the present work, we extend the method to deal with Bessel functions of general (non-integral) order, as well as to deal with the Y and K Bessel functions.

rawabstract:
A standard method for computing values of Bessel functions has been to use the well-known ascending series for small argument, and to use an asymptotic series for large argument; with the choice of the series changing at some appropriate argument magnitude, depending on the number of digits required. In a recent paper, D. Borwein, J. Borwein, and R. Crandall [D. Borwein, J. M. Borwein, R. Crandall, Effective Laguerre asymptotics, preprint at http://locutus.cs.dal.ca:8088/archive/00000334/] derived a series for an ``exp-arc'' integral which gave rise to an absolutely convergent series for the J and I Bessel functions with integral order. Such series can be rapidly evaluated via recursion and elementary operations, and provide a viable alternative to the conventional ascending-asymptotic switching. In the present work, we extend the method to deal with Bessel functions of general (non-integral) order, as well as to deal with the Y and K Bessel functions.

entry:
@Article{Borwein:2008:EBF,
  author =       "David Borwein and Jonathan M. Borwein and O-Yeat
                 Chan",
  title =        "The evaluation of {Bessel} functions via exp--arc
                 integrals",
  journal =      j-J-MATH-ANAL-APPL,
  volume =       "341",
  number =       "1",
  pages =        "478--500",
  month =        may,
  year =         "2008",
  CODEN =        "JMANAK",
  DOI =          "https://doi.org/10.1016/j.jmaa.2007.10.003",
  ISSN =         "0022-247X (print), 1096-0813 (electronic)",
  ISSN-L =       "0022-247X",
  MRclass =      "33C10 (33F05 65D20)",
  MRnumber =     "2394100",
  MRreviewer =   "Richard B. Paris",
  bibdate =      "Thu Aug 11 10:27:38 2016",
  bibsource =    "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://adsabs.harvard.edu/abs/2008JMAA..341..478B;
                 http://docserver.carma.newcastle.edu.au/1231/;
                 http://www.sciencedirect.com/science/article/pii/S0022247X07012346",
  abstract =     "A standard method for computing values of Bessel
                 functions has been to use the well-known ascending
                 series for small argument, and to use an asymptotic
                 series for large argument; with the choice of the
                 series changing at some appropriate argument magnitude,
                 depending on the number of digits required. In a recent
                 paper, D. Borwein, J. Borwein, and R. Crandall [D.
                 Borwein, J. M. Borwein, R. Crandall, Effective Laguerre
                 asymptotics, preprint at
                 http://locutus.cs.dal.ca:8088/archive/00000334/]
                 derived a series for an ``exp-arc'' integral which gave
                 rise to an absolutely convergent series for the J and I
                 Bessel functions with integral order. Such series can
                 be rapidly evaluated via recursion and elementary
                 operations, and provide a viable alternative to the
                 conventional ascending-asymptotic switching. In the
                 present work, we extend the method to deal with Bessel
                 functions of general (non-integral) order, as well as
                 to deal with the Y and K Bessel functions.",
  acknowledgement = ack-nhfb,
  fjournal =     "Journal of Mathematical Analysis and Applications",
  journal-URL =  "http://www.sciencedirect.com/science/journal/0022247X",
  keywords =     "Bessel function; Exponential-hyperbolic expansions;
                 Uniform series expansion",
  ORCID-numbers = "Borwein, Jonathan/0000-0002-1263-0646",
}


bibtype: article
filename: elefunt.bib
label: Borwein:2008:ELA
author: David Borwein and Jonathan M. Borwein and Richard E. Crandall
title: Effective Laguerre asymptotics
journal: j-SIAM-J-NUMER-ANAL
volume: 46
number: 6
pages: 3285--3312
year: 2008
CODEN: SJNAAM
DOI: https://doi.org/10.1137/07068031X
ISSN: 0036-1429 (print), 1095-7170 (electronic) OR 00361429 (print), 10957170 (electronic)
ISSNL: 0036-1429 OR 00361429
MRclass: 33C65 (30E20 34E05)
MRnumber: 2448665
MRreviewer: Yu-Qiu Zhao
bibdate: Wed Aug 10 11:09:47 2016
bibtimestamp: 2016.08.10 11:09:47 ???
URL: http://docserver.carma.newcastle.edu.au/334/
fjournal: SIAM Journal on Numerical Analysis
acknowledgement: ack-nhfb
journalURL: http://epubs.siam.org/sinum
rawauthor: David Borwein and Jonathan M. Borwein and Richard E. Crandall
rawtitle: Effective {Laguerre} asymptotics
ORCID: Borwein, Jonathan/0000-0002-1263-0646
bibsource:
https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

abstract:
It is known that the generalized Laguerre polynomials can enjoy subexponential growth for large primary index. In particular, for certain fixed parameter pairs (a, z) one has the large-n asymptotic behavior L-n((-a)) (-z) similar to C(a, z)(n)(-a)/2-1/ (4)e(2) root nz. We introduce a computationally motivated contour integral that allows efficient numerical Laguerre evaluations yet also leads to the complete asymptotic series over the full parameter domain of subexponential behavior. We present a fast algorithm for symbolic generation of the rather formidable expansion coefficients. Along the way we address the difficult problem of establishing effective (i. e., rigorous and explicit) error bounds on the general expansion. A primary tool for these developments is an ``exp-arc'' method giving a natural bridge between converging series and effective asymptotics.

rawabstract:
It is known that the generalized Laguerre polynomials can enjoy subexponential growth for large primary index. In particular, for certain fixed parameter pairs (a, z) one has the large-n asymptotic behavior L-n((-a)) (-z) similar to C(a, z)(n)(-a)/2-1/ (4)e(2) root nz. We introduce a computationally motivated contour integral that allows efficient numerical Laguerre evaluations yet also leads to the complete asymptotic series over the full parameter domain of subexponential behavior. We present a fast algorithm for symbolic generation of the rather formidable expansion coefficients. Along the way we address the difficult problem of establishing effective (i. e., rigorous and explicit) error bounds on the general expansion. A primary tool for these developments is an ``exp-arc'' method giving a natural bridge between converging series and effective asymptotics.

entry:
@Article{Borwein:2008:ELA,
  author =       "David Borwein and Jonathan M. Borwein and Richard E.
                 Crandall",
  title =        "Effective {Laguerre} asymptotics",
  journal =      j-SIAM-J-NUMER-ANAL,
  volume =       "46",
  number =       "6",
  pages =        "3285--3312",
  year =         "2008",
  CODEN =        "SJNAAM",
  DOI =          "https://doi.org/10.1137/07068031X",
  ISSN =         "0036-1429 (print), 1095-7170 (electronic)",
  ISSN-L =       "0036-1429",
  MRclass =      "33C65 (30E20 34E05)",
  MRnumber =     "2448665",
  MRreviewer =   "Yu-Qiu Zhao",
  bibdate =      "Wed Aug 10 11:09:47 2016",
  bibsource =    "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://docserver.carma.newcastle.edu.au/334/",
  abstract =     "It is known that the generalized Laguerre polynomials
                 can enjoy subexponential growth for large primary
                 index. In particular, for certain fixed parameter pairs
                 (a, z) one has the large-n asymptotic behavior
                 L-n((-a)) (-z) similar to C(a, z)(n)(-a)/2-1/ (4)e(2)
                 root nz. We introduce a computationally motivated
                 contour integral that allows efficient numerical
                 Laguerre evaluations yet also leads to the complete
                 asymptotic series over the full parameter domain of
                 subexponential behavior. We present a fast algorithm
                 for symbolic generation of the rather formidable
                 expansion coefficients. Along the way we address the
                 difficult problem of establishing effective (i. e.,
                 rigorous and explicit) error bounds on the general
                 expansion. A primary tool for these developments is an
                 ``exp-arc'' method giving a natural bridge between
                 converging series and effective asymptotics.",
  acknowledgement = ack-nhfb,
  fjournal =     "SIAM Journal on Numerical Analysis",
  journal-URL =  "http://epubs.siam.org/sinum",
  ORCID-numbers = "Borwein, Jonathan/0000-0002-1263-0646",
  researcherid-numbers = "Borwein, Jonathan/A-6082-2009",
  unique-id =    "Borwein:2008:ELA",
}


bibtype: inproceedings
filename: elefunt.bib
label: Brisebarre:2008:EME
title: An Efficient Method for Evaluating Polynomial and Rational Function Approximations
crossref: IEEE:2008:ICA
pages: 233--238
year: 2008
DOI: https://doi.org/10.1109/ASAP.2008.4580185
bibdate: Mon Feb 10 07:28:25 2020
bibtimestamp: 2020.02.10 07:28:25 ???
acknowledgement: ack-nhfb
rawtitle: An Efficient Method for Evaluating Polynomial and Rational Function Approximations
author:
Nicolas Brisebarre and Sylvain Chevillard and Milos D. Ercegovac and Jean-Michel Muller and Serge Torres

rawauthor:
Nicolas Brisebarre and Sylvain Chevillard and Milo{\v{s}} D. Ercegovac and Jean-Michel Muller and Serge Torres

entry:
@InProceedings{Brisebarre:2008:EME,
  author =       "Nicolas Brisebarre and Sylvain Chevillard and
                 Milo{\v{s}} D. Ercegovac and Jean-Michel Muller and
                 Serge Torres",
  title =        "An Efficient Method for Evaluating Polynomial and
                 Rational Function Approximations",
  crossref =     "IEEE:2008:ICA",
  pages =        "233--238",
  year =         "2008",
  DOI =          "https://doi.org/10.1109/ASAP.2008.4580185",
  bibdate =      "Mon Feb 10 07:28:25 2020",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: elefunt.bib
label: Chatterjee:2008:CNT
author: S. Chatterjee and D. Roy
title: A class of new transforms tailored for the hypergeometric series
journal: j-COMP-PHYS-COMM
volume: 179
number: 8
pages: 555--561
day: 15
month: oct
monthnumber: 10
year: 2008
CODEN: CPHCBZ
DOI: https://doi.org/10.1016/j.cpc.2008.05.001
ISSN: 0010-4655 (print), 1879-2944 (electronic) OR 00104655 (print), 18792944 (electronic)
ISSNL: 0010-4655 OR 00104655
bibdate: Thu Dec 01 09:09:57 2011
bibtimestamp: 2011.12.01 09:09:57 ???
fjournal: Computer Physics Communications
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00104655
rawauthor: S. Chatterjee and D. Roy
rawtitle: A class of new transforms tailored for the hypergeometric series
bibsource:
https://www.math.utah.edu/pub/tex/bib/compphyscomm2000.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

entry:
@Article{Chatterjee:2008:CNT,
  author =       "S. Chatterjee and D. Roy",
  title =        "A class of new transforms tailored for the
                 hypergeometric series",
  journal =      j-COMP-PHYS-COMM,
  volume =       "179",
  number =       "8",
  pages =        "555--561",
  day =          "15",
  month =        oct,
  year =         "2008",
  CODEN =        "CPHCBZ",
  DOI =          "https://doi.org/10.1016/j.cpc.2008.05.001",
  ISSN =         "0010-4655 (print), 1879-2944 (electronic)",
  ISSN-L =       "0010-4655",
  bibdate =      "Thu Dec 01 09:09:57 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/compphyscomm2000.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "Computer Physics Communications",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00104655",
  xxauthor =     "S. Charterjee and D. Roy",
}


bibtype: book
filename: elefunt.bib
label: Cuyt:2008:HCF
author: Annie Cuyt and Vigdis B. Petersen and Brigitte Verdonk and Haakon Waadeland and William B. Jones
title: Handbook of Continued Fractions for Special Functions
publisher: pub-SV
address: pub-SV:adr
pages: xx + 440
year: 2008
DOI: https://doi.org/10.1007/978-1-4020-6949-9
ISBN: 1-4020-6948-0 OR 1402069480
ISBN13: 978-1-4020-6948-2 OR 9781402069482
LCCN: QA295 .H275 2008
bibdate: Tue Jun 24 07:17:37 MDT 2008
bibtimestamp: 2008.06.24 07:17:37 MDT
acknowledgement: ack-nhfb
rawauthor: Annie Cuyt and Vigdis B. Petersen and Brigitte Verdonk and Haakon Waadeland and William B. Jones
rawtitle: Handbook of Continued Fractions for Special Functions
keywords:
applying the limit process; associated continued fraction; asymptotic series expansion; basic hypergeometric functions; canonical contraction; combination with property; complementary incomplete gamma function; complex error function; confluent hypergeometric series; continued fraction converges; continued fraction representations; fraction approximants; modified approximant; monic orthogonal polynomial sequence; normed field; nth approximant; nth denominator; nth numerator; nth tail; oval sequence theorem; parabola theorem; partial numerators; strong moment distribution function; successive approximants; truncation error bounds

TOC:
Preface / xi Notation / xiii 0 General considerations / 1 0.1 Part one / 1 0.2 Part two / 2 0.3 Part three / 2 Part I: Basic Theory 1 Basics / 9 1.1 Symbols and notation 1.2 Definitions / 10 1.3 Recurrence relations / 13 1.4 Equivalence transformations / 15 1.5 Contractions and extensions / 16 1.6 Continued fractions with prescribed approximants / 18 1.7 Connection between continued fractions and series / 19 1.8 Periodic and limit periodic continued fractions / 21 1.9 Tails of continued fractions / 23 1.10 Continued fractions over normed fields / 26 1.11 Generalisations of continued fractions / 28 2 Continued fraction representation of functions / 29 2.1 Symbols and notation / 29 2.2 Correspondence / 30 2.3 Families of continued fractions / 35 2.4 Correspondence of C-fractions / 39 2.5 Correspondence of P-fractions / 40 2.6 Correspondence of J-fractions and T-fractions / 41 2.7 Correspondence and three-term recurrences / 42 3 Convergence criteria / 45 3.1 Some classical theorems / 45 3.2 Convergence sets and value sets / 47 3.3 Parabola and oval theorems / 49 3.4 Correspondence and uniform convergence / 52 3.5 Periodic and limit periodic continued fractions / 53 3.6 Convergence and minimal solutions / 56 4 Pade approximants / 59 4.1 Definition and notation / 59 4.2 Fundamental properties / 60 4.3 Connection with regular C-fractions / 64 4.4 Connection with P-fractions / 65 4.5 Extension of the Pade table / 67 4.6 Connection with M-fractions and the M-table / 68 4.7 Convergence of Pade approximants / 70 4.8 Formal orthogonality property / 72 5 Moment theory and orthogonal functions / 77 5.1 Moment theory / 77 5.2 Stieltjes transforms / 85 5.3 Construction of solutions / 90 5.4 Orthogonal polynomials / 91 5.5 Monic orthogonal polynomials on $mathbb R$ and J-fractions / 92 5.6 Szego polynomials and PPC-fractions / 100 5.7 Orthogonal Laurent polynomials and APT-fractions / 102 Part II: Numerics 6 Continued fraction construction / 107 6.1 Regular C-fractions / 107 6.2 C-fractions / 113 6.3 S-fractions / 114 6.4 P-fractions / 114 6.5 J-fractions / 120 6.6 M-fractions / 122 6.7 Positive T-fractions / 124 6.8 Thiele fractions / 125 7 Truncation error bounds / 129 7.1 Parabola theorems / 129 7.2 The oval sequence theorem / 131 7.3 The interval sequence theorem / 136 7.4 Specific a priori bounds for S-fractions / 138 7.5 A posteriori truncation error bounds / 140 7.6 Tails and truncation error bounds / 143 7.7 Choice of modification / 143 8 Continued fraction evaluation / 149 8.1 The effect of finite precision arithmetic / 149 8.2 Evaluation of approximants / 152 8.3 The forward recurrence and minimal solutions / 154 8.4 Round-off error in the backward recurrence / 156 Part III: Special Functions 9 On tables and graphs / 163 9.1 Introduction / 163 9.2 Comparative tables / 163 9.3 Reliable graphs / 168 10 Mathematical constants / 175 10.1 Regular continued fractions / 175 10.2 Archimedes' constant, symbol $pi $ / 176 10.3 Euler's number, base of the natural logarithm / 178 10.4 Integer powers and roots of $pi $ and $e$ / 180 10.5 The natural logarithm, $ln (2)$ / 181 10.6 Pythagoras' constant, the square root of two / 183 10.7 The cube root of two / 183 10.8 Euler's constant, symbol $gamma $ / 185 10.9 Golden ratio, symbol $phi $ / 185 10.10 The rabbit constant, symbol $rho $ / 186 10.11 Apery's constant, $zeta (3)$ / 188 10.12 Catalan's constant, symbol $C$ / 189 10.13 Gompertz' constant, symbol $G$ / 190 10.14 Khinchin's constant, symbol $K$ / 190 11 Elementary functions / 193 11.1 The exponential function / 193 11.2 The natural logarithm / 196 11.3 Trigonometric functions / 200 11.4 Inverse trigonometric functions / 204 11.5 Hyperbolic functions / 210 11.6 Inverse hyperbolic functions / 213 11.7 The power function / 217 12 Gamma function and related functions / 221 12.1 Gamma function / 221 12.2 Binet function / 224 12.3 Polygamma functions / 229 12.4 Trigamma function / 232 12.5 Tetragamma function / 235 12.6 Incomplete gamma functions / 238 13 Error function and related integrals / 253 13.1 Error function and Dawson's integral / 253 13.2 Complementary and complex error function / 261 13.3 Repeated integrals / 268 13.4 Fresnel integrals / 269 14 Exponential integrals and related functions / 275 14.7 Exponential integrals / 275 14.2 Related functions / 285 15 Hypergeometric functions / 291 15.1 Definition and basic properties / 291 15.2 Stieltjes transform / 295 15.3 Continued fraction representations / 295 15.4 Pade approximants / 309 15.5 Monotonicity properties / 313 15.6 Hypergeometric series $_pF_q$ / 315 16 Confluent hypergeometric functions / 319 16.1 Kummer functions / 319 16.2 Confluent hypergeometric series $_2F_0$ / 330 16.3 Confluent hypergeometric limit function / 333 16.4 Whittaker functions / 334 16.5 Parabolic cylinder functions / 337 17 Bessel functions / 334 17.7 Bessel functions / 334 17.2 Modified Bessel functions / 356 18 Probability functions / 371 18.1 Definitions and elementary properties / 371 18.2 Normal and log-normal distributions / 373 18.3 Repeated integrals / 377 18.4 Gamma and chi-square distribution / 378 18.5 Beta, $F$- and Student's $t$-distributions / 382 19 Basic hypergeometric functions / 391 19.1 Definition and basic properties / 391 19.2 Continued fraction representations / 395 19.3 Higher order basic hypergeometric functions / 399 Bibliography / 401 Index / 421

shortTOC:
General considerations Part 1, Basic Theory 1. Basics 2. Continued fraction representation of functions 3. Convergence criteria 4. Pade approximants 5. Moment theory and orthogonal functions Part 2, Numerics 6. Continued fraction construction 7. Truncation error bounds 8. Continued fraction evaluation Part 3, Special Functions 9. On tables and graphs 10. Mathematical constants 11. Elementary functions 12. Gamma function and related functions 13. Error function and related integrals 14. Exponential integrals and related functions 15. Hypergeometric functions 16. Confluent hypergeometric functions, 17. Bessel functions 18. Probability functions 19. Basic hypergeometric functions

rawshortTOC:
General considerations \\ Part 1, Basic Theory \\ 1. Basics \\ 2. Continued fraction representation of functions \\ 3. Convergence criteria \\ 4. Pade approximants \\ 5. Moment theory and orthogonal functions \\ Part 2, Numerics \\ 6. Continued fraction construction \\ 7. Truncation error bounds \\ 8. Continued fraction evaluation \\ Part 3, Special Functions \\ 9. On tables and graphs \\ 10. Mathematical constants \\ 11. Elementary functions \\ 12. Gamma function and related functions \\ 13. Error function and related integrals \\ 14. Exponential integrals and related functions \\ 15. Hypergeometric functions \\ 16. Confluent hypergeometric functions, \\ 17. Bessel functions \\ 18. Probability functions \\ 19. Basic hypergeometric functions

rawTOC:
Preface / xi \\ Notation / xiii \\ 0 General considerations \\ / 1 \\ 0.1 Part one / 1 \\ 0.2 Part two / 2 \\ 0.3 Part three / 2 \\ \\ Part I: Basic Theory \\ \\ 1 Basics / 9 \\ 1.1 Symbols and notation \\ 1.2 Definitions / 10 \\ 1.3 Recurrence relations / 13 \\ 1.4 Equivalence transformations / 15 \\ 1.5 Contractions and extensions / 16 \\ 1.6 Continued fractions with prescribed approximants / 18 \\ 1.7 Connection between continued fractions and series / 19 \\ 1.8 Periodic and limit periodic continued fractions / 21 \\ 1.9 Tails of continued fractions / 23 \\ 1.10 Continued fractions over normed fields / 26 \\ 1.11 Generalisations of continued fractions / 28 \\ \\ 2 Continued fraction representation of functions / 29 \\ 2.1 Symbols and notation / 29 \\ 2.2 Correspondence / 30 \\ 2.3 Families of continued fractions / 35 \\ 2.4 Correspondence of C-fractions / 39 \\ 2.5 Correspondence of P-fractions / 40 \\ 2.6 Correspondence of J-fractions and T-fractions / 41 \\ 2.7 Correspondence and three-term recurrences / 42 \\ \\ 3 Convergence criteria / 45 \\ 3.1 Some classical theorems / 45 \\ 3.2 Convergence sets and value sets / 47 \\ 3.3 Parabola and oval theorems / 49 \\ 3.4 Correspondence and uniform convergence / 52 \\ 3.5 Periodic and limit periodic continued fractions / 53 \\ 3.6 Convergence and minimal solutions / 56 \\ \\ 4 Pad{\'e} approximants / 59 \\ 4.1 Definition and notation / 59 \\ 4.2 Fundamental properties / 60 \\ 4.3 Connection with regular C-fractions / 64 \\ 4.4 Connection with P-fractions / 65 \\ 4.5 Extension of the Pad{\'e} table / 67 \\ 4.6 Connection with M-fractions and the M-table / 68 \\ 4.7 Convergence of Pad{\'e} approximants / 70 \\ 4.8 Formal orthogonality property / 72 \\ \\ 5 Moment theory and orthogonal functions / 77 \\ 5.1 Moment theory / 77 \\ 5.2 Stieltjes transforms / 85 \\ 5.3 Construction of solutions / 90 \\ 5.4 Orthogonal polynomials / 91 \\ 5.5 Monic orthogonal polynomials on $\mathbb{R}$ and J-fractions / 92 \\ 5.6 Szeg{\H{o}} polynomials and PPC-fractions / 100 \\ 5.7 Orthogonal Laurent polynomials and APT-fractions / 102 \\ \\ Part II: Numerics \\ \\ 6 Continued fraction construction / 107 \\ 6.1 Regular C-fractions / 107 \\ 6.2 C-fractions / 113 \\ 6.3 S-fractions / 114 \\ 6.4 P-fractions / 114 \\ 6.5 J-fractions / 120 \\ 6.6 M-fractions / 122 \\ 6.7 Positive T-fractions / 124 \\ 6.8 Thiele fractions / 125 \\ \\ 7 Truncation error bounds / 129 \\ 7.1 Parabola theorems / 129 \\ 7.2 The oval sequence theorem / 131 \\ 7.3 The interval sequence theorem / 136 \\ 7.4 Specific a priori bounds for S-fractions / 138 \\ 7.5 A posteriori truncation error bounds / 140 \\ 7.6 Tails and truncation error bounds / 143 \\ 7.7 Choice of modification / 143 \\ \\ 8 Continued fraction evaluation / 149 \\ 8.1 The effect of finite precision arithmetic / 149 \\ 8.2 Evaluation of approximants / 152 \\ 8.3 The forward recurrence and minimal solutions / 154 \\ 8.4 Round-off error in the backward recurrence / 156 \\ \\ Part III: Special Functions \\ \\ 9 On tables and graphs / 163 \\ 9.1 Introduction / 163 \\ 9.2 Comparative tables / 163 \\ 9.3 Reliable graphs / 168 \\ \\ 10 Mathematical constants / 175 \\ 10.1 Regular continued fractions / 175 \\ 10.2 Archimedes' constant, symbol $\pi$ / 176 \\ 10.3 Euler's number, base of the natural logarithm / 178 \\ 10.4 Integer powers and roots of $\pi$ and $e$ / 180 \\ 10.5 The natural logarithm, $\ln(2)$ / 181 \\ 10.6 Pythagoras' constant, the square root of two / 183 \\ 10.7 The cube root of two / 183 \\ 10.8 Euler's constant, symbol $\gamma$ / 185 \\ 10.9 Golden ratio, symbol $\phi$ / 185 \\ 10.10 The rabbit constant, symbol $\rho$ / 186 \\ 10.11 Ap{\'e}ry's constant, $\zeta(3)$ / 188 \\ 10.12 Catalan's constant, symbol $C$ / 189 \\ 10.13 Gompertz' constant, symbol $G$ / 190 \\ 10.14 Khinchin's constant, symbol $K$ / 190 \\ \\ 11 Elementary functions / 193 \\ 11.1 The exponential function / 193 \\ 11.2 The natural logarithm / 196 \\ 11.3 Trigonometric functions / 200 \\ 11.4 Inverse trigonometric functions / 204 \\ 11.5 Hyperbolic functions / 210 \\ 11.6 Inverse hyperbolic functions / 213 \\ 11.7 The power function / 217 \\ \\ 12 Gamma function and related functions / 221 \\ 12.1 Gamma function / 221 \\ 12.2 Binet function / 224 \\ 12.3 Polygamma functions / 229 \\ 12.4 Trigamma function / 232 \\ 12.5 Tetragamma function / 235 \\ 12.6 Incomplete gamma functions / 238 \\ \\ 13 Error function and related integrals / 253 \\ 13.1 Error function and Dawson's integral / 253 \\ 13.2 Complementary and complex error function / 261 \\ 13.3 Repeated integrals / 268 \\ 13.4 Fresnel integrals / 269 \\ \\ 14 Exponential integrals and related functions / 275 \\ 14.7 Exponential integrals / 275 \\ 14.2 Related functions / 285 \\ \\ 15 Hypergeometric functions / 291 \\ 15.1 Definition and basic properties / 291 \\ 15.2 Stieltjes transform / 295 \\ 15.3 Continued fraction representations / 295 \\ 15.4 Pad{\'e} approximants / 309 \\ 15.5 Monotonicity properties / 313 \\ 15.6 Hypergeometric series $_pF_q$ / 315 \\ \\ 16 Confluent hypergeometric functions / 319 \\ 16.1 Kummer functions / 319 \\ 16.2 Confluent hypergeometric series $_2F_0$ / 330 \\ 16.3 Confluent hypergeometric limit function / 333 \\ 16.4 Whittaker functions / 334 \\ 16.5 Parabolic cylinder functions / 337 \\ \\ 17 Bessel functions / 334 \\ 17.7 Bessel functions / 334 \\ 17.2 Modified Bessel functions / 356 \\ \\ 18 Probability functions / 371 \\ 18.1 Definitions and elementary properties / 371 \\ 18.2 Normal and log-normal distributions / 373 \\ 18.3 Repeated integrals / 377 \\ 18.4 Gamma and chi-square distribution / 378 \\ 18.5 Beta, $F$- and Student's $t$-distributions / 382 \\ \\ 19 Basic hypergeometric functions / 391 \\ 19.1 Definition and basic properties / 391 \\ 19.2 Continued fraction representations / 395 \\ 19.3 Higher order basic hypergeometric functions / 399 \\ \\ Bibliography / 401 \\ \\ Index / 421

entry:
@Book{Cuyt:2008:HCF,
  author =       "Annie Cuyt and Vigdis B. Petersen and Brigitte Verdonk
                 and Haakon Waadeland and William B. Jones",
  title =        "Handbook of Continued Fractions for Special
                 Functions",
  publisher =    pub-SV,
  address =      pub-SV:adr,
  pages =        "xx + 440",
  year =         "2008",
  DOI =          "https://doi.org/10.1007/978-1-4020-6949-9",
  ISBN =         "1-4020-6948-0",
  ISBN-13 =      "978-1-4020-6948-2",
  LCCN =         "QA295 .H275 2008",
  bibdate =      "Tue Jun 24 07:17:37 MDT 2008",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/master.bib;
                 z3950.loc.gov:7090/Voyager",
  acknowledgement = ack-nhfb,
  keywords =     "applying the limit process; associated continued
                 fraction; asymptotic series expansion; basic
                 hypergeometric functions; canonical contraction;
                 combination with property; complementary incomplete
                 gamma function; complex error function; confluent
                 hypergeometric series; continued fraction converges;
                 continued fraction representations; fraction
                 approximants; modified approximant; monic orthogonal
                 polynomial sequence; normed field; nth approximant; nth
                 denominator; nth numerator; nth tail; oval sequence
                 theorem; parabola theorem; partial numerators; strong
                 moment distribution function; successive approximants;
                 truncation error bounds",
  shorttableofcontents = "General considerations \\
                 Part 1, Basic Theory \\
                 1. Basics \\
                 2. Continued fraction representation of functions \\
                 3. Convergence criteria \\
                 4. Pade approximants \\
                 5. Moment theory and orthogonal functions \\
                 Part 2, Numerics \\
                 6. Continued fraction construction \\
                 7. Truncation error bounds \\
                 8. Continued fraction evaluation \\
                 Part 3, Special Functions \\
                 9. On tables and graphs \\
                 10. Mathematical constants \\
                 11. Elementary functions \\
                 12. Gamma function and related functions \\
                 13. Error function and related integrals \\
                 14. Exponential integrals and related functions \\
                 15. Hypergeometric functions \\
                 16. Confluent hypergeometric functions, \\
                 17. Bessel functions \\
                 18. Probability functions \\
                 19. Basic hypergeometric functions",
  tableofcontents = "Preface / xi \\
                 Notation / xiii \\
                 0 General considerations \\
                 / 1 \\
                 0.1 Part one / 1 \\
                 0.2 Part two / 2 \\
                 0.3 Part three / 2 \\
                 \\
                 Part I: Basic Theory \\
                 \\
                 1 Basics / 9 \\
                 1.1 Symbols and notation \\
                 1.2 Definitions / 10 \\
                 1.3 Recurrence relations / 13 \\
                 1.4 Equivalence transformations / 15 \\
                 1.5 Contractions and extensions / 16 \\
                 1.6 Continued fractions with prescribed approximants /
                 18 \\
                 1.7 Connection between continued fractions and series /
                 19 \\
                 1.8 Periodic and limit periodic continued fractions /
                 21 \\
                 1.9 Tails of continued fractions / 23 \\
                 1.10 Continued fractions over normed fields / 26 \\
                 1.11 Generalisations of continued fractions / 28 \\
                 \\
                 2 Continued fraction representation of functions / 29
                 \\
                 2.1 Symbols and notation / 29 \\
                 2.2 Correspondence / 30 \\
                 2.3 Families of continued fractions / 35 \\
                 2.4 Correspondence of C-fractions / 39 \\
                 2.5 Correspondence of P-fractions / 40 \\
                 2.6 Correspondence of J-fractions and T-fractions / 41
                 \\
                 2.7 Correspondence and three-term recurrences / 42 \\
                 \\
                 3 Convergence criteria / 45 \\
                 3.1 Some classical theorems / 45 \\
                 3.2 Convergence sets and value sets / 47 \\
                 3.3 Parabola and oval theorems / 49 \\
                 3.4 Correspondence and uniform convergence / 52 \\
                 3.5 Periodic and limit periodic continued fractions /
                 53 \\
                 3.6 Convergence and minimal solutions / 56 \\
                 \\
                 4 Pad{\'e} approximants / 59 \\
                 4.1 Definition and notation / 59 \\
                 4.2 Fundamental properties / 60 \\
                 4.3 Connection with regular C-fractions / 64 \\
                 4.4 Connection with P-fractions / 65 \\
                 4.5 Extension of the Pad{\'e} table / 67 \\
                 4.6 Connection with M-fractions and the M-table / 68
                 \\
                 4.7 Convergence of Pad{\'e} approximants / 70 \\
                 4.8 Formal orthogonality property / 72 \\
                 \\
                 5 Moment theory and orthogonal functions / 77 \\
                 5.1 Moment theory / 77 \\
                 5.2 Stieltjes transforms / 85 \\
                 5.3 Construction of solutions / 90 \\
                 5.4 Orthogonal polynomials / 91 \\
                 5.5 Monic orthogonal polynomials on $\mathbb{R}$ and
                 J-fractions / 92 \\
                 5.6 Szeg{\H{o}} polynomials and PPC-fractions / 100 \\
                 5.7 Orthogonal Laurent polynomials and APT-fractions /
                 102 \\
                 \\
                 Part II: Numerics \\
                 \\
                 6 Continued fraction construction / 107 \\
                 6.1 Regular C-fractions / 107 \\
                 6.2 C-fractions / 113 \\
                 6.3 S-fractions / 114 \\
                 6.4 P-fractions / 114 \\
                 6.5 J-fractions / 120 \\
                 6.6 M-fractions / 122 \\
                 6.7 Positive T-fractions / 124 \\
                 6.8 Thiele fractions / 125 \\
                 \\
                 7 Truncation error bounds / 129 \\
                 7.1 Parabola theorems / 129 \\
                 7.2 The oval sequence theorem / 131 \\
                 7.3 The interval sequence theorem / 136 \\
                 7.4 Specific a priori bounds for S-fractions / 138 \\
                 7.5 A posteriori truncation error bounds / 140 \\
                 7.6 Tails and truncation error bounds / 143 \\
                 7.7 Choice of modification / 143 \\
                 \\
                 8 Continued fraction evaluation / 149 \\
                 8.1 The effect of finite precision arithmetic / 149 \\
                 8.2 Evaluation of approximants / 152 \\
                 8.3 The forward recurrence and minimal solutions / 154
                 \\
                 8.4 Round-off error in the backward recurrence / 156
                 \\
                 \\
                 Part III: Special Functions \\
                 \\
                 9 On tables and graphs / 163 \\
                 9.1 Introduction / 163 \\
                 9.2 Comparative tables / 163 \\
                 9.3 Reliable graphs / 168 \\
                 \\
                 10 Mathematical constants / 175 \\
                 10.1 Regular continued fractions / 175 \\
                 10.2 Archimedes' constant, symbol $\pi$ / 176 \\
                 10.3 Euler's number, base of the natural logarithm /
                 178 \\
                 10.4 Integer powers and roots of $\pi$ and $e$ / 180
                 \\
                 10.5 The natural logarithm, $\ln(2)$ / 181 \\
                 10.6 Pythagoras' constant, the square root of two / 183
                 \\
                 10.7 The cube root of two / 183 \\
                 10.8 Euler's constant, symbol $\gamma$ / 185 \\
                 10.9 Golden ratio, symbol $\phi$ / 185 \\
                 10.10 The rabbit constant, symbol $\rho$ / 186 \\
                 10.11 Ap{\'e}ry's constant, $\zeta(3)$ / 188 \\
                 10.12 Catalan's constant, symbol $C$ / 189 \\
                 10.13 Gompertz' constant, symbol $G$ / 190 \\
                 10.14 Khinchin's constant, symbol $K$ / 190 \\
                 \\
                 11 Elementary functions / 193 \\
                 11.1 The exponential function / 193 \\
                 11.2 The natural logarithm / 196 \\
                 11.3 Trigonometric functions / 200 \\
                 11.4 Inverse trigonometric functions / 204 \\
                 11.5 Hyperbolic functions / 210 \\
                 11.6 Inverse hyperbolic functions / 213 \\
                 11.7 The power function / 217 \\
                 \\
                 12 Gamma function and related functions / 221 \\
                 12.1 Gamma function / 221 \\
                 12.2 Binet function / 224 \\
                 12.3 Polygamma functions / 229 \\
                 12.4 Trigamma function / 232 \\
                 12.5 Tetragamma function / 235 \\
                 12.6 Incomplete gamma functions / 238 \\
                 \\
                 13 Error function and related integrals / 253 \\
                 13.1 Error function and Dawson's integral / 253 \\
                 13.2 Complementary and complex error function / 261 \\
                 13.3 Repeated integrals / 268 \\
                 13.4 Fresnel integrals / 269 \\
                 \\
                 14 Exponential integrals and related functions / 275
                 \\
                 14.7 Exponential integrals / 275 \\
                 14.2 Related functions / 285 \\
                 \\
                 15 Hypergeometric functions / 291 \\
                 15.1 Definition and basic properties / 291 \\
                 15.2 Stieltjes transform / 295 \\
                 15.3 Continued fraction representations / 295 \\
                 15.4 Pad{\'e} approximants / 309 \\
                 15.5 Monotonicity properties / 313 \\
                 15.6 Hypergeometric series $_pF_q$ / 315 \\
                 \\
                 16 Confluent hypergeometric functions / 319 \\
                 16.1 Kummer functions / 319 \\
                 16.2 Confluent hypergeometric series $_2F_0$ / 330 \\
                 16.3 Confluent hypergeometric limit function / 333 \\
                 16.4 Whittaker functions / 334 \\
                 16.5 Parabolic cylinder functions / 337 \\
                 \\
                 17 Bessel functions / 334 \\
                 17.7 Bessel functions / 334 \\
                 17.2 Modified Bessel functions / 356 \\
                 \\
                 18 Probability functions / 371 \\
                 18.1 Definitions and elementary properties / 371 \\
                 18.2 Normal and log-normal distributions / 373 \\
                 18.3 Repeated integrals / 377 \\
                 18.4 Gamma and chi-square distribution / 378 \\
                 18.5 Beta, $F$- and Student's $t$-distributions / 382
                 \\
                 \\
                 19 Basic hypergeometric functions / 391 \\
                 19.1 Definition and basic properties / 391 \\
                 19.2 Continued fraction representations / 395 \\
                 19.3 Higher order basic hypergeometric functions / 399
                 \\
                 \\
                 Bibliography / 401 \\
                 \\
                 Index / 421",
}


bibtype: article
filename: elefunt.bib
label: Dyer:2008:CCI
author: J. S. Dyer and S. A. Dyer
title: Corrections to, and comments on, ``An improved approximation for the Gaussian $Q$-Function''
journal: j-IEEE-COMMUN-LET
volume: 12
number: 4
pages: 231--231
month: apr
monthnumber: 4
year: 2008
CODEN: ICLEF6
DOI: https://doi.org/10.1109/lcomm.2008.080009
ISSN: 1089-7798 (print), 1558-2558 (electronic) OR 10897798 (print), 15582558 (electronic)
ISSNL: 1089-7798 OR 10897798
bibdate: Sat Dec 16 18:08:34 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2017.12.16 18:08:34 ???
note: See cite Karagiannidis:2007:IAG.
URL: https://ieeexplore.ieee.org/document/4489650/
fjournal: IEEE Communications Letters
acknowledgement: ack-nhfb
journalURL: https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=4234
rawauthor: J. S. Dyer and S. A. Dyer
rawnote: See \cite{Karagiannidis:2007:IAG}.
rawtitle: Corrections to, and comments on, {``An improved approximation for the Gaussian $Q$-Function''}
entry:
@Article{Dyer:2008:CCI,
  author =       "J. S. Dyer and S. A. Dyer",
  title =        "Corrections to, and comments on, {``An improved
                 approximation for the Gaussian $Q$-Function''}",
  journal =      j-IEEE-COMMUN-LET,
  volume =       "12",
  number =       "4",
  pages =        "231--231",
  month =        apr,
  year =         "2008",
  CODEN =        "ICLEF6",
  DOI =          "https://doi.org/10.1109/lcomm.2008.080009",
  ISSN =         "1089-7798 (print), 1558-2558 (electronic)",
  ISSN-L =       "1089-7798",
  bibdate =      "Sat Dec 16 18:08:34 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  note =         "See \cite{Karagiannidis:2007:IAG}.",
  URL =          "https://ieeexplore.ieee.org/document/4489650/",
  acknowledgement = ack-nhfb,
  fjournal =     "IEEE Communications Letters",
  journal-URL =  "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=4234",
}


bibtype: article
filename: elefunt.bib
label: Elbert:2008:ZCE
author: Arpad Elbert and Andrea Laforgia
title: The zeros of the complementary error function
journal: j-NUMER-ALGORITHMS
volume: 49
number: 1--4
pages: 153--157
month: dec
monthnumber: 12
year: 2008
CODEN: NUALEG
ISSN: 1017-1398 (print), 1572-9265 (electronic) OR 10171398 (print), 15729265 (electronic)
ISSNL: 1017-1398 OR 10171398
bibdate: Mon May 17 14:08:26 MDT 2010
bibtimestamp: 2010.05.17 14:08:26 MDT
URL: http://www.springerlink.com/openurl.asp?genre=article&issn=1017-1398&volume=49&issue=1&spage=153
fjournal: Numerical Algorithms
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/11075
rawauthor: {\'A}rp{\'a}d Elbert and Andrea Laforgia
rawtitle: The zeros of the complementary error function
bibsource:
http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1017-1398&volume=49&issue=1; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

entry:
@Article{Elbert:2008:ZCE,
  author =       "{\'A}rp{\'a}d Elbert and Andrea Laforgia",
  title =        "The zeros of the complementary error function",
  journal =      j-NUMER-ALGORITHMS,
  volume =       "49",
  number =       "1--4",
  pages =        "153--157",
  month =        dec,
  year =         "2008",
  CODEN =        "NUALEG",
  ISSN =         "1017-1398 (print), 1572-9265 (electronic)",
  ISSN-L =       "1017-1398",
  bibdate =      "Mon May 17 14:08:26 MDT 2010",
  bibsource =    "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1017-1398&volume=49&issue=1;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.springerlink.com/openurl.asp?genre=article&issn=1017-1398&volume=49&issue=1&spage=153",
  acknowledgement = ack-nhfb,
  fjournal =     "Numerical Algorithms",
  journal-URL =  "http://link.springer.com/journal/11075",
}


bibtype: article
filename: elefunt.bib
label: Gabutti:2008:EQG
author: Bruno Gabutti and Giampietro Allasia
title: Evaluation of $q$-gamma function and $q$-analogues by iterative algorithms
journal: j-NUMER-ALGORITHMS
volume: 49
number: 1--4
pages: 159--168
month: dec
monthnumber: 12
year: 2008
CODEN: NUALEG
ISSN: 1017-1398 (print), 1572-9265 (electronic) OR 10171398 (print), 15729265 (electronic)
ISSNL: 1017-1398 OR 10171398
bibdate: Mon May 17 15:07:19 MDT 2010
bibtimestamp: 2010.05.17 15:07:19 MDT
URL: http://www.springerlink.com/openurl.asp?genre=article&issn=1017-1398&volume=49&issue=1&spage=159
fjournal: Numerical Algorithms
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/11075
rawauthor: Bruno Gabutti and Giampietro Allasia
rawtitle: Evaluation of $q$-gamma function and $q$-analogues by iterative algorithms
bibsource:
http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1017-1398&volume=49&issue=1; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

entry:
@Article{Gabutti:2008:EQG,
  author =       "Bruno Gabutti and Giampietro Allasia",
  title =        "Evaluation of $q$-gamma function and $q$-analogues by
                 iterative algorithms",
  journal =      j-NUMER-ALGORITHMS,
  volume =       "49",
  number =       "1--4",
  pages =        "159--168",
  month =        dec,
  year =         "2008",
  CODEN =        "NUALEG",
  ISSN =         "1017-1398 (print), 1572-9265 (electronic)",
  ISSN-L =       "1017-1398",
  bibdate =      "Mon May 17 15:07:19 MDT 2010",
  bibsource =    "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1017-1398&volume=49&issue=1;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.springerlink.com/openurl.asp?genre=article&issn=1017-1398&volume=49&issue=1&spage=159",
  acknowledgement = ack-nhfb,
  fjournal =     "Numerical Algorithms",
  journal-URL =  "http://link.springer.com/journal/11075",
}


bibtype: article
filename: elefunt.bib
label: Gautschi:2008:LGW
author: Walter Gautschi and Carla Giordano
title: Luigi Gatteschi's work on asymptotics of special functions and their zeros
journal: j-NUMER-ALGORITHMS
volume: 49
number: 1--4
pages: 11--31
month: dec
monthnumber: 12
year: 2008
CODEN: NUALEG
ISSN: 1017-1398 (print), 1572-9265 (electronic) OR 10171398 (print), 15729265 (electronic)
ISSNL: 1017-1398 OR 10171398
bibdate: Mon May 17 14:08:26 MDT 2010
bibtimestamp: 2010.05.17 14:08:26 MDT
URL: http://www.springerlink.com/openurl.asp?genre=article&issn=1017-1398&volume=49&issue=1&spage=11
fjournal: Numerical Algorithms
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/11075
rawauthor: Walter Gautschi and Carla Giordano
rawtitle: {Luigi Gatteschi}'s work on asymptotics of special functions and their zeros
entry:
@Article{Gautschi:2008:LGW,
  author =       "Walter Gautschi and Carla Giordano",
  title =        "{Luigi Gatteschi}'s work on asymptotics of special
                 functions and their zeros",
  journal =      j-NUMER-ALGORITHMS,
  volume =       "49",
  number =       "1--4",
  pages =        "11--31",
  month =        dec,
  year =         "2008",
  CODEN =        "NUALEG",
  ISSN =         "1017-1398 (print), 1572-9265 (electronic)",
  ISSN-L =       "1017-1398",
  bibdate =      "Mon May 17 14:08:26 MDT 2010",
  bibsource =    "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1017-1398&volume=49&issue=1;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.springerlink.com/openurl.asp?genre=article&issn=1017-1398&volume=49&issue=1&spage=11",
  acknowledgement = ack-nhfb,
  fjournal =     "Numerical Algorithms",
  journal-URL =  "http://link.springer.com/journal/11075",
}


bibtype: article
filename: elefunt.bib
label: Harris:2008:IBG
author: Frank E. Harris
title: Incomplete Bessel, generalized incomplete gamma, or leaky aquifer functions
journal: j-J-COMPUT-APPL-MATH
volume: 215
number: 1
pages: 260--269
year: 2008
CODEN: JCAMDI
DOI: https://doi.org/10.1016/j.cam.2007.04.008
ISSN: 0377-0427 (print), 1879-1778 (electronic) OR 03770427 (print), 18791778 (electronic)
ISSNL: 0377-0427 OR 03770427
MRclass: 33B10 (33C10 41A58); 33B20
MRnumber: 2400632 (2009d:33003)
MRreviewer: Necdet Batir
bibdate: Wed Dec 4 07:03:09 2013
bibtimestamp: 2013.12.04 07:03:09 ???
URL: http://www.sciencedirect.com/science/article/pii/S0377042707002014
fjournal: Journal of Computational and Applied Mathematics
ZMnumber: Zbl 1135.33002
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/03770427
rawauthor: Frank E. Harris
rawtitle: Incomplete {Bessel}, generalized incomplete gamma, or leaky aquifer functions
bibsource:
https://www.math.utah.edu/pub/bibnet/authors/h/harris-frank-e.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

entry:
@Article{Harris:2008:IBG,
  author =       "Frank E. Harris",
  title =        "Incomplete {Bessel}, generalized incomplete gamma, or
                 leaky aquifer functions",
  journal =      j-J-COMPUT-APPL-MATH,
  volume =       "215",
  number =       "1",
  pages =        "260--269",
  year =         "2008",
  CODEN =        "JCAMDI",
  DOI =          "https://doi.org/10.1016/j.cam.2007.04.008",
  ISSN =         "0377-0427 (print), 1879-1778 (electronic)",
  ISSN-L =       "0377-0427",
  MRclass =      "33B10 (33C10 41A58); 33B20",
  MRnumber =     "2400632 (2009d:33003)",
  MRreviewer =   "Necdet Batir",
  bibdate =      "Wed Dec 4 07:03:09 2013",
  bibsource =    "https://www.math.utah.edu/pub/bibnet/authors/h/harris-frank-e.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0377042707002014",
  ZMnumber =     "Zbl 1135.33002",
  acknowledgement = ack-nhfb,
  fjournal =     "Journal of Computational and Applied Mathematics",
  journal-URL =  "http://www.sciencedirect.com/science/journal/03770427",
}


bibtype: article
filename: elefunt.bib
label: Isukapalli:2008:ATA
author: Yogananda Isukapalli and Bhaskar D. Rao
title: An Analytically Tractable Approximation for the Gaussian $Q$-Function
journal: j-IEEE-COMMUN-LET
volume: 12
number: 9
pages: 669--671
month: sep
monthnumber: 9
year: 2008
CODEN: ICLEF6
DOI: https://doi.org/10.1109/lcomm.2008.080815
ISSN: 1089-7798 (print), 1558-2558 (electronic) OR 10897798 (print), 15582558 (electronic)
ISSNL: 1089-7798 OR 10897798
bibdate: Sat Dec 16 16:44:42 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2017.12.16 16:44:42 ???
fjournal: IEEE Communications Letters
acknowledgement: ack-nhfb
journalURL: https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=4234
rawauthor: Yogananda Isukapalli and Bhaskar D. Rao
rawtitle: An Analytically Tractable Approximation for the {Gaussian} {$Q$}-Function
entry:
@Article{Isukapalli:2008:ATA,
  author =       "Yogananda Isukapalli and Bhaskar D. Rao",
  title =        "An Analytically Tractable Approximation for the
                 {Gaussian} {$Q$}-Function",
  journal =      j-IEEE-COMMUN-LET,
  volume =       "12",
  number =       "9",
  pages =        "669--671",
  month =        sep,
  year =         "2008",
  CODEN =        "ICLEF6",
  DOI =          "https://doi.org/10.1109/lcomm.2008.080815",
  ISSN =         "1089-7798 (print), 1558-2558 (electronic)",
  ISSN-L =       "1089-7798",
  bibdate =      "Sat Dec 16 16:44:42 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "IEEE Communications Letters",
  journal-URL =  "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=4234",
}


bibtype: book
filename: elefunt.bib
label: Jeffrey:2008:HMF
author: Alan Jeffrey and Hui-Hui Dai
title: Handbook of Mathematical Formulas and Integrals
publisher: pub-ELSEVIER-ACADEMIC
address: pub-ELSEVIER-ACADEMIC:adr
edition: Fourth
pages: xlv + 541
year: 2008
ISBN: 0-12-374288-9 (paperback), 0-08-055684-1 (e-book) OR 0123742889 (paperback), 0080556841 (ebook)
LCCN: QA47 .J38 2008
bibdate: Thu May 8 16:02:52 MDT 2008
bibtimestamp: 2008.05.08 16:02:52 MDT
subject: mathematics; tables; formulae
acknowledgement: ack-nhfb
rawauthor: Alan Jeffrey and Hui-Hui Dai
rawtitle: Handbook of Mathematical Formulas and Integrals
ISBN13:
978-0-12-374288-9 (paperback), 978-0-08-055684-0 (e-book) OR 9780123742889 (paperback), 9780080556840 (ebook)

bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/master.bib; https://www.math.utah.edu/pub/tex/bib/mathcw.bib; melvyl.cdlib.org:210/CDL90

URL:
https://shop.elsevier.com/books/handbook-of-mathematical-formulas-and-integrals/jeffrey/978-0-12-374288-9

TOC:
Preface Preface to the Fourth Edition Notes for Handbook Users Index of Special Functions and Notations 0. Quick Reference List of Frequently Used Data 1. Numerical, Algebraic, and Analytical Results for Series and Calculus 2. Functions and Identities 3. Derivatives of Elementary Functions 4. Indefinite Integrals of Algebraic Functions 5. Indefinite Integrals of Exponential Functions 6. Indefinite Integrals of Logarithmic Functions 7. Indefinite Integrals of Hyperbolic Functions 8. Indefinite Integrals Involving Inverse Hyperbolic Functions 9. Indefinite Integrals of Trigonometric Functions 10. Indefinite Integrals of Inverse Trigonometric Functions 11. The Gamma, Beta, Pi, and Psi Functions, and the Incomplete Gamma Functions 12. Elliptic Integrals and Functions 13. Probability Distributions and Integrals, and the Error Function 14. Fresnel Integrals, Sine and Cosine Integrals 15. Definite Integrals 16. Different Forms of Fourier Series 17. Bessel Functions 18. Orthogonal Polynomials 19. Laplace Transformation 20. Fourier Transforms 21. Numerical Integration 22. Solutions of Standard Ordinary Differential Equations 23. Vector Analysis 24. Systems of Orthogonal Coordinates 25. Partial Differential Equations and Special Functions 26. Qualitative Properties of the Heat and Laplace Equation 27. Solutions of Elliptic, Parabolic, and Hyperbolic Equations 28. The z-Transform 29. Numerical Approximation 30. Conformal Mapping and Boundary Value Problems Short Classified Reference List Index

rawTOC:
Preface \\ Preface to the Fourth Edition \\ Notes for Handbook Users \\ Index of Special Functions and Notations \\ 0. Quick Reference List of Frequently Used Data \\ 1. Numerical, Algebraic, and Analytical Results for Series and Calculus \\ 2. Functions and Identities \\ 3. Derivatives of Elementary Functions \\ 4. Indefinite Integrals of Algebraic Functions \\ 5. Indefinite Integrals of Exponential Functions \\ 6. Indefinite Integrals of Logarithmic Functions \\ 7. Indefinite Integrals of Hyperbolic Functions \\ 8. Indefinite Integrals Involving Inverse Hyperbolic Functions \\ 9. Indefinite Integrals of Trigonometric Functions \\ 10. Indefinite Integrals of Inverse Trigonometric Functions \\ 11. The Gamma, Beta, Pi, and Psi Functions, and the Incomplete Gamma Functions \\ 12. Elliptic Integrals and Functions \\ 13. Probability Distributions and Integrals, and the Error Function \\ 14. Fresnel Integrals, Sine and Cosine Integrals \\ 15. Definite Integrals \\ 16. Different Forms of Fourier Series \\ 17. Bessel Functions \\ 18. Orthogonal Polynomials \\ 19. Laplace Transformation \\ 20. Fourier Transforms \\ 21. Numerical Integration \\ 22. Solutions of Standard Ordinary Differential Equations \\ 23. Vector Analysis \\ 24. Systems of Orthogonal Coordinates \\ 25. Partial Differential Equations and Special Functions \\ 26. Qualitative Properties of the Heat and Laplace Equation \\ 27. Solutions of Elliptic, Parabolic, and Hyperbolic Equations \\ 28. The z-Transform \\ 29. Numerical Approximation \\ 30. Conformal Mapping and Boundary Value Problems \\ Short Classified Reference List \\ Index

entry:
@Book{Jeffrey:2008:HMF,
  author =       "Alan Jeffrey and Hui-Hui Dai",
  title =        "Handbook of Mathematical Formulas and Integrals",
  publisher =    pub-ELSEVIER-ACADEMIC,
  address =      pub-ELSEVIER-ACADEMIC:adr,
  edition =      "Fourth",
  pages =        "xlv + 541",
  year =         "2008",
  ISBN =         "0-12-374288-9 (paperback), 0-08-055684-1 (e-book)",
  ISBN-13 =      "978-0-12-374288-9 (paperback), 978-0-08-055684-0
                 (e-book)",
  LCCN =         "QA47 .J38 2008",
  bibdate =      "Thu May 8 16:02:52 MDT 2008",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/master.bib;
                 https://www.math.utah.edu/pub/tex/bib/mathcw.bib;
                 melvyl.cdlib.org:210/CDL90",
  URL =          "https://shop.elsevier.com/books/handbook-of-mathematical-formulas-and-integrals/jeffrey/978-0-12-374288-9",
  acknowledgement = ack-nhfb,
  subject =      "mathematics; tables; formulae",
  tableofcontents = "Preface \\
                 Preface to the Fourth Edition \\
                 Notes for Handbook Users \\
                 Index of Special Functions and Notations \\
                 0. Quick Reference List of Frequently Used Data \\
                 1. Numerical, Algebraic, and Analytical Results for
                 Series and Calculus \\
                 2. Functions and Identities \\
                 3. Derivatives of Elementary Functions \\
                 4. Indefinite Integrals of Algebraic Functions \\
                 5. Indefinite Integrals of Exponential Functions \\
                 6. Indefinite Integrals of Logarithmic Functions \\
                 7. Indefinite Integrals of Hyperbolic Functions \\
                 8. Indefinite Integrals Involving Inverse Hyperbolic
                 Functions \\
                 9. Indefinite Integrals of Trigonometric Functions \\
                 10. Indefinite Integrals of Inverse Trigonometric
                 Functions \\
                 11. The Gamma, Beta, Pi, and Psi Functions, and the
                 Incomplete Gamma Functions \\
                 12. Elliptic Integrals and Functions \\
                 13. Probability Distributions and Integrals, and the
                 Error Function \\
                 14. Fresnel Integrals, Sine and Cosine Integrals \\
                 15. Definite Integrals \\
                 16. Different Forms of Fourier Series \\
                 17. Bessel Functions \\
                 18. Orthogonal Polynomials \\
                 19. Laplace Transformation \\
                 20. Fourier Transforms \\
                 21. Numerical Integration \\
                 22. Solutions of Standard Ordinary Differential
                 Equations \\
                 23. Vector Analysis \\
                 24. Systems of Orthogonal Coordinates \\
                 25. Partial Differential Equations and Special
                 Functions \\
                 26. Qualitative Properties of the Heat and Laplace
                 Equation \\
                 27. Solutions of Elliptic, Parabolic, and Hyperbolic
                 Equations \\
                 28. The z-Transform \\
                 29. Numerical Approximation \\
                 30. Conformal Mapping and Boundary Value Problems \\
                 Short Classified Reference List \\
                 Index",
}


bibtype: article
filename: elefunt.bib
label: Kiani:2008:AND
author: M. Kiani and J. Panaretos and S. Psarakis and M. Saleem
journal: J. Iran. Stat. Soc.
volume: 7
number: 1
pages: 57--72
year: 2008
bibdate: Sat Dec 16 16:56:03 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2017.12.16 16:56:03 ???
acknowledgement: ack-nhfb
rawauthor: M. Kiani and J. Panaretos and S. Psarakis and M. Saleem
title:
Approximations to the normal distribution function and an extended table for the mean range of the normal variables

rawtitle:
Approximations to the normal distribution function and an extended table for the mean range of the normal variables

entry:
@Article{Kiani:2008:AND,
  author =       "M. Kiani and J. Panaretos and S. Psarakis and M.
                 Saleem",
  title =        "Approximations to the normal distribution function and
                 an extended table for the mean range of the normal
                 variables",
  journal =      "J. Iran. Stat. Soc.",
  volume =       "7",
  number =       "1",
  pages =        "57--72",
  month =        "????",
  year =         "2008",
  DOI =          "",
  bibdate =      "Sat Dec 16 16:56:03 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: elefunt.bib
label: Kodama:2008:ASP
author: Masao Kodama
journal: j-TOMS
volume: 34
number: 4
pages: 22:1--22:21
month: jul
monthnumber: 7
year: 2008
CODEN: ACMSCU
DOI: https://doi.org/10.1145/1377596.1377602
ISSN: 0098-3500 (print), 1557-7295 (electronic) OR 00983500 (print), 15577295 (electronic)
ISSNL: 0098-3500 OR 00983500
bibdate: Wed Jul 16 11:30:01 MDT 2008
bibtimestamp: 2008.07.16 11:30:01 MDT
fjournal: ACM Transactions on Mathematical Software (TOMS)
acknowledgement: ack-nhfb
articleno: 22
journalURL: https://dl.acm.org/loi/toms
rawauthor: Masao Kodama
title:
Algorithm 877: a Subroutine Package for Cylindrical Functions of Complex Order and Nonnegative Argument

abstract:
The algorithm presented provides a package of subroutines for calculating the cylindrical functions $ J_nu (x) $, $ N_nu (x) $, $ H_nu ^1 (x) $, $ H_nu ^2 (x) $ where the order $ nu $ is complex and the real argument $x$ is nonnegative. The algorithm is written in Fortran 95 and calculates the functions using single, double, or quadruple precision according to the value of a parameter defined in the algorithm. The methods of calculating the functions are based on a series expansion, Debye's asymptotic expansions, Olver's asymptotic expansions, and recurrence methods (Miller's algorithms). The relative errors of the functional values computed by this algorithm using double precision are less than $ 2.4 times 10^-13 $ in the region $ 0 leq mbox Re(nu ) leq 64 $, $ 0 leq mbox Im(nu ) leq 63 $, $ 0.024 leq x leq 97 $.

keywords:
Bessel functions; complex order; Cylindrical functions; Debye's asymptotic expansions; Hankel functions; Miller's algorithms; Neumann functions; nonnegative argument; numerical calculation; Olver's asymptotic expansions

rawabstract:
The algorithm presented provides a package of subroutines for calculating the cylindrical functions $ J_\nu (x) $, $ N_\nu (x) $, $ H_\nu^1 (x) $, $ H_\nu^2 (x) $ where the order $ \nu $ is complex and the real argument $x$ is nonnegative. The algorithm is written in Fortran 95 and calculates the functions using single, double, or quadruple precision according to the value of a parameter defined in the algorithm. The methods of calculating the functions are based on a series expansion, Debye's asymptotic expansions, Olver's asymptotic expansions, and recurrence methods (Miller's algorithms). The relative errors of the functional values computed by this algorithm using double precision are less than $ 2.4 \times 10^{-13} $ in the region $ 0 \leq \mbox {Re}(\nu) \leq 64 $, $ 0 \leq \mbox {Im}(\nu) \leq 63 $, $ 0.024 \leq x \leq 97 $.

rawtitle:
{Algorithm 877}: a Subroutine Package for Cylindrical Functions of Complex Order and Nonnegative Argument

entry:
@Article{Kodama:2008:ASP,
  author =       "Masao Kodama",
  title =        "{Algorithm 877}: a Subroutine Package for Cylindrical
                 Functions of Complex Order and Nonnegative Argument",
  journal =      j-TOMS,
  volume =       "34",
  number =       "4",
  pages =        "22:1--22:21",
  month =        jul,
  year =         "2008",
  CODEN =        "ACMSCU",
  DOI =          "https://doi.org/10.1145/1377596.1377602",
  ISSN =         "0098-3500 (print), 1557-7295 (electronic)",
  ISSN-L =       "0098-3500",
  bibdate =      "Wed Jul 16 11:30:01 MDT 2008",
  bibsource =    "http://www.acm.org/pubs/contents/journals/toms/;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/toms.bib",
  abstract =     "The algorithm presented provides a package of
                 subroutines for calculating the cylindrical functions $
                 J_\nu (x) $, $ N_\nu (x) $, $ H_\nu^1 (x) $, $ H_\nu^2
                 (x) $ where the order $ \nu $ is complex and the real
                 argument $x$ is nonnegative. The algorithm is written
                 in Fortran 95 and calculates the functions using
                 single, double, or quadruple precision according to the
                 value of a parameter defined in the algorithm. The
                 methods of calculating the functions are based on a
                 series expansion, Debye's asymptotic expansions,
                 Olver's asymptotic expansions, and recurrence methods
                 (Miller's algorithms). The relative errors of the
                 functional values computed by this algorithm using
                 double precision are less than $ 2.4 \times 10^{-13} $
                 in the region $ 0 \leq \mbox {Re}(\nu) \leq 64 $, $ 0
                 \leq \mbox {Im}(\nu) \leq 63 $, $ 0.024 \leq x \leq 97
                 $.",
  acknowledgement = ack-nhfb,
  articleno =    "22",
  fjournal =     "ACM Transactions on Mathematical Software (TOMS)",
  journal-URL =  "https://dl.acm.org/loi/toms",
  keywords =     "Bessel functions; complex order; Cylindrical
                 functions; Debye's asymptotic expansions; Hankel
                 functions; Miller's algorithms; Neumann functions;
                 nonnegative argument; numerical calculation; Olver's
                 asymptotic expansions",
}


bibtype: article
filename: elefunt.bib
label: Lefevre:2008:WCE
author: Vincent Lefevre and Damien Stehle and Paul Zimmermann
title: Worst Cases for the Exponential Function in the IEEE 754r decimal64 Format
journal: j-LECT-NOTES-COMP-SCI
volume: 5045
pages: 114--126
year: 2008
CODEN: LNCSD9
DOI: https://doi.org/10.1007/978-3-540-85521-7_7
ISSN: 0302-9743 (print), 1611-3349 (electronic) OR 03029743 (print), 16113349 (electronic)
ISSNL: 0302-9743 OR 03029743
bibdate: Thu Oct 1 11:29:36 MDT 2015
bibtimestamp: 2015.10.01 11:29:36 MDT
URL: http://link.springer.com/content/pdf/10.1007/978-3-540-85521-7_7.pdf
fjournal: Lecture Notes in Computer Science
acknowledgement: ack-nhfb
bookDOI: https://doi.org/10.1007/978-3-540-85521-7
bookURL: http://www.springerlink.com/content/978-3-540-85521-7
journalURL: http://link.springer.com/bookseries/558
rawauthor: Vincent Lef{\`e}vre and Damien Stehl{\'e} and Paul Zimmermann
rawtitle: Worst Cases for the Exponential Function in the {IEEE 754r decimal64} Format
bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/lncs2008a.bib

remark:
From the abstract: ``the worst case for $ |x| geq 3 times 10^-11 $ is exp(9.407822313572878e-2) = 1.09864568206633850000000000000000278.''

entry:
@Article{Lefevre:2008:WCE,
  author =       "Vincent Lef{\`e}vre and Damien Stehl{\'e} and Paul
                 Zimmermann",
  title =        "Worst Cases for the Exponential Function in the {IEEE
                 754r decimal64} Format",
  journal =      j-LECT-NOTES-COMP-SCI,
  volume =       "5045",
  pages =        "114--126",
  year =         "2008",
  CODEN =        "LNCSD9",
  DOI =          "https://doi.org/10.1007/978-3-540-85521-7_7",
  ISSN =         "0302-9743 (print), 1611-3349 (electronic)",
  ISSN-L =       "0302-9743",
  bibdate =      "Thu Oct 1 11:29:36 MDT 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/lncs2008a.bib",
  URL =          "http://link.springer.com/content/pdf/10.1007/978-3-540-85521-7_7.pdf",
  acknowledgement = ack-nhfb,
  book-DOI =     "https://doi.org/10.1007/978-3-540-85521-7",
  book-URL =     "http://www.springerlink.com/content/978-3-540-85521-7",
  fjournal =     "Lecture Notes in Computer Science",
  journal-URL =  "http://link.springer.com/bookseries/558",
  remark =       "From the abstract: ``the worst case for $ |x| \geq 3
                 \times 10^{-11} $ is exp(9.407822313572878e-2) =
                 1.09864568206633850000000000000000278.''",
}


bibtype: article
filename: elefunt.bib
label: Lorch:2008:MSR
author: Lee Lorch and Martin E. Muldoon
title: Monotonic sequences related to zeros of Bessel functions
journal: j-NUMER-ALGORITHMS
volume: 49
number: 1--4
pages: 221--233
month: dec
monthnumber: 12
year: 2008
CODEN: NUALEG
ISSN: 1017-1398 (print), 1572-9265 (electronic) OR 10171398 (print), 15729265 (electronic)
ISSNL: 1017-1398 OR 10171398
bibdate: Mon May 17 15:07:19 MDT 2010
bibtimestamp: 2010.05.17 15:07:19 MDT
URL: http://www.springerlink.com/openurl.asp?genre=article&issn=1017-1398&volume=49&issue=1&spage=221
fjournal: Numerical Algorithms
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/11075
rawauthor: Lee Lorch and Martin E. Muldoon
rawtitle: Monotonic sequences related to zeros of {Bessel} functions
entry:
@Article{Lorch:2008:MSR,
  author =       "Lee Lorch and Martin E. Muldoon",
  title =        "Monotonic sequences related to zeros of {Bessel}
                 functions",
  journal =      j-NUMER-ALGORITHMS,
  volume =       "49",
  number =       "1--4",
  pages =        "221--233",
  month =        dec,
  year =         "2008",
  CODEN =        "NUALEG",
  ISSN =         "1017-1398 (print), 1572-9265 (electronic)",
  ISSN-L =       "1017-1398",
  bibdate =      "Mon May 17 15:07:19 MDT 2010",
  bibsource =    "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1017-1398&volume=49&issue=1;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.springerlink.com/openurl.asp?genre=article&issn=1017-1398&volume=49&issue=1&spage=221",
  acknowledgement = ack-nhfb,
  fjournal =     "Numerical Algorithms",
  journal-URL =  "http://link.springer.com/journal/11075",
}


bibtype: book
filename: elefunt.bib
label: Mathai:2008:SFA
author: A. M. Mathai and H. J. Haubold
title: Special Functions for Applied Scientists
publisher: Springer Science+Business Media
address: New York, NY, USA
pages: xxv + 464
year: 2008
DOI: https://doi.org/10.1007/978-0-387-75894-7
ISBN: 0-387-75893-3 OR 0387758933
ISBN13: 978-0-387-75893-0 OR 9780387758930
LCCN: QA351 .M37X 2008
bibdate: Sat Oct 30 17:02:02 MDT 2010
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib; prodorbis.library.yale.edu:7090/voyager
bibtimestamp: 2010.10.30 17:02:02 MDT
subject: Functions, Special; Fractional calculus; Wavelets (Mathematics)
acknowledgement: ack-nhfb
rawauthor: A. M. Mathai and H. J. Haubold
rawtitle: Special Functions for Applied Scientists
TOC:
1 Basic Ideas of Special Functions and Statistical Distributions / 1 1.0 Introduction / 1 1.1 Gamma Function / 3 1.1.1 Some basic properties of gamma functions / 4 1.1.2 Wedge product and Jacobians of transformations / 6 1.1.3 Multiplication formula for gamma functions / 8 1.1.4 Asymptotic formula for a gamma function / 9 1.1.5 Bernoulli polynomials / 9 1.1.6 Some basic properties of generalized Bernoulli polynomials / 9 1.1.7 The first three generalized Bernoulli polynomials / 10 Exercises 1.1 / 11 1.2 The Psi and Zeta Functions / 12 1.2.1 Generalized zeta function / 12 Exercises 1.2 / 13 1.3 Integral Transforms / 14 1.3.1 Mellin transform / 14 1.3.2 Laplace transform / 17 Exercises 1.3 / 18 1.4 Some Statistical Preliminaries / 19 Exercises 1.4 / 26 1.5 Some Properties of Random Variables / 28 1.5.1 Multivariate analogues / 30 1.5.2 Marginal and conditional densities / 31 Exercises 1.5 / 33 1.6 Beta and Related Functions / 34 1.6.1 Dirichlet integrals and Dirichlet densities / 37 Exercises 1.6 / 42 1.7 Hypergeometric Series / 42 1.7.1 Evaluation of some contour integrals / 45 1.7.2 Residues when several gammas are involved / 46 Exercises 1.7 / 48 1.8 Meijer's $G$-function / 49 Exercises 1.8 / 54 1.9 The $H$-function / 54 Exercises 1.9 / 57 1.10 Lauricella Functions and Appell's Functions / 58 1.10.1 Some properties of $f_A$ / 59 1.10.2 Some properties of $f_B$ / 60 1.10.3 Some properties of $f_C$ / 60 1.10.4 Some properties of $f_D$ / 61 Exercises 1.10 / 64 1.11 Special Functions as Solutions of Differential Equations and Applications / 64 1.11.0 Introduction / 64 1.11.1 Sine, cosine and exponential functions / 64 Exercises 1.11 / 66 1.11.2 Linear second order differential equations / 66 1.11.3 Hypergeometric function / 67 Exercises 1.11 / 69 1.11.4 Confluent hypergeometric function / 70 Exercises 1.11 / 71 1.11.5 Hermite polynomials / 71 Exercises 1.11 / 72 1.11.6 Bessel functions / 72 Exercises 1.11 / 73 1.11.7 Laguerre polynomial / 74 1.11.8 Legendre polynomial / 74 Exercises 1.11 / 74 1.11.9 Generalized hypergeometric function / 75 1.11.10 $G$-function / 76 Exercises 1.11 / 77 References / 77 2 Mittag-Leffler Functions and Fractional Calculus / 79 2.0 Introduction / 79 2.1 Mittag-Leffler Function / 79 Revision Exercises 2.1 / 81 2.2 Basic Properties of Mittag-Leffler Function / 82 2.2.1 Mittag-Leffler functions of rational order / 84 2.2.2 Euler transform of Mittag-Leffler function / 84 2.2.3 Laplace transform of Mittag-Leffler function / 85 2.2.4 Application of Laplace transform / 87 2.2.5 Mittag-Leffler functions and the $H$-function / 88 Exercises 2.2 / 90 2.3 Generalized Mittag-Leffler Function / 91 2.3.1 Special cases of $E_beta , gamma ^delta (z)$ / 92 2.3.2 Mellin--Barnes integral representation / 92 2.3.3 Relations with the H-function and Wright function / 92 2.3.4 Cases of reducibility / 93 2.3.5 Differentiation of generalized Mittag-Leffler function / 94 2.3.6 Integral property of generalized Mittag-Leffler function / 94 2.3.7 Integral transform of $E_beta , gamma ^delta (z)$ / 95 Exercises 2.3 / 96 2.4 Fractional Integrals / 97 2.4.1 Riemann--Liouville fractional integrals of arbitrary order / 98 2.4.2 Riemann--Liouville fractional integrals of order $alpha $ / 99 2.4.3 Basic properties of fractional integrals / 100 2.4.4 A useful integral / 101 2.4.5 The Weyl integral / 103 2.4.6 Basic properties of Weyl integral / 104 Exercises 2.4 / 105 2.4.7 Laplace transform of the fractional integral / 107 2.4.8 Laplace transform of the fractional derivative / 107 2.4.9 Laplace transform of Caputo derivative / 108 Exercises 2.4 / 109 2.5 Mellin Transform of the Fractional Integrals and the Fractional Derivatives / 109 2.5.1 Mellin transform / 109 2.5.2 Mellin transform of the fractional integral / 110 2.5.3 Mellin transform of the fractional derivative / 111 Exercises 2.5 / 111 2.6 Kober Operators / 111 Exercises 2.6 / 113 2.7 Generalized Kober Operators / 114 Exercises 2.7 / 118 2.8 Compositions of Riemann--Liouville Fractional Calculus Operators and Generalized Mittag-Leffler Functions / 121 2.8.1 Composition relations between R-L operators and $E_beta , gamma ^delta (z)$ / 121 Exercises 2.8 / 126 2.9 Fractional Differential Equations / 126 2.9.1 Fractional relaxation / 127 Exercises 2.9 / 130 2.9.2 Fractional diffusion / 131 Exercises 2.9 / 132 References / 133 3 An Introduction to $q$-Series / 135 3.0 Introduction / 135 3.1 Hypergeometric Series / 135 Exercises 3.1 / 136 3.2 Basic Hypergeometric Series ($q$-Series) / 138 Exercises 3.2 / 140 3.2.1 The $q$-binomial theorem / 142 Exercises 3.2 / 146 3.2.2 The $q$-binomial coefficients / 146 Exercises 3.2 / 148 3.3 $q$-Calculus / 149 Exercises 3.3 / 151 3.4 The $q$-Gamma and $q$-Beta Functions / 151 3.5 Transformation and Summation Formulas for $q$-Series / 152 Exercises 3.5 / 154 3.6 Jacobi's Triple Product and Rogers--Ramanujan Identities / 154 References / 157 4 Ramanujan's Theories of Theta and Elliptic Functions / 159 4.0 Introduction / 159 4.1 Ramanujan's Theory of Classical Theta Functions / 159 4.1.1 Series definition and additive results / 159 4.2 Ramanujan's $_1Psi _1$ Summation Formula and Multiplicative Results for Theta Functions / 163 4.3 Modular Equations / 170 4.4 Inversion Formulas and Evaluations / 172 4.5 Modular Identities (Classical Theory) / 175 4.6 Ramanujan's Theory of Cubic Theta Functions / 177 4.6.1 The cubic theta functions / 177 4.6.2 Inversion formulas and evaluations (cubic theory) / 179 4.6.3 Triplication and trimediation formulas / 182 4.6.4 Further evaluations / 183 4.6.5 Evaluations of Ramanujan--Eisenstein series ($L$, $M$, $N$ or $P$, $Q$, $R$) / 185 4.6.6 The cubic analogue of the Jacobian elliptic functions / 186 Test on Ramanujan's work / 187 4.7 The One-variable Cubic Theta Functions / 188 4.7.1 Cubic theta functions and some properties / 189 4.7.2 Product representations for $b(q)$ and $c(q)$ / 190 4.7.3 The cubic analogue of Jacobi's quartic modular equations / 191 4.8 The Two-variable Cubic Theta Functions / 193 4.8.1 Series definitions and some properties / 193 4.8.2 Product representations for $b(q,z)$ and $c(q,z)$ / 195 4.8.3 A two-variable cubic counterpart of Jacobi's quartic modular equation / 200 4.9 The Three-variable Cubic Theta Functions / 200 4.9.1 Unification of one and two-variable cubic theta functions / 200 Exercises 4.9 / 200 4.9.2 Generalization of Hirschhorn--Garvan--Borwein identity / 202 4.9.3 Laurent's expansions for two-parameter cubic theta functions / 205 References / 209 5 Lie Group and Special Functions / 211 5.1 General Introduction to Group Theory / 211 5.1.1 Isomorphisms / 213 5.1.2 Symmetry groups / 213 5.1.3 Isometries of the Euclidean plane / 214 5.1.4 Finite groups of motion / 215 5.1.5 Discrete groups of motions / 216 5.2 Lie Group and Special Functions / 217 5.2.1 Subspace of a vector space / 218 5.3 Lie Algebra / 219 5.4 Representations of Lie Algebra / 222 Exercises 5.4 / 225 5.5 Special Functions / 226 5.5.1 Gauss hypergeometric function / 226 5.5.2 Differential equation satisfied by $_2F_1$ / 228 5.5.3 Integral representation of $_F_q(alpha _1,alpha _2, ldots , alpha _p beta _1, beta _2, ldots , beta _p; z)$ / 230 5.6 Laguerre Polynomial $L^(alpha )_n(x)$ / 231 5.6.1 Laguerre polynomial and Lie algebra / 232 Exercises 5.6 / 232 5.7 Helmholtz Equation / 234 5.7.1 Helmholtz equation in three variables / 238 5.8 Lie Group / 241 5.8.1 Basis of the Lie algebra of the Lie group SL(2) / 242 Exercises 5.8 / 243 Test on Lie Theory and Special Functions / 244 6 Applications to Stochastic Process and Time Series / 247 6.0 Introduction / 247 6.1 Stochastic Processes / 247 6.1.1 Classical types of stochastic processes / 252 6.1.2 Processes with stationary independent increments / 252 6.1.3 Stationary processes / 253 6.1.4 Gaussian processes and stationarity / 253 6.1.5 Brownian processes / 257 6.1.6 Markov chains / 258 Exercises 6.1 / 261 6.2 Modern Concepts in Distribution Theory / 262 6.2.1 Introduction / 262 6.2.2 Geometric infinite divisibility / 263 6.2.3 Bernstein functions / 264 6.2.4 Self-decomposability / 265 6.2.5 Stable distributions / 265 6.2.6 Geometrically strictly stable distributions / 266 6.2.7 Mittag-Leffler distribution / 266 6.2.8 $alpha $-Laplace distribution / 267 6.2.9 Semi-Pareto distribution / 268 Exercises 6.2 / 268 6.3 Stationary Time Series Models / 269 6.3.1 Introduction / 269 6.3.2 Autoregressive models / 270 6.3.3 A general solution / 270 6.3.4 Extension to a $k$-th order autoregressive model / 272 6.3.5 Mittag-Leffler autoregressive structure / 273 Exercises 6.3 / 274 6.4 A Structural Relationship and New Processes / 275 6.4.1 The TMLAR(1) process / 276 6.4.2 The NEAR(1) model / 277 6.4.3 New Mittag-Leffler autoregressive models / 278 6.4.4 The NSMLAR(1) process / 280 Exercises 6.4 / 281 6.5 Tailed Processes / 281 6.5.1 The exponential tailed autoregressive process [ETAR(1)] / 282 6.5.2 The Mittag-Leffler tailed autoregressive process [MLTAR(1)] / 283 Exercises 6.5 / 286 6.6 Marshall--Olkin Weibull Time Series Models / 286 6.6.1 Introduction / 286 6.6.2 Marshall--Olkin semi-Weibull distribution / 287 6.6.3 An AR(1) model with MOSW marginal distribution / 289 6.6.4 Marshall--Olkin generalized Weibull distribution / 291 6.6.5 An AR(1) model with MOGW marginal distribution / 292 6.6.6 Case study / 293 Exercises 6.6 / 293 References / 294 7 Applications to Density Estimation / 297 7.0 Density Estimation and Orthogonal Polynomials / 297 7.1 Introduction / 297 7.2 Approximants Based on Legendre Polynomials / 299 7.3 Approximants Based on Laguerre Polynomials / 301 7.4 A Unified Methodology / 304 7.5 Approximants Expressed in Terms of Orthogonal Polynomials / 305 7.5.1 Approximants expressed in terms of Laguerre polynomials / 306 7.5.2 Approximants expressed in terms of Legendre polynomials / 306 7.5.3 Approximants expressed in terms of Jacobi polynomials / 307 7.5.4 Approximants expressed in terms of Hermite polynomials / 307 References / 308 8 Applications to Order Statistics / 311 8.0 Introduction / 311 8.1 Distribution Function / 311 8.1.1 Density of the $r$-th order statistic / 313 8.1.2 Joint distribution function of two order statistics / 314 8.1.3 Joint density of two order statistics / 314 8.1.4 Moments of order statistics / 315 8.1.5 Recurrence relations for moments / 317 8.1.6 Recurrence relations on the product moments / 318 8.1.7 Order statistics from symmetric distributions / 319 8.2 Discrete Order Statistics / 319 8.2.1 Probability function of discrete order statistics / 320 8.2.2 Joint probability function of two order statistics / 321 8.2.3 Bernoulli order statistics / 321 8.3 Independent Random Variables / 322 8.3.1 Distribution of a single order statistic / 322 8.3.2 Joint distribution of two order statistics / 324 Test on Order Statistics / 325 8.4 On Concomitants of Order Statistics / 326 8.4.1 Application of concomitants of order statistics / 326 8.4.2 Application in estimation / 328 8.4.3 Concomitants of record values and estimation problems / 333 References / 339 9 Applications to Astrophysics Problems / 341 9.0 Introduction / 341 9.1 Entropy: Boltzmann, Planck, and Einstein on W / 342 9.1.1 Entropic functional / 342 9.1.2 Entropy and probability / 342 9.1.3 Boltzmann--Gibbs / 343 9.2 Gravitationally Stabilized Fusion Reactor: The Sun / 345 9.2.1 Internal solar structure / 345 9.2.2 Solar fusion plasma / 349 9.2.3 Estimation of central temperature in the Sun / 349 9.3 Crucial Astrophysical Experiments: Data Analysis / 351 9.3.1 The experiments / 351 9.3.2 Analysis of the time series / 352 9.4 Fundamental Equations for Nonequilibrium Processes / 355 9.4.1 Chapman--Kolmogorov equation / 355 9.4.2 Master equation / 355 9.4.3 Fokker--Planck equation / 356 9.4.4 Langevin equation / 357 9.4.5 Reaction-diffusion equation / 357 9.5 Fractional Calculus / 360 9.6 Nonextensive Statistical Mechanics / 362 9.7 Standard and Fractional Reaction / 363 9.7.1 Boltzmann--Gibbs statistical mechanics / 363 9.7.2 Generalized Boltzmann--Gibbs statistical mechanics / 366 9.7.3 Fractional reaction / 369 9.7.4 Thermonuclear reaction coefficient / 371 9.8 Standard and Fractional Diffusion / 380 9.8.1 Fick's first law of diffusion / 380 9.8.2 Einstein's approach to diffusion / 381 9.8.3 Fractional diffusion / 381 9.8.4 Spatio-temporal pattern formation / 383 References / 384 10 An Introduction to Wavelet Analysis / 389 10.0 Introduction / 389 10.1 Fourier Analysis to Wavelet Analysis / 390 10.2 Construction of Orthonormal Wavelets / 393 10.3 Classification of Wavelets and Multiresolution Analysis / 400 10.4 Spline Wavelets / 405 10.5 A Variant of Construction of Orthonormal Wavelets / 407 Exercises 10.5 / 408 References / 408 11 Jacobians of Matrix Transformations / 409 11.0 Introduction / 409 11.1 Jacobians of Linear Matrix Transformations / 410 Exercises 11.1 / 415 11.2 Jacobians in Some Nonlinear Transformations / 417 Exercises 11.2 / 421 11.3 Transformations Involving Orthonormal Matrices / 423 Exercises 11.3 / 426 References / 428 12 Special Functions of Matrix Argument / 429 12.0 Introduction / 429 12.1 Real Matrix-Variate Scalar Functions / 429 12.1.1 Real matrix-variate gamma / 430 12.1.2 Real matrix-variate gamma density / 430 Exercises 12.1 / 434 12.2 The Laplace Transform in the Matrix Case / 435 12.2.1 A convolution property for Laplace transforms / 436 Exercises 12.2 / 439 12.3 Hypergeometric Functions with Matrix Argument / 439 12.3.1 Hypergeometric function through Laplace transform / 440 12.3.2 Hypergeometric function through zonal polynomials / 441 12.3.3 Hypergeometric functions through $M$-transforms / 443 12.3.4 A convolution theorem for $M$-transforms / 445 Exercises 12.3 / 446 12.4 A Pathway Model / 448 12.4.1 The pathway density / 448 12.4.2 A general density / 451 12.4.3 Arbitrary moments / 451 12.4.4 Quadratic forms / 452 12.4.5 Generalized quadratic form / 452 12.4.6 Applications to random volumes / 453 Exercises 12.4 / 453 References / 454 Author Index / 457 Subject Index / 461

shortTOC:
Basic ideas of special functions and statistical distributions Mittag-Leffler functions and fractional calculus An introduction to q-series Ramanujan's theories of theta and elliptic functions Lie group and special functions Applications to stochastic process and time series Applications to density estimation Applications to order statistics Applications to astrophysics problems An introduction to wavelet analysis Jacobians of matrix formations Special functions of matrix argument

rawshortTOC:
Basic ideas of special functions and statistical distributions \\ Mittag-Leffler functions and fractional calculus \\ An introduction to q-series \\ Ramanujan's theories of theta and elliptic functions \\ Lie group and special functions \\ Applications to stochastic process and time series \\ Applications to density estimation \\ Applications to order statistics \\ Applications to astrophysics problems \\ An introduction to wavelet analysis \\ Jacobians of matrix formations \\ Special functions of matrix argument

rawTOC:
1 Basic Ideas of Special Functions and Statistical Distributions / 1 \\ 1.0 Introduction / 1 \\ 1.1 Gamma Function / 3 \\ 1.1.1 Some basic properties of gamma functions / 4 \\ 1.1.2 Wedge product and Jacobians of transformations / 6 \\ 1.1.3 Multiplication formula for gamma functions / 8 \\ 1.1.4 Asymptotic formula for a gamma function / 9 \\ 1.1.5 Bernoulli polynomials / 9 \\ 1.1.6 Some basic properties of generalized Bernoulli polynomials / 9 \\ 1.1.7 The first three generalized Bernoulli polynomials / 10 \\ Exercises 1.1 / 11 \\ 1.2 The Psi and Zeta Functions / 12 \\ 1.2.1 Generalized zeta function / 12 \\ Exercises 1.2 / 13 \\ 1.3 Integral Transforms / 14 \\ 1.3.1 Mellin transform / 14 \\ 1.3.2 Laplace transform / 17 \\ Exercises 1.3 / 18 \\ 1.4 Some Statistical Preliminaries / 19 \\ Exercises 1.4 / 26 \\ 1.5 Some Properties of Random Variables / 28 \\ 1.5.1 Multivariate analogues / 30 \\ 1.5.2 Marginal and conditional densities / 31 \\ Exercises 1.5 / 33 \\ 1.6 Beta and Related Functions / 34 \\ 1.6.1 Dirichlet integrals and Dirichlet densities / 37 \\ Exercises 1.6 / 42 \\ 1.7 Hypergeometric Series / 42 \\ 1.7.1 Evaluation of some contour integrals / 45 \\ 1.7.2 Residues when several gammas are involved / 46 \\ Exercises 1.7 / 48 \\ 1.8 Meijer's $G$-function / 49 \\ Exercises 1.8 / 54 \\ 1.9 The $H$-function / 54 \\ Exercises 1.9 / 57 \\ 1.10 Lauricella Functions and Appell's Functions / 58 \\ 1.10.1 Some properties of $f_A$ / 59 \\ 1.10.2 Some properties of $f_B$ / 60 \\ 1.10.3 Some properties of $f_C$ / 60 \\ 1.10.4 Some properties of $f_D$ / 61 \\ Exercises 1.10 / 64 \\ 1.11 Special Functions as Solutions of Differential Equations and Applications / 64 \\ 1.11.0 Introduction / 64 \\ 1.11.1 Sine, cosine and exponential functions / 64 \\ Exercises 1.11 / 66 \\ 1.11.2 Linear second order differential equations / 66 \\ 1.11.3 Hypergeometric function / 67 \\ Exercises 1.11 / 69 \\ 1.11.4 Confluent hypergeometric function / 70 \\ Exercises 1.11 / 71 \\ 1.11.5 Hermite polynomials / 71 \\ Exercises 1.11 / 72 \\ 1.11.6 Bessel functions / 72 \\ Exercises 1.11 / 73 \\ 1.11.7 Laguerre polynomial / 74 \\ 1.11.8 Legendre polynomial / 74 \\ Exercises 1.11 / 74 \\ 1.11.9 Generalized hypergeometric function / 75 \\ 1.11.10 $G$-function / 76 \\ Exercises 1.11 / 77 \\ References / 77 \\ 2 Mittag-Leffler Functions and Fractional Calculus / 79 \\ 2.0 Introduction / 79 \\ 2.1 Mittag-Leffler Function / 79 \\ Revision Exercises 2.1 / 81 \\ 2.2 Basic Properties of Mittag-Leffler Function / 82 \\ 2.2.1 Mittag-Leffler functions of rational order / 84 \\ 2.2.2 Euler transform of Mittag-Leffler function / 84 \\ 2.2.3 Laplace transform of Mittag-Leffler function / 85 \\ 2.2.4 Application of Laplace transform / 87 \\ 2.2.5 Mittag-Leffler functions and the $H$-function / 88 \\ Exercises 2.2 / 90 \\ 2.3 Generalized Mittag-Leffler Function / 91 \\ 2.3.1 Special cases of $E_{\beta, \gamma}^\delta (z)$ / 92 \\ 2.3.2 Mellin--Barnes integral representation / 92 \\ 2.3.3 Relations with the H-function and Wright function / 92 \\ 2.3.4 Cases of reducibility / 93 \\ 2.3.5 Differentiation of generalized Mittag-Leffler function / 94 \\ 2.3.6 Integral property of generalized Mittag-Leffler function / 94 \\ 2.3.7 Integral transform of $E_{\beta, \gamma}^\delta (z)$ / 95 \\ Exercises 2.3 / 96 \\ 2.4 Fractional Integrals / 97 \\ 2.4.1 Riemann--Liouville fractional integrals of arbitrary order / 98 \\ 2.4.2 Riemann--Liouville fractional integrals of order $\alpha$ / 99 \\ 2.4.3 Basic properties of fractional integrals / 100 \\ 2.4.4 A useful integral / 101 \\ 2.4.5 The Weyl integral / 103 \\ 2.4.6 Basic properties of Weyl integral / 104 \\ Exercises 2.4 / 105 \\ 2.4.7 Laplace transform of the fractional integral / 107 \\ 2.4.8 Laplace transform of the fractional derivative / 107 \\ 2.4.9 Laplace transform of Caputo derivative / 108 \\ Exercises 2.4 / 109 \\ 2.5 Mellin Transform of the Fractional Integrals and the Fractional Derivatives / 109 \\ 2.5.1 Mellin transform / 109 \\ 2.5.2 Mellin transform of the fractional integral / 110 \\ 2.5.3 Mellin transform of the fractional derivative / 111 \\ Exercises 2.5 / 111 \\ 2.6 Kober Operators / 111 \\ Exercises 2.6 / 113 \\ 2.7 Generalized Kober Operators / 114 \\ Exercises 2.7 / 118 \\ 2.8 Compositions of Riemann--Liouville Fractional Calculus Operators and Generalized Mittag-Leffler Functions / 121 \\ 2.8.1 Composition relations between R-L operators and $E_{\beta, \gamma}^\delta (z)$ / 121 \\ Exercises 2.8 / 126 \\ 2.9 Fractional Differential Equations / 126 \\ 2.9.1 Fractional relaxation / 127 \\ Exercises 2.9 / 130 \\ 2.9.2 Fractional diffusion / 131 \\ Exercises 2.9 / 132 \\ References / 133 \\ 3 An Introduction to $q$-Series / 135 \\ 3.0 Introduction / 135 \\ 3.1 Hypergeometric Series / 135 \\ Exercises 3.1 / 136 \\ 3.2 Basic Hypergeometric Series ($q$-Series) / 138 \\ Exercises 3.2 / 140 \\ 3.2.1 The $q$-binomial theorem / 142 \\ Exercises 3.2 / 146 \\ 3.2.2 The $q$-binomial coefficients / 146 \\ Exercises 3.2 / 148 \\ 3.3 $q$-Calculus / 149 \\ Exercises 3.3 / 151 \\ 3.4 The $q$-Gamma and $q$-Beta Functions / 151 \\ 3.5 Transformation and Summation Formulas for $q$-Series / 152 \\ Exercises 3.5 / 154 \\ 3.6 Jacobi's Triple Product and Rogers--Ramanujan Identities / 154 \\ References / 157 \\ 4 Ramanujan's Theories of Theta and Elliptic Functions / 159 \\ 4.0 Introduction / 159 \\ 4.1 Ramanujan's Theory of Classical Theta Functions / 159 \\ 4.1.1 Series definition and additive results / 159 \\ 4.2 Ramanujan's $_1\Psi_1$ Summation Formula and Multiplicative Results for Theta Functions / 163 \\ 4.3 Modular Equations / 170 \\ 4.4 Inversion Formulas and Evaluations / 172 \\ 4.5 Modular Identities (Classical Theory) / 175 \\ 4.6 Ramanujan's Theory of Cubic Theta Functions / 177 \\ 4.6.1 The cubic theta functions / 177 \\ 4.6.2 Inversion formulas and evaluations (cubic theory) / 179 \\ 4.6.3 Triplication and trimediation formulas / 182 \\ 4.6.4 Further evaluations / 183 \\ 4.6.5 Evaluations of Ramanujan--Eisenstein series ($L$, $M$, $N$ or $P$, $Q$, $R$) / 185 \\ 4.6.6 The cubic analogue of the Jacobian elliptic functions / 186 \\ Test on Ramanujan's work / 187 \\ 4.7 The One-variable Cubic Theta Functions / 188 \\ 4.7.1 Cubic theta functions and some properties / 189 \\ 4.7.2 Product representations for $b(q)$ and $c(q)$ / 190 \\ 4.7.3 The cubic analogue of Jacobi's quartic modular equations / 191 \\ 4.8 The Two-variable Cubic Theta Functions / 193 \\ 4.8.1 Series definitions and some properties / 193 \\ 4.8.2 Product representations for $b(q,z)$ and $c(q,z)$ / 195 \\ 4.8.3 A two-variable cubic counterpart of Jacobi's quartic modular equation / 200 \\ 4.9 The Three-variable Cubic Theta Functions / 200 \\ 4.9.1 Unification of one and two-variable cubic theta functions / 200 \\ Exercises 4.9 / 200 \\ 4.9.2 Generalization of Hirschhorn--Garvan--Borwein identity / 202 \\ 4.9.3 Laurent's expansions for two-parameter cubic theta functions / 205 \\ References / 209 \\ 5 Lie Group and Special Functions / 211 \\ 5.1 General Introduction to Group Theory / 211 \\ 5.1.1 Isomorphisms / 213 \\ 5.1.2 Symmetry groups / 213 \\ 5.1.3 Isometries of the Euclidean plane / 214 \\ 5.1.4 Finite groups of motion / 215 \\ 5.1.5 Discrete groups of motions / 216 \\ 5.2 Lie Group and Special Functions / 217 \\ 5.2.1 Subspace of a vector space / 218 \\ 5.3 Lie Algebra / 219 \\ 5.4 Representations of Lie Algebra / 222 \\ Exercises 5.4 / 225 \\ 5.5 Special Functions / 226 \\ 5.5.1 Gauss hypergeometric function / 226 \\ 5.5.2 Differential equation satisfied by $_2F_1$ / 228 \\ 5.5.3 Integral representation of $_F_q(\alpha_1,\alpha_2, \ldots{}, \alpha_p \beta_1, \beta_2, \ldots{}, \beta_p; z)$ / 230 \\ 5.6 Laguerre Polynomial $L^{(\alpha)}_n(x)$ / 231 \\ 5.6.1 Laguerre polynomial and Lie algebra / 232 \\ Exercises 5.6 / 232 \\ 5.7 Helmholtz Equation / 234 \\ 5.7.1 Helmholtz equation in three variables / 238 \\ 5.8 Lie Group / 241 \\ 5.8.1 Basis of the Lie algebra of the Lie group SL(2) / 242 \\ Exercises 5.8 / 243 \\ Test on Lie Theory and Special Functions / 244 \\ 6 Applications to Stochastic Process and Time Series / 247 \\ 6.0 Introduction / 247 \\ 6.1 Stochastic Processes / 247 \\ 6.1.1 Classical types of stochastic processes / 252 \\ 6.1.2 Processes with stationary independent increments / 252 \\ 6.1.3 Stationary processes / 253 \\ 6.1.4 Gaussian processes and stationarity / 253 \\ 6.1.5 Brownian processes / 257 \\ 6.1.6 Markov chains / 258 \\ Exercises 6.1 / 261 \\ 6.2 Modern Concepts in Distribution Theory / 262 \\ 6.2.1 Introduction / 262 \\ 6.2.2 Geometric infinite divisibility / 263 \\ 6.2.3 Bernstein functions / 264 \\ 6.2.4 Self-decomposability / 265 \\ 6.2.5 Stable distributions / 265 \\ 6.2.6 Geometrically strictly stable distributions / 266 \\ 6.2.7 Mittag-Leffler distribution / 266 \\ 6.2.8 $\alpha$-Laplace distribution / 267 \\ 6.2.9 Semi-Pareto distribution / 268 \\ Exercises 6.2 / 268 \\ 6.3 Stationary Time Series Models / 269 \\ 6.3.1 Introduction / 269 \\ 6.3.2 Autoregressive models / 270 \\ 6.3.3 A general solution / 270 \\ 6.3.4 Extension to a $k$-th order autoregressive model / 272 \\ 6.3.5 Mittag-Leffler autoregressive structure / 273 \\ Exercises 6.3 / 274 \\ 6.4 A Structural Relationship and New Processes / 275 \\ 6.4.1 The TMLAR(1) process / 276 \\ 6.4.2 The NEAR(1) model / 277 \\ 6.4.3 New Mittag-Leffler autoregressive models / 278 \\ 6.4.4 The NSMLAR(1) process / 280 \\ Exercises 6.4 / 281 \\ 6.5 Tailed Processes / 281 \\ 6.5.1 The exponential tailed autoregressive process [ETAR(1)] / 282 \\ 6.5.2 The Mittag-Leffler tailed autoregressive process [MLTAR(1)] / 283 \\ Exercises 6.5 / 286 \\ 6.6 Marshall--Olkin Weibull Time Series Models / 286 \\ 6.6.1 Introduction / 286 \\ 6.6.2 Marshall--Olkin semi-Weibull distribution / 287 \\ 6.6.3 An AR(1) model with MOSW marginal distribution / 289 \\ 6.6.4 Marshall--Olkin generalized Weibull distribution / 291 \\ 6.6.5 An AR(1) model with MOGW marginal distribution / 292 \\ 6.6.6 Case study / 293 \\ Exercises 6.6 / 293 \\ References / 294 \\ 7 Applications to Density Estimation / 297 \\ 7.0 Density Estimation and Orthogonal Polynomials / 297 \\ 7.1 Introduction / 297 \\ 7.2 Approximants Based on Legendre Polynomials / 299 \\ 7.3 Approximants Based on Laguerre Polynomials / 301 \\ 7.4 A Unified Methodology / 304 \\ 7.5 Approximants Expressed in Terms of Orthogonal Polynomials / 305 \\ 7.5.1 Approximants expressed in terms of Laguerre polynomials / 306 \\ 7.5.2 Approximants expressed in terms of Legendre polynomials / 306 \\ 7.5.3 Approximants expressed in terms of Jacobi polynomials / 307 \\ 7.5.4 Approximants expressed in terms of Hermite polynomials / 307 \\ References / 308 \\ 8 Applications to Order Statistics / 311 \\ 8.0 Introduction / 311 \\ 8.1 Distribution Function / 311 \\ 8.1.1 Density of the $r$-th order statistic / 313 \\ 8.1.2 Joint distribution function of two order statistics / 314 \\ 8.1.3 Joint density of two order statistics / 314 \\ 8.1.4 Moments of order statistics / 315 \\ 8.1.5 Recurrence relations for moments / 317 \\ 8.1.6 Recurrence relations on the product moments / 318 \\ 8.1.7 Order statistics from symmetric distributions / 319 \\ 8.2 Discrete Order Statistics / 319 \\ 8.2.1 Probability function of discrete order statistics / 320 \\ 8.2.2 Joint probability function of two order statistics / 321 \\ 8.2.3 Bernoulli order statistics / 321 \\ 8.3 Independent Random Variables / 322 \\ 8.3.1 Distribution of a single order statistic / 322 \\ 8.3.2 Joint distribution of two order statistics / 324 \\ Test on Order Statistics / 325 \\ 8.4 On Concomitants of Order Statistics / 326 \\ 8.4.1 Application of concomitants of order statistics / 326 \\ 8.4.2 Application in estimation / 328 \\ 8.4.3 Concomitants of record values and estimation problems / 333 \\ References / 339 \\ 9 Applications to Astrophysics Problems / 341 \\ 9.0 Introduction / 341 \\ 9.1 Entropy: Boltzmann, Planck, and Einstein on W / 342 \\ 9.1.1 Entropic functional / 342 \\ 9.1.2 Entropy and probability / 342 \\ 9.1.3 Boltzmann--Gibbs / 343 \\ 9.2 Gravitationally Stabilized Fusion Reactor: The Sun / 345 \\ 9.2.1 Internal solar structure / 345 \\ 9.2.2 Solar fusion plasma / 349 \\ 9.2.3 Estimation of central temperature in the Sun / 349 \\ 9.3 Crucial Astrophysical Experiments: Data Analysis / 351 \\ 9.3.1 The experiments / 351 \\ 9.3.2 Analysis of the time series / 352 \\ 9.4 Fundamental Equations for Nonequilibrium Processes / 355 \\ 9.4.1 Chapman--Kolmogorov equation / 355 \\ 9.4.2 Master equation / 355 \\ 9.4.3 Fokker--Planck equation / 356 \\ 9.4.4 Langevin equation / 357 \\ 9.4.5 Reaction-diffusion equation / 357 \\ 9.5 Fractional Calculus / 360 \\ 9.6 Nonextensive Statistical Mechanics / 362 \\ 9.7 Standard and Fractional Reaction / 363 \\ 9.7.1 Boltzmann--Gibbs statistical mechanics / 363 \\ 9.7.2 Generalized Boltzmann--Gibbs statistical mechanics / 366 \\ 9.7.3 Fractional reaction / 369 \\ 9.7.4 Thermonuclear reaction coefficient / 371 \\ 9.8 Standard and Fractional Diffusion / 380 \\ 9.8.1 Fick's first law of diffusion / 380 \\ 9.8.2 Einstein's approach to diffusion / 381 \\ 9.8.3 Fractional diffusion / 381 \\ 9.8.4 Spatio-temporal pattern formation / 383 \\ References / 384 \\ 10 An Introduction to Wavelet Analysis / 389 \\ 10.0 Introduction / 389 \\ 10.1 Fourier Analysis to Wavelet Analysis / 390 \\ 10.2 Construction of Orthonormal Wavelets / 393 \\ 10.3 Classification of Wavelets and Multiresolution Analysis / 400 \\ 10.4 Spline Wavelets / 405 \\ 10.5 A Variant of Construction of Orthonormal Wavelets / 407 \\ Exercises 10.5 / 408 \\ References / 408 \\ 11 Jacobians of Matrix Transformations / 409 \\ 11.0 Introduction / 409 \\ 11.1 Jacobians of Linear Matrix Transformations / 410 \\ Exercises 11.1 / 415 \\ 11.2 Jacobians in Some Nonlinear Transformations / 417 \\ Exercises 11.2 / 421 \\ 11.3 Transformations Involving Orthonormal Matrices / 423 \\ Exercises 11.3 / 426 \\ References / 428 \\ 12 Special Functions of Matrix Argument / 429 \\ 12.0 Introduction / 429 \\ 12.1 Real Matrix-Variate Scalar Functions / 429 \\ 12.1.1 Real matrix-variate gamma / 430 \\ 12.1.2 Real matrix-variate gamma density / 430 \\ Exercises 12.1 / 434 \\ 12.2 The Laplace Transform in the Matrix Case / 435 \\ 12.2.1 A convolution property for Laplace transforms / 436 \\ Exercises 12.2 / 439 \\ 12.3 Hypergeometric Functions with Matrix Argument / 439 \\ 12.3.1 Hypergeometric function through Laplace transform / 440 \\ 12.3.2 Hypergeometric function through zonal polynomials / 441 \\ 12.3.3 Hypergeometric functions through $M$-transforms / 443 \\ 12.3.4 A convolution theorem for $M$-transforms / 445 \\ Exercises 12.3 / 446 \\ 12.4 A Pathway Model / 448 \\ 12.4.1 The pathway density / 448 \\ 12.4.2 A general density / 451 \\ 12.4.3 Arbitrary moments / 451 \\ 12.4.4 Quadratic forms / 452 \\ 12.4.5 Generalized quadratic form / 452 \\ 12.4.6 Applications to random volumes / 453 \\ Exercises 12.4 / 453 \\ References / 454 \\ Author Index / 457 \\ Subject Index / 461

entry:
@Book{Mathai:2008:SFA,
  author =       "A. M. Mathai and H. J. Haubold",
  title =        "Special Functions for Applied Scientists",
  publisher =    "Springer Science+Business Media",
  address =      "New York, NY, USA",
  pages =        "xxv + 464",
  year =         "2008",
  DOI =          "https://doi.org/10.1007/978-0-387-75894-7",
  ISBN =         "0-387-75893-3",
  ISBN-13 =      "978-0-387-75893-0",
  LCCN =         "QA351 .M37X 2008",
  bibdate =      "Sat Oct 30 17:02:02 MDT 2010",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 prodorbis.library.yale.edu:7090/voyager",
  acknowledgement = ack-nhfb,
  shorttableofcontents = "Basic ideas of special functions and
                 statistical distributions \\
                 Mittag-Leffler functions and fractional calculus \\
                 An introduction to q-series \\
                 Ramanujan's theories of theta and elliptic functions
                 \\
                 Lie group and special functions \\
                 Applications to stochastic process and time series \\
                 Applications to density estimation \\
                 Applications to order statistics \\
                 Applications to astrophysics problems \\
                 An introduction to wavelet analysis \\
                 Jacobians of matrix formations \\
                 Special functions of matrix argument",
  subject =      "Functions, Special; Fractional calculus; Wavelets
                 (Mathematics)",
  tableofcontents = "1 Basic Ideas of Special Functions and Statistical
                 Distributions / 1 \\
                 1.0 Introduction / 1 \\
                 1.1 Gamma Function / 3 \\
                 1.1.1 Some basic properties of gamma functions / 4 \\
                 1.1.2 Wedge product and Jacobians of transformations /
                 6 \\
                 1.1.3 Multiplication formula for gamma functions / 8
                 \\
                 1.1.4 Asymptotic formula for a gamma function / 9 \\
                 1.1.5 Bernoulli polynomials / 9 \\
                 1.1.6 Some basic properties of generalized Bernoulli
                 polynomials / 9 \\
                 1.1.7 The first three generalized Bernoulli polynomials
                 / 10 \\
                 Exercises 1.1 / 11 \\
                 1.2 The Psi and Zeta Functions / 12 \\
                 1.2.1 Generalized zeta function / 12 \\
                 Exercises 1.2 / 13 \\
                 1.3 Integral Transforms / 14 \\
                 1.3.1 Mellin transform / 14 \\
                 1.3.2 Laplace transform / 17 \\
                 Exercises 1.3 / 18 \\
                 1.4 Some Statistical Preliminaries / 19 \\
                 Exercises 1.4 / 26 \\
                 1.5 Some Properties of Random Variables / 28 \\
                 1.5.1 Multivariate analogues / 30 \\
                 1.5.2 Marginal and conditional densities / 31 \\
                 Exercises 1.5 / 33 \\
                 1.6 Beta and Related Functions / 34 \\
                 1.6.1 Dirichlet integrals and Dirichlet densities / 37
                 \\
                 Exercises 1.6 / 42 \\
                 1.7 Hypergeometric Series / 42 \\
                 1.7.1 Evaluation of some contour integrals / 45 \\
                 1.7.2 Residues when several gammas are involved / 46
                 \\
                 Exercises 1.7 / 48 \\
                 1.8 Meijer's $G$-function / 49 \\
                 Exercises 1.8 / 54 \\
                 1.9 The $H$-function / 54 \\
                 Exercises 1.9 / 57 \\
                 1.10 Lauricella Functions and Appell's Functions / 58
                 \\
                 1.10.1 Some properties of $f_A$ / 59 \\
                 1.10.2 Some properties of $f_B$ / 60 \\
                 1.10.3 Some properties of $f_C$ / 60 \\
                 1.10.4 Some properties of $f_D$ / 61 \\
                 Exercises 1.10 / 64 \\
                 1.11 Special Functions as Solutions of Differential
                 Equations and Applications / 64 \\
                 1.11.0 Introduction / 64 \\
                 1.11.1 Sine, cosine and exponential functions / 64 \\
                 Exercises 1.11 / 66 \\
                 1.11.2 Linear second order differential equations / 66
                 \\
                 1.11.3 Hypergeometric function / 67 \\
                 Exercises 1.11 / 69 \\
                 1.11.4 Confluent hypergeometric function / 70 \\
                 Exercises 1.11 / 71 \\
                 1.11.5 Hermite polynomials / 71 \\
                 Exercises 1.11 / 72 \\
                 1.11.6 Bessel functions / 72 \\
                 Exercises 1.11 / 73 \\
                 1.11.7 Laguerre polynomial / 74 \\
                 1.11.8 Legendre polynomial / 74 \\
                 Exercises 1.11 / 74 \\
                 1.11.9 Generalized hypergeometric function / 75 \\
                 1.11.10 $G$-function / 76 \\
                 Exercises 1.11 / 77 \\
                 References / 77 \\
                 2 Mittag-Leffler Functions and Fractional Calculus / 79
                 \\
                 2.0 Introduction / 79 \\
                 2.1 Mittag-Leffler Function / 79 \\
                 Revision Exercises 2.1 / 81 \\
                 2.2 Basic Properties of Mittag-Leffler Function / 82
                 \\
                 2.2.1 Mittag-Leffler functions of rational order / 84
                 \\
                 2.2.2 Euler transform of Mittag-Leffler function / 84
                 \\
                 2.2.3 Laplace transform of Mittag-Leffler function / 85
                 \\
                 2.2.4 Application of Laplace transform / 87 \\
                 2.2.5 Mittag-Leffler functions and the $H$-function /
                 88 \\
                 Exercises 2.2 / 90 \\
                 2.3 Generalized Mittag-Leffler Function / 91 \\
                 2.3.1 Special cases of $E_{\beta, \gamma}^\delta (z)$ /
                 92 \\
                 2.3.2 Mellin--Barnes integral representation / 92 \\
                 2.3.3 Relations with the H-function and Wright function
                 / 92 \\
                 2.3.4 Cases of reducibility / 93 \\
                 2.3.5 Differentiation of generalized Mittag-Leffler
                 function / 94 \\
                 2.3.6 Integral property of generalized Mittag-Leffler
                 function / 94 \\
                 2.3.7 Integral transform of $E_{\beta, \gamma}^\delta
                 (z)$ / 95 \\
                 Exercises 2.3 / 96 \\
                 2.4 Fractional Integrals / 97 \\
                 2.4.1 Riemann--Liouville fractional integrals of
                 arbitrary order / 98 \\
                 2.4.2 Riemann--Liouville fractional integrals of order
                 $\alpha$ / 99 \\
                 2.4.3 Basic properties of fractional integrals / 100
                 \\
                 2.4.4 A useful integral / 101 \\
                 2.4.5 The Weyl integral / 103 \\
                 2.4.6 Basic properties of Weyl integral / 104 \\
                 Exercises 2.4 / 105 \\
                 2.4.7 Laplace transform of the fractional integral /
                 107 \\
                 2.4.8 Laplace transform of the fractional derivative /
                 107 \\
                 2.4.9 Laplace transform of Caputo derivative / 108 \\
                 Exercises 2.4 / 109 \\
                 2.5 Mellin Transform of the Fractional Integrals and
                 the Fractional Derivatives / 109 \\
                 2.5.1 Mellin transform / 109 \\
                 2.5.2 Mellin transform of the fractional integral / 110
                 \\
                 2.5.3 Mellin transform of the fractional derivative /
                 111 \\
                 Exercises 2.5 / 111 \\
                 2.6 Kober Operators / 111 \\
                 Exercises 2.6 / 113 \\
                 2.7 Generalized Kober Operators / 114 \\
                 Exercises 2.7 / 118 \\
                 2.8 Compositions of Riemann--Liouville Fractional
                 Calculus Operators and Generalized Mittag-Leffler
                 Functions / 121 \\
                 2.8.1 Composition relations between R-L operators and
                 $E_{\beta, \gamma}^\delta (z)$ / 121 \\
                 Exercises 2.8 / 126 \\
                 2.9 Fractional Differential Equations / 126 \\
                 2.9.1 Fractional relaxation / 127 \\
                 Exercises 2.9 / 130 \\
                 2.9.2 Fractional diffusion / 131 \\
                 Exercises 2.9 / 132 \\
                 References / 133 \\
                 3 An Introduction to $q$-Series / 135 \\
                 3.0 Introduction / 135 \\
                 3.1 Hypergeometric Series / 135 \\
                 Exercises 3.1 / 136 \\
                 3.2 Basic Hypergeometric Series ($q$-Series) / 138 \\
                 Exercises 3.2 / 140 \\
                 3.2.1 The $q$-binomial theorem / 142 \\
                 Exercises 3.2 / 146 \\
                 3.2.2 The $q$-binomial coefficients / 146 \\
                 Exercises 3.2 / 148 \\
                 3.3 $q$-Calculus / 149 \\
                 Exercises 3.3 / 151 \\
                 3.4 The $q$-Gamma and $q$-Beta Functions / 151 \\
                 3.5 Transformation and Summation Formulas for
                 $q$-Series / 152 \\
                 Exercises 3.5 / 154 \\
                 3.6 Jacobi's Triple Product and Rogers--Ramanujan
                 Identities / 154 \\
                 References / 157 \\
                 4 Ramanujan's Theories of Theta and Elliptic Functions
                 / 159 \\
                 4.0 Introduction / 159 \\
                 4.1 Ramanujan's Theory of Classical Theta Functions /
                 159 \\
                 4.1.1 Series definition and additive results / 159 \\
                 4.2 Ramanujan's $_1\Psi_1$ Summation Formula and
                 Multiplicative Results for Theta Functions / 163 \\
                 4.3 Modular Equations / 170 \\
                 4.4 Inversion Formulas and Evaluations / 172 \\
                 4.5 Modular Identities (Classical Theory) / 175 \\
                 4.6 Ramanujan's Theory of Cubic Theta Functions / 177
                 \\
                 4.6.1 The cubic theta functions / 177 \\
                 4.6.2 Inversion formulas and evaluations (cubic theory)
                 / 179 \\
                 4.6.3 Triplication and trimediation formulas / 182 \\
                 4.6.4 Further evaluations / 183 \\
                 4.6.5 Evaluations of Ramanujan--Eisenstein series ($L$,
                 $M$, $N$ or $P$, $Q$, $R$) / 185 \\
                 4.6.6 The cubic analogue of the Jacobian elliptic
                 functions / 186 \\
                 Test on Ramanujan's work / 187 \\
                 4.7 The One-variable Cubic Theta Functions / 188 \\
                 4.7.1 Cubic theta functions and some properties / 189
                 \\
                 4.7.2 Product representations for $b(q)$ and $c(q)$ /
                 190 \\
                 4.7.3 The cubic analogue of Jacobi's quartic modular
                 equations / 191 \\
                 4.8 The Two-variable Cubic Theta Functions / 193 \\
                 4.8.1 Series definitions and some properties / 193 \\
                 4.8.2 Product representations for $b(q,z)$ and $c(q,z)$
                 / 195 \\
                 4.8.3 A two-variable cubic counterpart of Jacobi's
                 quartic modular equation / 200 \\
                 4.9 The Three-variable Cubic Theta Functions / 200 \\
                 4.9.1 Unification of one and two-variable cubic theta
                 functions / 200 \\
                 Exercises 4.9 / 200 \\
                 4.9.2 Generalization of Hirschhorn--Garvan--Borwein
                 identity / 202 \\
                 4.9.3 Laurent's expansions for two-parameter cubic
                 theta functions / 205 \\
                 References / 209 \\
                 5 Lie Group and Special Functions / 211 \\
                 5.1 General Introduction to Group Theory / 211 \\
                 5.1.1 Isomorphisms / 213 \\
                 5.1.2 Symmetry groups / 213 \\
                 5.1.3 Isometries of the Euclidean plane / 214 \\
                 5.1.4 Finite groups of motion / 215 \\
                 5.1.5 Discrete groups of motions / 216 \\
                 5.2 Lie Group and Special Functions / 217 \\
                 5.2.1 Subspace of a vector space / 218 \\
                 5.3 Lie Algebra / 219 \\
                 5.4 Representations of Lie Algebra / 222 \\
                 Exercises 5.4 / 225 \\
                 5.5 Special Functions / 226 \\
                 5.5.1 Gauss hypergeometric function / 226 \\
                 5.5.2 Differential equation satisfied by $_2F_1$ / 228
                 \\
                 5.5.3 Integral representation of
                 $_F_q(\alpha_1,\alpha_2, \ldots{}, \alpha_p \beta_1,
                 \beta_2, \ldots{}, \beta_p; z)$ / 230 \\
                 5.6 Laguerre Polynomial $L^{(\alpha)}_n(x)$ / 231 \\
                 5.6.1 Laguerre polynomial and Lie algebra / 232 \\
                 Exercises 5.6 / 232 \\
                 5.7 Helmholtz Equation / 234 \\
                 5.7.1 Helmholtz equation in three variables / 238 \\
                 5.8 Lie Group / 241 \\
                 5.8.1 Basis of the Lie algebra of the Lie group SL(2) /
                 242 \\
                 Exercises 5.8 / 243 \\
                 Test on Lie Theory and Special Functions / 244 \\
                 6 Applications to Stochastic Process and Time Series /
                 247 \\
                 6.0 Introduction / 247 \\
                 6.1 Stochastic Processes / 247 \\
                 6.1.1 Classical types of stochastic processes / 252 \\
                 6.1.2 Processes with stationary independent increments
                 / 252 \\
                 6.1.3 Stationary processes / 253 \\
                 6.1.4 Gaussian processes and stationarity / 253 \\
                 6.1.5 Brownian processes / 257 \\
                 6.1.6 Markov chains / 258 \\
                 Exercises 6.1 / 261 \\
                 6.2 Modern Concepts in Distribution Theory / 262 \\
                 6.2.1 Introduction / 262 \\
                 6.2.2 Geometric infinite divisibility / 263 \\
                 6.2.3 Bernstein functions / 264 \\
                 6.2.4 Self-decomposability / 265 \\
                 6.2.5 Stable distributions / 265 \\
                 6.2.6 Geometrically strictly stable distributions / 266
                 \\
                 6.2.7 Mittag-Leffler distribution / 266 \\
                 6.2.8 $\alpha$-Laplace distribution / 267 \\
                 6.2.9 Semi-Pareto distribution / 268 \\
                 Exercises 6.2 / 268 \\
                 6.3 Stationary Time Series Models / 269 \\
                 6.3.1 Introduction / 269 \\
                 6.3.2 Autoregressive models / 270 \\
                 6.3.3 A general solution / 270 \\
                 6.3.4 Extension to a $k$-th order autoregressive model
                 / 272 \\
                 6.3.5 Mittag-Leffler autoregressive structure / 273 \\
                 Exercises 6.3 / 274 \\
                 6.4 A Structural Relationship and New Processes / 275
                 \\
                 6.4.1 The TMLAR(1) process / 276 \\
                 6.4.2 The NEAR(1) model / 277 \\
                 6.4.3 New Mittag-Leffler autoregressive models / 278
                 \\
                 6.4.4 The NSMLAR(1) process / 280 \\
                 Exercises 6.4 / 281 \\
                 6.5 Tailed Processes / 281 \\
                 6.5.1 The exponential tailed autoregressive process
                 [ETAR(1)] / 282 \\
                 6.5.2 The Mittag-Leffler tailed autoregressive process
                 [MLTAR(1)] / 283 \\
                 Exercises 6.5 / 286 \\
                 6.6 Marshall--Olkin Weibull Time Series Models / 286
                 \\
                 6.6.1 Introduction / 286 \\
                 6.6.2 Marshall--Olkin semi-Weibull distribution / 287
                 \\
                 6.6.3 An AR(1) model with MOSW marginal distribution /
                 289 \\
                 6.6.4 Marshall--Olkin generalized Weibull distribution
                 / 291 \\
                 6.6.5 An AR(1) model with MOGW marginal distribution /
                 292 \\
                 6.6.6 Case study / 293 \\
                 Exercises 6.6 / 293 \\
                 References / 294 \\
                 7 Applications to Density Estimation / 297 \\
                 7.0 Density Estimation and Orthogonal Polynomials / 297
                 \\
                 7.1 Introduction / 297 \\
                 7.2 Approximants Based on Legendre Polynomials / 299
                 \\
                 7.3 Approximants Based on Laguerre Polynomials / 301
                 \\
                 7.4 A Unified Methodology / 304 \\
                 7.5 Approximants Expressed in Terms of Orthogonal
                 Polynomials / 305 \\
                 7.5.1 Approximants expressed in terms of Laguerre
                 polynomials / 306 \\
                 7.5.2 Approximants expressed in terms of Legendre
                 polynomials / 306 \\
                 7.5.3 Approximants expressed in terms of Jacobi
                 polynomials / 307 \\
                 7.5.4 Approximants expressed in terms of Hermite
                 polynomials / 307 \\
                 References / 308 \\
                 8 Applications to Order Statistics / 311 \\
                 8.0 Introduction / 311 \\
                 8.1 Distribution Function / 311 \\
                 8.1.1 Density of the $r$-th order statistic / 313 \\
                 8.1.2 Joint distribution function of two order
                 statistics / 314 \\
                 8.1.3 Joint density of two order statistics / 314 \\
                 8.1.4 Moments of order statistics / 315 \\
                 8.1.5 Recurrence relations for moments / 317 \\
                 8.1.6 Recurrence relations on the product moments / 318
                 \\
                 8.1.7 Order statistics from symmetric distributions /
                 319 \\
                 8.2 Discrete Order Statistics / 319 \\
                 8.2.1 Probability function of discrete order statistics
                 / 320 \\
                 8.2.2 Joint probability function of two order
                 statistics / 321 \\
                 8.2.3 Bernoulli order statistics / 321 \\
                 8.3 Independent Random Variables / 322 \\
                 8.3.1 Distribution of a single order statistic / 322
                 \\
                 8.3.2 Joint distribution of two order statistics / 324
                 \\
                 Test on Order Statistics / 325 \\
                 8.4 On Concomitants of Order Statistics / 326 \\
                 8.4.1 Application of concomitants of order statistics /
                 326 \\
                 8.4.2 Application in estimation / 328 \\
                 8.4.3 Concomitants of record values and estimation
                 problems / 333 \\
                 References / 339 \\
                 9 Applications to Astrophysics Problems / 341 \\
                 9.0 Introduction / 341 \\
                 9.1 Entropy: Boltzmann, Planck, and Einstein on W / 342
                 \\
                 9.1.1 Entropic functional / 342 \\
                 9.1.2 Entropy and probability / 342 \\
                 9.1.3 Boltzmann--Gibbs / 343 \\
                 9.2 Gravitationally Stabilized Fusion Reactor: The Sun
                 / 345 \\
                 9.2.1 Internal solar structure / 345 \\
                 9.2.2 Solar fusion plasma / 349 \\
                 9.2.3 Estimation of central temperature in the Sun /
                 349 \\
                 9.3 Crucial Astrophysical Experiments: Data Analysis /
                 351 \\
                 9.3.1 The experiments / 351 \\
                 9.3.2 Analysis of the time series / 352 \\
                 9.4 Fundamental Equations for Nonequilibrium Processes
                 / 355 \\
                 9.4.1 Chapman--Kolmogorov equation / 355 \\
                 9.4.2 Master equation / 355 \\
                 9.4.3 Fokker--Planck equation / 356 \\
                 9.4.4 Langevin equation / 357 \\
                 9.4.5 Reaction-diffusion equation / 357 \\
                 9.5 Fractional Calculus / 360 \\
                 9.6 Nonextensive Statistical Mechanics / 362 \\
                 9.7 Standard and Fractional Reaction / 363 \\
                 9.7.1 Boltzmann--Gibbs statistical mechanics / 363 \\
                 9.7.2 Generalized Boltzmann--Gibbs statistical
                 mechanics / 366 \\
                 9.7.3 Fractional reaction / 369 \\
                 9.7.4 Thermonuclear reaction coefficient / 371 \\
                 9.8 Standard and Fractional Diffusion / 380 \\
                 9.8.1 Fick's first law of diffusion / 380 \\
                 9.8.2 Einstein's approach to diffusion / 381 \\
                 9.8.3 Fractional diffusion / 381 \\
                 9.8.4 Spatio-temporal pattern formation / 383 \\
                 References / 384 \\
                 10 An Introduction to Wavelet Analysis / 389 \\
                 10.0 Introduction / 389 \\
                 10.1 Fourier Analysis to Wavelet Analysis / 390 \\
                 10.2 Construction of Orthonormal Wavelets / 393 \\
                 10.3 Classification of Wavelets and Multiresolution
                 Analysis / 400 \\
                 10.4 Spline Wavelets / 405 \\
                 10.5 A Variant of Construction of Orthonormal Wavelets
                 / 407 \\
                 Exercises 10.5 / 408 \\
                 References / 408 \\
                 11 Jacobians of Matrix Transformations / 409 \\
                 11.0 Introduction / 409 \\
                 11.1 Jacobians of Linear Matrix Transformations / 410
                 \\
                 Exercises 11.1 / 415 \\
                 11.2 Jacobians in Some Nonlinear Transformations / 417
                 \\
                 Exercises 11.2 / 421 \\
                 11.3 Transformations Involving Orthonormal Matrices /
                 423 \\
                 Exercises 11.3 / 426 \\
                 References / 428 \\
                 12 Special Functions of Matrix Argument / 429 \\
                 12.0 Introduction / 429 \\
                 12.1 Real Matrix-Variate Scalar Functions / 429 \\
                 12.1.1 Real matrix-variate gamma / 430 \\
                 12.1.2 Real matrix-variate gamma density / 430 \\
                 Exercises 12.1 / 434 \\
                 12.2 The Laplace Transform in the Matrix Case / 435 \\
                 12.2.1 A convolution property for Laplace transforms /
                 436 \\
                 Exercises 12.2 / 439 \\
                 12.3 Hypergeometric Functions with Matrix Argument /
                 439 \\
                 12.3.1 Hypergeometric function through Laplace
                 transform / 440 \\
                 12.3.2 Hypergeometric function through zonal
                 polynomials / 441 \\
                 12.3.3 Hypergeometric functions through $M$-transforms
                 / 443 \\
                 12.3.4 A convolution theorem for $M$-transforms / 445
                 \\
                 Exercises 12.3 / 446 \\
                 12.4 A Pathway Model / 448 \\
                 12.4.1 The pathway density / 448 \\
                 12.4.2 A general density / 451 \\
                 12.4.3 Arbitrary moments / 451 \\
                 12.4.4 Quadratic forms / 452 \\
                 12.4.5 Generalized quadratic form / 452 \\
                 12.4.6 Applications to random volumes / 453 \\
                 Exercises 12.4 / 453 \\
                 References / 454 \\
                 Author Index / 457 \\
                 Subject Index / 461",
}


bibtype: article
filename: elefunt.bib
label: Nam:2008:PAE
author: Byeong-Gyu Nam and Hyejung Kim and Hoi-Jun Yoo
journal: j-IEEE-TRANS-COMPUT
volume: 57
number: 4
pages: 490--504
month: apr
monthnumber: 4
year: 2008
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2008.12
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Mon Jul 4 12:17:40 MDT 2011
bibtimestamp: 2011.07.04 12:17:40 MDT
URL: https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4432232
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: Byeong-Gyu Nam and Hyejung Kim and Hoi-Jun Yoo
title:
Power and Area-Efficient Unified Computation of Vector and Elementary Functions for Handheld $3$D Graphics Systems

abstract:
A unified computation method of vector and elementary functions is proposed for handheld 3D graphics systems. It unifies vector operations like vector multiply, multiply-and-add, divide, divide-by-square-root, and dot product and elementary functions like trigonometric, inverse trigonometric, hyperbolic, inverse hyperbolic, power ($ x^y $ with two variables), and logarithm to an arbitrary base into a single four-way arithmetic platform. A number system called the fixed-point hybrid number system (FXP-HNS), which combines the fixed-point number system (FXP) and the logarithmic number system (LNS), is proposed for the power and area-efficient unification. Power and area-efficient logarithmic and antilogarithmic conversion schemes are also proposed for the data conversions between fixed-point and logarithmic numbers in the FXP-HNS and achieve 0.41 percent and 0.08 percent maximum conversion errors, respectively. The unified arithmetic unit based on the proposed schemes is presented with less than 6.3 percent operation error. Its fully pipelined architecture achieves single-cycle throughput with maximum four-cycle latency for all of the supported operations. Comparison results show that the proposed arithmetic unit achieves 30 percent power and 10.9 percent area reductions and runs two times faster than the previous approach.

rawabstract:
A unified computation method of vector and elementary functions is proposed for handheld 3D graphics systems. It unifies vector operations like vector multiply, multiply-and-add, divide, divide-by-square-root, and dot product and elementary functions like trigonometric, inverse trigonometric, hyperbolic, inverse hyperbolic, power ($ x^y $ with two variables), and logarithm to an arbitrary base into a single four-way arithmetic platform. A number system called the fixed-point hybrid number system (FXP-HNS), which combines the fixed-point number system (FXP) and the logarithmic number system (LNS), is proposed for the power and area-efficient unification. Power and area-efficient logarithmic and antilogarithmic conversion schemes are also proposed for the data conversions between fixed-point and logarithmic numbers in the FXP-HNS and achieve 0.41 percent and 0.08 percent maximum conversion errors, respectively. The unified arithmetic unit based on the proposed schemes is presented with less than 6.3 percent operation error. Its fully pipelined architecture achieves single-cycle throughput with maximum four-cycle latency for all of the supported operations. Comparison results show that the proposed arithmetic unit achieves 30 percent power and 10.9 percent area reductions and runs two times faster than the previous approach.

rawtitle:
Power and Area-Efficient Unified Computation of Vector and Elementary Functions for Handheld {$3$D} Graphics Systems

entry:
@Article{Nam:2008:PAE,
  author =       "Byeong-Gyu Nam and Hyejung Kim and Hoi-Jun Yoo",
  title =        "Power and Area-Efficient Unified Computation of Vector
                 and Elementary Functions for Handheld {$3$D} Graphics
                 Systems",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "57",
  number =       "4",
  pages =        "490--504",
  month =        apr,
  year =         "2008",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2008.12",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Mon Jul 4 12:17:40 MDT 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib",
  URL =          "https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4432232",
  abstract =     "A unified computation method of vector and elementary
                 functions is proposed for handheld 3D graphics systems.
                 It unifies vector operations like vector multiply,
                 multiply-and-add, divide, divide-by-square-root, and
                 dot product and elementary functions like
                 trigonometric, inverse trigonometric, hyperbolic,
                 inverse hyperbolic, power ($ x^y $ with two variables),
                 and logarithm to an arbitrary base into a single
                 four-way arithmetic platform. A number system called
                 the fixed-point hybrid number system (FXP-HNS), which
                 combines the fixed-point number system (FXP) and the
                 logarithmic number system (LNS), is proposed for the
                 power and area-efficient unification. Power and
                 area-efficient logarithmic and antilogarithmic
                 conversion schemes are also proposed for the data
                 conversions between fixed-point and logarithmic numbers
                 in the FXP-HNS and achieve 0.41 percent and 0.08
                 percent maximum conversion errors, respectively. The
                 unified arithmetic unit based on the proposed schemes
                 is presented with less than 6.3 percent operation
                 error. Its fully pipelined architecture achieves
                 single-cycle throughput with maximum four-cycle latency
                 for all of the supported operations. Comparison results
                 show that the proposed arithmetic unit achieves 30
                 percent power and 10.9 percent area reductions and runs
                 two times faster than the previous approach.",
  acknowledgement = ack-nhfb,
  ajournal =     "IEEE Trans. Comput.",
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
}


bibtype: article
filename: elefunt.bib
label: Paszkowski:2008:CAO
author: Stefan Paszkowski
title: Convergence acceleration of orthogonal series
journal: j-NUMER-ALGORITHMS
volume: 47
number: 1
pages: 35--62
month: jan
monthnumber: 1
year: 2008
CODEN: NUALEG
DOI: https://doi.org/10.1007/s11075-007-9146-7
ISSN: 1017-1398 (print), 1572-9265 (electronic) OR 10171398 (print), 15729265 (electronic)
ISSNL: 1017-1398 OR 10171398
MRclass: subject classification (2000); 33C45; 42A32; 42C10; 42C20; 65B10
bibdate: Tue Jul 8 19:14:30 MDT 2008
bibtimestamp: 2008.07.08 19:14:30 MDT
URL: http://www.springerlink.com/openurl.asp?genre=article&issn=1017-1398&volume=47&issue=1&spage=35
fjournal: Numerical Algorithms
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/11075
rawauthor: Stefan Paszkowski
rawtitle: Convergence acceleration of orthogonal series
bibsource:
http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1017-1398&volume=47&issue=1; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

keywords:
Convergence acceleration; convergence acceleration; Orthogonal polynomials; Orthogonal series; Trigonometric series

entry:
@Article{Paszkowski:2008:CAO,
  author =       "Stefan Paszkowski",
  title =        "Convergence acceleration of orthogonal series",
  journal =      j-NUMER-ALGORITHMS,
  volume =       "47",
  number =       "1",
  pages =        "35--62",
  month =        jan,
  year =         "2008",
  CODEN =        "NUALEG",
  DOI =          "https://doi.org/10.1007/s11075-007-9146-7",
  ISSN =         "1017-1398 (print), 1572-9265 (electronic)",
  ISSN-L =       "1017-1398",
  MRclass =      "subject classification (2000); 33C45; 42A32; 42C10;
                 42C20; 65B10",
  bibdate =      "Tue Jul 8 19:14:30 MDT 2008",
  bibsource =    "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1017-1398&volume=47&issue=1;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.springerlink.com/openurl.asp?genre=article&issn=1017-1398&volume=47&issue=1&spage=35",
  acknowledgement = ack-nhfb,
  fjournal =     "Numerical Algorithms",
  journal-URL =  "http://link.springer.com/journal/11075",
  keywords =     "Convergence acceleration; convergence acceleration;
                 Orthogonal polynomials; Orthogonal series;
                 Trigonometric series",
}


bibtype: article
filename: elefunt.bib
label: Pinchon:2008:NEL
author: Didier Pinchon and Philip E. Hoggan and Frank E. Harris
title: A new expansion of the leaky aquifer function
journal: j-IJQC
volume: 108
number: 15
pages: 3042--3046
year: 2008
CODEN: IJQCB2
DOI: https://doi.org/10.1002/qua.21448; https://doi.org/10.1002/qua.21835
ISSN: 0020-7608 (print), 1097-461X (electronic) OR 00207608 (print), 1097461X (electronic)
ISSNL: 0020-7608 OR 00207608
MRclass: 86A05 80A20 33C10 82B80
bibdate: Sat Oct 1 14:02:23 MDT 2011
bibtimestamp: 2011.10.01 14:02:23 MDT
fjournal: International Journal of Quantum Chemistry
ZMnumber: Zbl 1189.86005
acknowledgement: ack-nhfb
ajournal: Int. J. Quantum Chem.
journalURL: http://www.interscience.wiley.com/jpages/0020-7608/
onlinedate: 4 Aug 2008
rawauthor: Didier Pinchon and Philip E. Hoggan and Frank E. Harris
rawtitle: A new expansion of the leaky aquifer function
bibsource:
https://www.math.utah.edu/pub/bibnet/authors/h/harris-frank-e.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/ijqc2000.bib

entry:
@Article{Pinchon:2008:NEL,
  author =       "Didier Pinchon and Philip E. Hoggan and Frank E.
                 Harris",
  title =        "A new expansion of the leaky aquifer function",
  journal =      j-IJQC,
  volume =       "108",
  number =       "15",
  pages =        "3042--3046",
  month =        "????",
  year =         "2008",
  CODEN =        "IJQCB2",
  DOI =          "https://doi.org/10.1002/qua.21448;
                 https://doi.org/10.1002/qua.21835",
  ISSN =         "0020-7608 (print), 1097-461X (electronic)",
  ISSN-L =       "0020-7608",
  MRclass =      "86A05 80A20 33C10 82B80",
  bibdate =      "Sat Oct 1 14:02:23 MDT 2011",
  bibsource =    "https://www.math.utah.edu/pub/bibnet/authors/h/harris-frank-e.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/ijqc2000.bib",
  ZMnumber =     "Zbl 1189.86005",
  acknowledgement = ack-nhfb,
  ajournal =     "Int. J. Quantum Chem.",
  fjournal =     "International Journal of Quantum Chemistry",
  journal-URL =  "http://www.interscience.wiley.com/jpages/0020-7608/",
  onlinedate =   "4 Aug 2008",
}


bibtype: article
filename: elefunt.bib
label: Pineiro:2008:RDD
author: J.-A. Pineiro and J. D. Bruguera and F. Lamberti and P. Montuschi
title: A Radix-2 Digit-by-Digit Architecture for Cube Root
journal: j-IEEE-TRANS-COMPUT
volume: 57
number: 4
pages: 562--566
month: apr
monthnumber: 4
year: 2008
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2007.70848
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Mon Jul 4 12:17:41 MDT 2011
bibtimestamp: 2011.07.04 12:17:41 MDT
URL: https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4407683
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: J.-A. Pineiro and J. D. Bruguera and F. Lamberti and P. Montuschi
rawtitle: A Radix-2 Digit-by-Digit Architecture for Cube Root
abstract:
A radix-2 digit-recurrence algorithm and architecture for the computation of the cube root are presented in this paper. The original recurrence based on the concept of completing the cube is modified to allow an efficient implementation of the algorithm and the cycle time and area cost of the resulting architecture are estimated as 7.5 times the delay of a full adder and around 9,000 nand2 cells, respectively, for double-precision computations.

rawabstract:
A radix-2 digit-recurrence algorithm and architecture for the computation of the cube root are presented in this paper. The original recurrence based on the concept of completing the cube is modified to allow an efficient implementation of the algorithm and the cycle time and area cost of the resulting architecture are estimated as 7.5 times the delay of a full adder and around 9,000 nand2 cells, respectively, for double-precision computations.

entry:
@Article{Pineiro:2008:RDD,
  author =       "J.-A. Pineiro and J. D. Bruguera and F. Lamberti and
                 P. Montuschi",
  title =        "A Radix-2 Digit-by-Digit Architecture for Cube Root",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "57",
  number =       "4",
  pages =        "562--566",
  month =        apr,
  year =         "2008",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2007.70848",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Mon Jul 4 12:17:41 MDT 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib",
  URL =          "https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4407683",
  abstract =     "A radix-2 digit-recurrence algorithm and architecture
                 for the computation of the cube root are presented in
                 this paper. The original recurrence based on the
                 concept of completing the cube is modified to allow an
                 efficient implementation of the algorithm and the cycle
                 time and area cost of the resulting architecture are
                 estimated as 7.5 times the delay of a full adder and
                 around 9,000 nand2 cells, respectively, for
                 double-precision computations.",
  acknowledgement = ack-nhfb,
  ajournal =     "IEEE Trans. Comput.",
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
}


bibtype: inproceedings
filename: elefunt.bib
label: Piso:2008:NRA
author: D. Piso and J. D. Bruguera
editor: Luca Fanucci
title: A New Rounding Algorithm for Variable Latency Division and Square Root Implementations
publisher: pub-IEEE
address: pub-IEEE:adr
pages: 760--767
year: 2008
DOI: https://doi.org/10.1109/DSD.2008.28.
ISBN: 0-7695-3277-2 OR 0769532772
ISBN13: 978-0-7695-3277-6 OR 9780769532776
bibdate: Sun Dec 10 13:55:38 2023
bibtimestamp: 2023.12.10 13:55:38 ???
acknowledgement: ack-nhfb
rawauthor: D. Piso and J. D. Bruguera
raweditor: Luca Fanucci
rawtitle: A New Rounding Algorithm for Variable Latency Division and Square Root Implementations
booktitle:
Proceedings: 11th Euromicro Symposium on Digital Systems Design: Architectures, Methods and Tools (DSD 2008), Parma, Italy, September 3--5, 2008

abstract:
The aim of this work is to present a method for rounding quadratically converging algorithms that improves their performance. This method is able to reduce significantly the number of cases where the remainder calculation is necessary. It is based on previous methods and incorporates additional bits of the result approximation to be checked. This work includes the result of exhaustive simulations that permit us to measure exactly how many calculations are avoided. Using these simulations, it is concluded that the presented method is able to reduce by half the number of remainder calculations. Using adequate result approximations the remainder calculation is necessary in only 5% of the total cases

rawabstract:
The aim of this work is to present a method for rounding quadratically converging algorithms that improves their performance. This method is able to reduce significantly the number of cases where the remainder calculation is necessary. It is based on previous methods and incorporates additional bits of the result approximation to be checked. This work includes the result of exhaustive simulations that permit us to measure exactly how many calculations are avoided. Using these simulations, it is concluded that the presented method is able to reduce by half the number of remainder calculations. Using adequate result approximations the remainder calculation is necessary in only 5\% of the total cases

rawbooktitle:
Proceedings: {11th Euromicro Symposium on Digital Systems Design: Architectures, Methods and Tools (DSD 2008), Parma, Italy, September 3--5, 2008}

entry:
@InProceedings{Piso:2008:NRA,
  author =       "D. Piso and J. D. Bruguera",
  editor =       "Luca Fanucci",
  booktitle =    "Proceedings: {11th Euromicro Symposium on Digital
                 Systems Design: Architectures, Methods and Tools (DSD
                 2008), Parma, Italy, September 3--5, 2008}",
  title =        "A New Rounding Algorithm for Variable Latency Division
                 and Square Root Implementations",
  publisher =    pub-IEEE,
  address =      pub-IEEE:adr,
  pages =        "760--767",
  year =         "2008",
  DOI =          "https://doi.org/10.1109/DSD.2008.28.",
  ISBN =         "0-7695-3277-2",
  ISBN-13 =      "978-0-7695-3277-6",
  bibdate =      "Sun Dec 10 13:55:38 2023",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  abstract =     "The aim of this work is to present a method for
                 rounding quadratically converging algorithms that
                 improves their performance. This method is able to
                 reduce significantly the number of cases where the
                 remainder calculation is necessary. It is based on
                 previous methods and incorporates additional bits of
                 the result approximation to be checked. This work
                 includes the result of exhaustive simulations that
                 permit us to measure exactly how many calculations are
                 avoided. Using these simulations, it is concluded that
                 the presented method is able to reduce by half the
                 number of remainder calculations. Using adequate result
                 approximations the remainder calculation is necessary
                 in only 5\% of the total cases",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: elefunt.bib
label: Rodriguez-Henriquez:2008:LCB
author: F. Rodriguez-Henriquez and G. Morales-Luna and J. Lopez
title: Low-Complexity Bit-Parallel Square Root Computation over $ mathrm GF(2^m) $ for All Trinomials
journal: j-IEEE-TRANS-COMPUT
volume: 57
number: 4
pages: 472--480
month: apr
monthnumber: 4
year: 2008
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2007.70822
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Mon Jul 4 12:17:40 MDT 2011
bibtimestamp: 2011.07.04 12:17:40 MDT
URL: https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4358282
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: F. Rodriguez-Henriquez and G. Morales-Luna and J. Lopez
rawtitle: Low-Complexity Bit-Parallel Square Root Computation over {$ \mathrm {GF}(2^m) $} for All Trinomials
bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2000.bib

entry:
@Article{Rodriguez-Henriquez:2008:LCB,
  author =       "F. Rodriguez-Henriquez and G. Morales-Luna and J.
                 Lopez",
  title =        "Low-Complexity Bit-Parallel Square Root Computation
                 over {$ \mathrm {GF}(2^m) $} for All Trinomials",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "57",
  number =       "4",
  pages =        "472--480",
  month =        apr,
  year =         "2008",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2007.70822",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Mon Jul 4 12:17:40 MDT 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2000.bib",
  URL =          "https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4358282",
  acknowledgement = ack-nhfb,
  ajournal =     "IEEE Trans. Comput.",
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
}


bibtype: article
filename: elefunt.bib
label: Sablonniere:2008:BSH
author: Paul Sablonniere
title: B-splines and Hermite--Pade approximants to the exponential function
journal: j-J-COMPUT-APPL-MATH
volume: 219
number: 2
pages: 509--517
day: 1
month: oct
monthnumber: 10
year: 2008
CODEN: JCAMDI
ISSN: 0377-0427 (print), 1879-1778 (electronic) OR 03770427 (print), 18791778 (electronic)
ISSNL: 0377-0427 OR 03770427
bibdate: Sat Feb 25 13:13:26 MST 2017
bibtimestamp: 2017.02.25 13:13:26 MST
URL: http://www.sciencedirect.com/science/article/pii/S037704270700252X
fjournal: Journal of Computational and Applied Mathematics
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/03770427
rawauthor: Paul Sablonni{\`e}re
rawtitle: {B}-splines and {Hermite--Pad{\'e}} approximants to the exponential function
entry:
@Article{Sablonniere:2008:BSH,
  author =       "Paul Sablonni{\`e}re",
  title =        "{B}-splines and {Hermite--Pad{\'e}} approximants to
                 the exponential function",
  journal =      j-J-COMPUT-APPL-MATH,
  volume =       "219",
  number =       "2",
  pages =        "509--517",
  day =          "1",
  month =        oct,
  year =         "2008",
  CODEN =        "JCAMDI",
  ISSN =         "0377-0427 (print), 1879-1778 (electronic)",
  ISSN-L =       "0377-0427",
  bibdate =      "Sat Feb 25 13:13:26 MST 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2005.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S037704270700252X",
  acknowledgement = ack-nhfb,
  fjournal =     "Journal of Computational and Applied Mathematics",
  journal-URL =  "http://www.sciencedirect.com/science/journal/03770427",
}


bibtype: article
filename: elefunt.bib
label: Schreier:2008:OIR
author: Franz Schreier and Dieter Kohlert
journal: j-COMP-PHYS-COMM
volume: 179
number: 7
pages: 457--465
day: 1
month: oct
monthnumber: 10
year: 2008
CODEN: CPHCBZ
DOI: https://doi.org/10.1016/j.cpc.2008.04.012
ISSN: 0010-4655 (print), 1879-2944 (electronic) OR 00104655 (print), 18792944 (electronic)
ISSNL: 0010-4655 OR 00104655
bibdate: Mon Feb 13 23:42:36 MST 2012
bibtimestamp: 2012.02.13 23:42:36 MST
URL: http://www.sciencedirect.com/science/article/pii/S0010465508001495
fjournal: Computer Physics Communications
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00104655
rawauthor: Franz Schreier and Dieter Kohlert
title:
Optimized implementations of rational approximations --- a case study on the Voigt and complex error function

bibsource:
https://www.math.utah.edu/pub/tex/bib/compphyscomm2000.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

rawtitle:
Optimized implementations of rational approximations --- a case study on the {Voigt} and complex error function

entry:
@Article{Schreier:2008:OIR,
  author =       "Franz Schreier and Dieter Kohlert",
  title =        "Optimized implementations of rational approximations
                 --- a case study on the {Voigt} and complex error
                 function",
  journal =      j-COMP-PHYS-COMM,
  volume =       "179",
  number =       "7",
  pages =        "457--465",
  day =          "1",
  month =        oct,
  year =         "2008",
  CODEN =        "CPHCBZ",
  DOI =          "https://doi.org/10.1016/j.cpc.2008.04.012",
  ISSN =         "0010-4655 (print), 1879-2944 (electronic)",
  ISSN-L =       "0010-4655",
  bibdate =      "Mon Feb 13 23:42:36 MST 2012",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/compphyscomm2000.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0010465508001495",
  acknowledgement = ack-nhfb,
  fjournal =     "Computer Physics Communications",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00104655",
}


bibtype: article
filename: elefunt.bib
label: Segura:2008:IZC
author: Javier Segura
title: Interlacing of the zeros of contiguous hypergeometric functions
journal: j-NUMER-ALGORITHMS
volume: 49
number: 1--4
pages: 387--407
month: dec
monthnumber: 12
year: 2008
CODEN: NUALEG
ISSN: 1017-1398 (print), 1572-9265 (electronic) OR 10171398 (print), 15729265 (electronic)
ISSNL: 1017-1398 OR 10171398
bibdate: Mon May 17 15:07:19 MDT 2010
bibtimestamp: 2010.05.17 15:07:19 MDT
URL: http://www.springerlink.com/openurl.asp?genre=article&issn=1017-1398&volume=49&issue=1&spage=387
fjournal: Numerical Algorithms
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/11075
rawauthor: Javier Segura
rawtitle: Interlacing of the zeros of contiguous hypergeometric functions
bibsource:
http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1017-1398&volume=49&issue=1; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib

entry:
@Article{Segura:2008:IZC,
  author =       "Javier Segura",
  title =        "Interlacing of the zeros of contiguous hypergeometric
                 functions",
  journal =      j-NUMER-ALGORITHMS,
  volume =       "49",
  number =       "1--4",
  pages =        "387--407",
  month =        dec,
  year =         "2008",
  CODEN =        "NUALEG",
  ISSN =         "1017-1398 (print), 1572-9265 (electronic)",
  ISSN-L =       "1017-1398",
  bibdate =      "Mon May 17 15:07:19 MDT 2010",
  bibsource =    "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1017-1398&volume=49&issue=1;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib",
  URL =          "http://www.springerlink.com/openurl.asp?genre=article&issn=1017-1398&volume=49&issue=1&spage=387",
  acknowledgement = ack-nhfb,
  fjournal =     "Numerical Algorithms",
  journal-URL =  "http://link.springer.com/journal/11075",
}


bibtype: inproceedings
filename: elefunt.bib
label: Sima:2008:RAT
author: Mihai Sima and Michael McGuire and Scott Miller
title: Reconfigurable Array for Transcendental Functions Calculation
publisher: pub-IEEE
address: pub-IEEE:adr
pages: 49--56
month: dec
monthnumber: 12
year: 2008
DOI: https://doi.org/10.1109/fpt.2008.4762365
bibdate: Mon Nov 10 06:57:23 2025
bibsource: https://www.math.utah.edu/pub/tex/bib/cordic.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2025.11.10 06:57:23 ???
keywords: acos; asin; cosine; exp; log; sine; sqrt
acknowledgement: ack-nhfb
rawauthor: Mihai Sima and Michael McGuire and Scott Miller
rawtitle: Reconfigurable Array for Transcendental Functions Calculation
booktitle:
Proceedings of the 2005 International Conference on Engineering of Reconfigurable Systems and Algorithms (ERSA 2005)

rawbooktitle:
Proceedings of the {2005 International Conference on Engineering of Reconfigurable Systems and Algorithms (ERSA 2005)}

entry:
@InProceedings{Sima:2008:RAT,
  author =       "Mihai Sima and Michael McGuire and Scott Miller",
  booktitle =    "Proceedings of the {2005 International Conference on
                 Engineering of Reconfigurable Systems and Algorithms
                 (ERSA 2005)}",
  title =        "Reconfigurable Array for Transcendental Functions
                 Calculation",
  publisher =    pub-IEEE,
  address =      pub-IEEE:adr,
  pages =        "49--56",
  month =        dec,
  year =         "2008",
  DOI =          "https://doi.org/10.1109/fpt.2008.4762365",
  bibdate =      "Mon Nov 10 06:57:23 2025",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/cordic.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
  keywords =     "acos; asin; cosine; exp; log; sine; sqrt",
}


bibtype: misc
filename: elefunt.bib
label: Steele:2008:FPSb
author: Guy L. Steele Jr.
title: Floating point square root provider with embedded status information
pages: 49--56
day: 30
month: sep
monthnumber: 9
year: 2008
bibdate: Tue Dec 23 15:06:43 2008
bibtimestamp: 2008.12.23 15:06:43 ???
URL: http://www.patentstorm.us/patents/7430576/fulltext.html
acknowledgement: ack-nhfb
howpublished: US Patent 7430576
rawauthor: Guy L. {Steele Jr.}
rawtitle: Floating point square root provider with embedded status information
abstract:
A system for providing a floating point square root comprises an analyzer circuit configured to determine a first status of a first floating point operand based upon data within the first floating point operand. In addition, the system comprises a results circuit coupled to the analyzer circuit. The results circuit is configured to assert a resulting floating point operand containing the square root of the first floating point operand and a resulting status embedded within the resulting floating point operand.

rawabstract:
A system for providing a floating point square root comprises an analyzer circuit configured to determine a first status of a first floating point operand based upon data within the first floating point operand. In addition, the system comprises a results circuit coupled to the analyzer circuit. The results circuit is configured to assert a resulting floating point operand containing the square root of the first floating point operand and a resulting status embedded within the resulting floating point operand.

entry:
@Misc{Steele:2008:FPSb,
  author =       "Guy L. {Steele Jr.}",
  title =        "Floating point square root provider with embedded
                 status information",
  howpublished = "US Patent 7430576",
  day =          "30",
  month =        sep,
  year =         "2008",
  bibdate =      "Tue Dec 23 15:06:43 2008",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://www.patentstorm.us/patents/7430576/fulltext.html",
  abstract =     "A system for providing a floating point square root
                 comprises an analyzer circuit configured to determine a
                 first status of a first floating point operand based
                 upon data within the first floating point operand. In
                 addition, the system comprises a results circuit
                 coupled to the analyzer circuit. The results circuit is
                 configured to assert a resulting floating point operand
                 containing the square root of the first floating point
                 operand and a resulting status embedded within the
                 resulting floating point operand.",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: elefunt.bib
label: Vepstas:2008:EAA
author: Linas Vepstas
journal: j-NUMER-ALGORITHMS
volume: 47
number: 3
pages: 211--252
month: mar
monthnumber: 3
year: 2008
CODEN: NUALEG
DOI: https://doi.org/10.1007/s11075-007-9153-8
ISSN: 1017-1398 (print), 1572-9265 (electronic) OR 10171398 (print), 15729265 (electronic)
ISSNL: 1017-1398 OR 10171398
bibdate: Mon May 17 15:49:34 MDT 2010
bibtimestamp: 2010.05.17 15:49:34 MDT
URL: http://www.springerlink.com/openurl.asp?genre=article&issn=1017-1398&volume=47&issue=3&spage=211
fjournal: Numerical Algorithms
keywords: convergence acceleration
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/11075
rawauthor: Linas Vepstas
title:
An efficient algorithm for accelerating the convergence of oscillatory series, useful for computing the polylogarithm and Hurwitz zeta functions

bibsource:
http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1017-1398&volume=47&issue=3; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

abstract:
This paper sketches a technique for improving the rate of convergence of a general oscillatory sequence, and then applies this series acceleration algorithm to the polylogarithm and the Hurwitz zeta function. As such, it may be taken as an extension of the techniques given by Borwein's ``An efficient algorithm for computing the Riemann zeta function'' by Borwein for computing the Riemann zeta function, to more general series. The algorithm provides a rapid means of evaluating $ operatorname Li_s(z) $ for general values of complex $s$ and a kidney-shaped region of complex $z$ values given by $ |z^2 / (z - 1)| < 4 $. By using the duplication formula and the inversion formula, the range of convergence for the polylogarithm may be extended to the entire complex $z$-plane, and so the algorithms described here allow for the evaluation of the polylogarithm for all complex $s$ and $z$ values. Alternatively, the Hurwitz zeta can be very rapidly evaluated by means of an Euler Maclaurin series. The polylogarithm and the Hurwitz zeta are related, in that two evaluations of the one can be used to obtain a value of the other; thus, either algorithm can be used to evaluate either function. The Euler Maclaurin series is a clear performance winner for the Hurwitz zeta, while the Borwein algorithm is superior for evaluating the polylogarithm in the kidney-shaped region. Both algorithms are superior to the simple Taylor's series or direct summation. The primary, concrete result of this paper is an algorithm allows the exploration of the Hurwitz zeta in the critical strip, where fast algorithms are otherwise unavailable. A discussion of the monodromy group of the polylogarithm is included.

rawabstract:
This paper sketches a technique for improving the rate of convergence of a general oscillatory sequence, and then applies this series acceleration algorithm to the polylogarithm and the Hurwitz zeta function. As such, it may be taken as an extension of the techniques given by Borwein's ``An efficient algorithm for computing the Riemann zeta function'' by Borwein for computing the Riemann zeta function, to more general series. The algorithm provides a rapid means of evaluating $ \operatorname {Li}_s(z) $ for general values of complex $s$ and a kidney-shaped region of complex $z$ values given by $ |z^2 / (z - 1)| < 4 $. By using the duplication formula and the inversion formula, the range of convergence for the polylogarithm may be extended to the entire complex $z$-plane, and so the algorithms described here allow for the evaluation of the polylogarithm for all complex $s$ and $z$ values. Alternatively, the Hurwitz zeta can be very rapidly evaluated by means of an Euler Maclaurin series. The polylogarithm and the Hurwitz zeta are related, in that two evaluations of the one can be used to obtain a value of the other; thus, either algorithm can be used to evaluate either function. The Euler Maclaurin series is a clear performance winner for the Hurwitz zeta, while the Borwein algorithm is superior for evaluating the polylogarithm in the kidney-shaped region. Both algorithms are superior to the simple Taylor's series or direct summation. The primary, concrete result of this paper is an algorithm allows the exploration of the Hurwitz zeta in the critical strip, where fast algorithms are otherwise unavailable. A discussion of the monodromy group of the polylogarithm is included.

rawtitle:
An efficient algorithm for accelerating the convergence of oscillatory series, useful for computing the polylogarithm and {Hurwitz} zeta functions

entry:
@Article{Vepstas:2008:EAA,
  author =       "Linas Vepstas",
  title =        "An efficient algorithm for accelerating the
                 convergence of oscillatory series, useful for computing
                 the polylogarithm and {Hurwitz} zeta functions",
  journal =      j-NUMER-ALGORITHMS,
  volume =       "47",
  number =       "3",
  pages =        "211--252",
  month =        mar,
  year =         "2008",
  CODEN =        "NUALEG",
  DOI =          "https://doi.org/10.1007/s11075-007-9153-8",
  ISSN =         "1017-1398 (print), 1572-9265 (electronic)",
  ISSN-L =       "1017-1398",
  bibdate =      "Mon May 17 15:49:34 MDT 2010",
  bibsource =    "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1017-1398&volume=47&issue=3;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.springerlink.com/openurl.asp?genre=article&issn=1017-1398&volume=47&issue=3&spage=211",
  abstract =     "This paper sketches a technique for improving the rate
                 of convergence of a general oscillatory sequence, and
                 then applies this series acceleration algorithm to the
                 polylogarithm and the Hurwitz zeta function. As such,
                 it may be taken as an extension of the techniques given
                 by Borwein's ``An efficient algorithm for computing the
                 Riemann zeta function'' by Borwein for computing the
                 Riemann zeta function, to more general series. The
                 algorithm provides a rapid means of evaluating $
                 \operatorname {Li}_s(z) $ for general values of complex
                 $s$ and a kidney-shaped region of complex $z$ values
                 given by $ |z^2 / (z - 1)| < 4 $. By using the
                 duplication formula and the inversion formula, the
                 range of convergence for the polylogarithm may be
                 extended to the entire complex $z$-plane, and so the
                 algorithms described here allow for the evaluation of
                 the polylogarithm for all complex $s$ and $z$ values.
                 Alternatively, the Hurwitz zeta can be very rapidly
                 evaluated by means of an Euler Maclaurin series. The
                 polylogarithm and the Hurwitz zeta are related, in that
                 two evaluations of the one can be used to obtain a
                 value of the other; thus, either algorithm can be used
                 to evaluate either function. The Euler Maclaurin series
                 is a clear performance winner for the Hurwitz zeta,
                 while the Borwein algorithm is superior for evaluating
                 the polylogarithm in the kidney-shaped region. Both
                 algorithms are superior to the simple Taylor's series
                 or direct summation. The primary, concrete result of
                 this paper is an algorithm allows the exploration of
                 the Hurwitz zeta in the critical strip, where fast
                 algorithms are otherwise unavailable. A discussion of
                 the monodromy group of the polylogarithm is included.",
  acknowledgement = ack-nhfb,
  fjournal =     "Numerical Algorithms",
  journal-URL =  "http://link.springer.com/journal/11075",
  keywords =     "convergence acceleration",
}


bibtype: book
filename: elefunt.bib
label: Ware:2008:RIE
author: Willis H. Ware
title: RAND and the information evolution: a history in essays and vignettes
publisher: Rand Corporation
address: Santa Monica, CA
pages: xxvi + 201
year: 2008
DOI: https://doi.org/10.7249/cp537rc
ISBN: 0-8330-4513-X, 0-8330-4816-3, 1-282-45123-5 OR 083304513X, 0833048163, 1282451235
LCCN: QA76.27
bibdate: Tue Jun 2 19:14:18 MDT 2020
bibtimestamp: 2020.06.02 19:14:18 MDT
keywords: JOHNNIAC; JOSS; JOSS-1; JOSS-2; RAND tablet
acknowledgement: ack-nhfb
rawauthor: Willis H. Ware
rawtitle: {RAND} and the information evolution: a history in essays and vignettes
ISBN13:
978-0-8330-4513-3, 978-0-8330-4816-5, 978-1-282-45123-0 OR 9780833045133, 9780833048165, 9781282451230

bibsource:
fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/bibnet/authors/v/von-neumann-john.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/prng.bib; https://www.math.utah.edu/pub/tex/bib/unix.bib

URL:
http://www.jstor.org/stable/10.7249/cp537rc; https://www.rand.org/content/dam/rand/pubs/corporate_pubs/2008/RAND_CP537.pdf

abstract:
This professional memoir describes RAND's contributions to the evolution of computer science, particularly during the first decades following World War II, when digital computers succeeded slide rules, mechanical desk calculators, electric accounting machines, and analog computers. The memoir includes photographs and vignettes that reveal the collegial, creative, and often playful spirit in which the groundbreaking research was conducted at RAND.

TOC:
Dedication / v Preface / vii Figures / xiii Photographs / xv Tables / xvii Acknowledgments / xix Abbreviations / xxiii CHAPTER ONE Introduction / 1 Purpose and Scope / 1 Organization of the Document / 3 CHAPTER TWO The Department / 5 The Genesis of RAND / 5 The Need for a New Kind of Organization / 6 The Douglas Years / 7 An Independent, Private Nonprofit Organization / 8 The Nature of RAND's Contributions / 9 RAND Contributions to the Development of Computing / 10 In the Beginning / 10 An Early Computing Success / 11 The Move to Electronic Machines / 11 The Middle Years / 14 The JOHNNIAC Open-Shop System / 15 The Tablet / 16 Videographic System / 16 The Later Years / 17 RAND and the USAF Computing Evolution / 18 The Bottom Line / 19 CHAPTER THREE RAND's First Computer People / 21 The Legacy of Wartime Collaboration / 21 Early RAND Leaders / 22 Early Technical Staff / 24 The Douglas Thread / 24 The Wartime Thread / 26 The University Thread / 28 The Recruiting Thread / 30 Departmental Growth / 36 CHAPTER FOUR RAND's Early Computers / 45 Mid-20th Century Computation / 45 Reeves Electronic Analog Computer / 47 Plug-Board Interconnections / 50 Chopper-Stabilized Amplifiers / 50 Arbitrary Function Input / 51 The JOHNNIAC Digital Computer / 53 JOHNNIAC's ``Obituary'' / 63 IBM Mainframes / 64 Other Machinery. / 66 CHAPTER FIVE A Building for People with Computers / 67 A New Building and Campus. / 68 The Machine Room. / 72 Two-Story Installation / 72 REAC Installation. / 73 Raised-Floor Installation / 73 Air Conditioning. / 74 Configurations of the Machine Room / 75 Open House. / 75 Later Enhancements / 79 The Camera / 79 Kevershan's Trough / 80 Programmer-Alert Lights / 80 CHAPTER SIX Project Essays / 83 Approximations / 83 Random Digits and Normal Deviates / 87 The Bombing Simulator (aka Pinball Machine) / 90 The Air-Combat Room / 94 System Research Laboratory / 94 The RAND Tablet, Videographics, and Related Projects / 98 The RAND Tablet / 98 Handwriting Recognition / 99 Chinese-Character Lookup / 100 Map Annotation / 100 Videographic System / 103 GRAIL / 105 BIOMOD / 105 CLINFO / 107 Time-Shared Computing: JOSS / 109 JOSS-1 / 110 JOSS-2 / 113 Networked Computing: Packet Switching and Distributed Communications / 115 The Beginnings of Packet Switching: Some Underlying Concepts / 116 Text Editors (NED and e) / 122 Word Processing / 126 The Mail Handler / 128 The Original MH-Proposal Memorandum / 129 Implementation / 132 Another Perspective / 134 A User's Perspective / 135 The Developers' Present Views / 137 Artificial-Intelligence Research / 138 The Beginnings of Artificial Intelligence / 138 Newell, Shaw, and Simon: The Development of List-Processing Languages / 138 Expert Systems / 140 Knowledge-Based Simulation / 142 Computational Linguistics / 143 The Perfect Buddy / 144 Department of Defense Computer Institute / 147 Officer Career Paths / 149 Software / 150 Security and Privacy / 152 Security / 152 Privacy / 154 Fair Information Practices / 155 CHAPTER SEVEN Lore, Snippets, and Snapshots / 159 The Great Machine Fire / 159 The Gavel Caper / 159 Department-Head-Office Decor / 161 Oliver Alfred Gross and JOSS-1 / 162 The Soviet ``Threat'' / 163 Social Events / 164 The One-Way Wire / 166 Soviet Cybernetics / 166 Inter/Exhume / 167 The RAND Computer Symposia / 168 Professional Societies / 169 Microvignettes / 170 The Marchant March / 170 Getting Out the Documents / 171 Hero of the Week / 171 The Chiquita Banana War / 171 The Mengel Joint / 171 John Williams' Jaguar / 172 Programmer Sweepstakes / 173 CHAPTER EIGHT Epilogue / 175 Bibliography / 177 Index / 191

rawabstract:
This professional memoir describes RAND's contributions to the evolution of computer science, particularly during the first decades following World War II, when digital computers succeeded slide rules, mechanical desk calculators, electric accounting machines, and analog computers. The memoir includes photographs and vignettes that reveal the collegial, creative, and often playful spirit in which the groundbreaking research was conducted at RAND.

shortTOC:
Introduction The department RAND's first computer people RAND's early computers A building for people with computers Project essays Lore, snippets, and snapshots Epilogue

rawshortTOC:
Introduction \\ The department \\ RAND's first computer people \\ RAND's early computers \\ A building for people with computers \\ Project essays \\ Lore, snippets, and snapshots \\ Epilogue

rawTOC:
Dedication / v \\ Preface / vii \\ Figures / xiii \\ Photographs / xv \\ Tables / xvii \\ Acknowledgments / xix \\ Abbreviations / xxiii \\ CHAPTER ONE \\ Introduction / 1 \\ Purpose and Scope / 1 \\ Organization of the Document / 3 \\ CHAPTER TWO \\ The Department / 5 \\ The Genesis of RAND / 5 \\ The Need for a New Kind of Organization / 6 \\ The Douglas Years / 7 \\ An Independent, Private Nonprofit Organization / 8 \\ The Nature of RAND's Contributions / 9 \\ RAND Contributions to the Development of Computing / 10 \\ In the Beginning / 10 \\ An Early Computing Success / 11 \\ The Move to Electronic Machines / 11 \\ The Middle Years / 14 \\ The JOHNNIAC Open-Shop System / 15 \\ The Tablet / 16 \\ Videographic System / 16 \\ The Later Years / 17 \\ RAND and the USAF Computing Evolution / 18 \\ The Bottom Line / 19 \\ CHAPTER THREE \\ RAND's First Computer People / 21 \\ The Legacy of Wartime Collaboration / 21 \\ Early RAND Leaders / 22 \\ Early Technical Staff / 24 \\ The Douglas Thread / 24 \\ The Wartime Thread / 26 \\ The University Thread / 28 \\ The Recruiting Thread / 30 \\ Departmental Growth / 36 \\ CHAPTER FOUR \\ RAND's Early Computers / 45 \\ Mid-20th Century Computation / 45 \\ Reeves Electronic Analog Computer / 47 \\ Plug-Board Interconnections / 50 \\ Chopper-Stabilized Amplifiers / 50 \\ Arbitrary Function Input / 51 \\ The JOHNNIAC Digital Computer / 53 \\ JOHNNIAC's ``Obituary'' / 63 \\ IBM Mainframes / 64 \\ Other Machinery. / 66 \\ CHAPTER FIVE \\ A Building for People with Computers / 67 \\ A New Building and Campus. / 68 \\ The Machine Room. / 72 \\ Two-Story Installation / 72 \\ REAC Installation. / 73 \\ Raised-Floor Installation / 73 \\ Air Conditioning. / 74 \\ Configurations of the Machine Room / 75 \\ Open House. / 75 \\ Later Enhancements / 79 \\ The Camera / 79 \\ Kevershan's Trough / 80 \\ Programmer-Alert Lights / 80 \\ CHAPTER SIX \\ Project Essays / 83 \\ Approximations / 83 \\ Random Digits and Normal Deviates / 87 \\ The Bombing Simulator (aka Pinball Machine) / 90 \\ The Air-Combat Room / 94 \\ System Research Laboratory / 94 \\ The RAND Tablet, Videographics, and Related Projects / 98 \\ The RAND Tablet / 98 \\ Handwriting Recognition / 99 \\ Chinese-Character Lookup / 100 \\ Map Annotation / 100 \\ Videographic System / 103 \\ GRAIL / 105 \\ BIOMOD / 105 \\ CLINFO / 107 \\ Time-Shared Computing: JOSS / 109 \\ JOSS-1 / 110 \\ JOSS-2 / 113 \\ Networked Computing: Packet Switching and Distributed Communications / 115 \\ The Beginnings of Packet Switching: Some Underlying Concepts / 116 \\ Text Editors (NED and e) / 122 \\ Word Processing / 126 \\ The Mail Handler / 128 \\ The Original MH-Proposal Memorandum / 129 \\ Implementation / 132 \\ Another Perspective / 134 \\ A User's Perspective / 135 \\ The Developers' Present Views / 137 \\ Artificial-Intelligence Research / 138 \\ The Beginnings of Artificial Intelligence / 138 \\ Newell, Shaw, and Simon: The Development of List-Processing Languages / 138 \\ Expert Systems / 140 \\ Knowledge-Based Simulation / 142 \\ Computational Linguistics / 143 \\ The Perfect Buddy / 144 \\ Department of Defense Computer Institute / 147 \\ Officer Career Paths / 149 \\ Software / 150 \\ Security and Privacy / 152 \\ Security / 152 \\ Privacy / 154 \\ Fair Information Practices / 155 \\ CHAPTER SEVEN \\ Lore, Snippets, and Snapshots / 159 \\ The Great Machine Fire / 159 \\ The Gavel Caper / 159 \\ Department-Head-Office Decor / 161 \\ Oliver Alfred Gross and JOSS-1 / 162 \\ The Soviet ``Threat'' / 163 \\ Social Events / 164 \\ The One-Way Wire / 166 \\ Soviet Cybernetics / 166 \\ Inter/Exhume / 167 \\ The RAND Computer Symposia / 168 \\ Professional Societies / 169 \\ Microvignettes / 170 \\ The Marchant March / 170 \\ Getting Out the Documents / 171 \\ Hero of the Week / 171 \\ The Chiquita Banana War / 171 \\ The Mengel Joint / 171 \\ John Williams' Jaguar / 172 \\ Programmer Sweepstakes / 173 \\ CHAPTER EIGHT \\ Epilogue / 175 \\ Bibliography / 177 \\ Index / 191

entry:
@Book{Ware:2008:RIE,
  author =       "Willis H. Ware",
  title =        "{RAND} and the information evolution: a history in
                 essays and vignettes",
  publisher =    "Rand Corporation",
  address =      "Santa Monica, CA",
  pages =        "xxvi + 201",
  year =         "2008",
  DOI =          "https://doi.org/10.7249/cp537rc",
  ISBN =         "0-8330-4513-X, 0-8330-4816-3, 1-282-45123-5",
  ISBN-13 =      "978-0-8330-4513-3, 978-0-8330-4816-5,
                 978-1-282-45123-0",
  LCCN =         "QA76.27",
  bibdate =      "Tue Jun 2 19:14:18 MDT 2020",
  bibsource =    "fsz3950.oclc.org:210/WorldCat;
                 https://www.math.utah.edu/pub/bibnet/authors/v/von-neumann-john.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/prng.bib;
                 https://www.math.utah.edu/pub/tex/bib/unix.bib",
  URL =          "http://www.jstor.org/stable/10.7249/cp537rc;
                 https://www.rand.org/content/dam/rand/pubs/corporate_pubs/2008/RAND_CP537.pdf",
  abstract =     "This professional memoir describes RAND's
                 contributions to the evolution of computer science,
                 particularly during the first decades following World
                 War II, when digital computers succeeded slide rules,
                 mechanical desk calculators, electric accounting
                 machines, and analog computers. The memoir includes
                 photographs and vignettes that reveal the collegial,
                 creative, and often playful spirit in which the
                 groundbreaking research was conducted at RAND.",
  acknowledgement = ack-nhfb,
  keywords =     "JOHNNIAC; JOSS; JOSS-1; JOSS-2; RAND tablet",
  remark-1 =     "Page 13 has a photograph of the JOHNNIAC, and on the
                 wall of its room, a photograph of John von Neumann.",
  remark-2 =     "From page 15: ``\ldots{} the JOHNNIAC, which
                 nonetheless was the basis of a continuing series of
                 engineering advances, each making important
                 contributions to the art of the time. Among them were
                 the first commercially produced magnetic core memory,
                 which, for a while, was the largest in existence [4096
                 40-bit words]; a transistor-based adder and logic which
                 caused the JOHNNIAC to become a hybrid
                 transistor-vacuum tube device; the first high-speed
                 impact printer 140 columns wide (manufactured by
                 Anderson--Nichols, an engineering contracting firm);
                 and the first machine with extensive trouble-diagnostic
                 capability from the operating console.''",
  remark-3 =     "From page 53: ``the only bright spot was the Princeton
                 development at IAS, and thus it was that a working
                 alliance between RAND and IAS came into being. RAND
                 would build a machine patterned in the likeness of the
                 Princeton one. So JOHNNIAC came from an illustrious
                 ancestor --- the so-called von Neumann machine
                 developed at Princeton's IAS.''",
  remark-4 =     "Page 57 has a photograph of the JOHNNIAC's 256-word
                 Selectron high-speed memory. Page 59, a picture of its
                 140-column drum printer. Page 61 has an inside view of
                 the JOHNNIAC. Page 73 shows a step in the installation
                 of the JOHNNIAC. Page 162 has a photograph of the
                 JOHNNIAC console.",
  remark-5 =     "From page 66: ``RAND purchased the first commercially
                 available license for UNIX.''",
  remark-6 =     "Page 84 has a photo of a young Cecil Hastings, an
                 early pioneer of function approximation on digital
                 computers, and a few paragraphs about his work and its
                 influence.",
  remark-7 =     "Pages 87--90 discuss the preparation of RAND's famous
                 book of one million random digits, computed in Spring
                 1947, tested for two years after that before
                 publication in 1955. About 7000 copies of the book were
                 sold over three printings and fifteen years, and the
                 book was reprinted in 1966 and 2001.",
  remark-8 =     "From page 138: ``In the 1950s, RAND was involved in
                 designing and building one of the first stored-program
                 digital computers, the JOHNNIAC (named after John von
                 Neumann, a RAND consultant in the late 1940s and early
                 1950s). It was in operation from 1953 to 1966,
                 \ldots{}.''",
  shorttableofcontents = "Introduction \\
                 The department \\
                 RAND's first computer people \\
                 RAND's early computers \\
                 A building for people with computers \\
                 Project essays \\
                 Lore, snippets, and snapshots \\
                 Epilogue",
  tableofcontents = "Dedication / v \\
                 Preface / vii \\
                 Figures / xiii \\
                 Photographs / xv \\
                 Tables / xvii \\
                 Acknowledgments / xix \\
                 Abbreviations / xxiii \\
                 CHAPTER ONE \\
                 Introduction / 1 \\
                 Purpose and Scope / 1 \\
                 Organization of the Document / 3 \\
                 CHAPTER TWO \\
                 The Department / 5 \\
                 The Genesis of RAND / 5 \\
                 The Need for a New Kind of Organization / 6 \\
                 The Douglas Years / 7 \\
                 An Independent, Private Nonprofit Organization / 8 \\
                 The Nature of RAND's Contributions / 9 \\
                 RAND Contributions to the Development of Computing / 10
                 \\
                 In the Beginning / 10 \\
                 An Early Computing Success / 11 \\
                 The Move to Electronic Machines / 11 \\
                 The Middle Years / 14 \\
                 The JOHNNIAC Open-Shop System / 15 \\
                 The Tablet / 16 \\
                 Videographic System / 16 \\
                 The Later Years / 17 \\
                 RAND and the USAF Computing Evolution / 18 \\
                 The Bottom Line / 19 \\
                 CHAPTER THREE \\
                 RAND's First Computer People / 21 \\
                 The Legacy of Wartime Collaboration / 21 \\
                 Early RAND Leaders / 22 \\
                 Early Technical Staff / 24 \\
                 The Douglas Thread / 24 \\
                 The Wartime Thread / 26 \\
                 The University Thread / 28 \\
                 The Recruiting Thread / 30 \\
                 Departmental Growth / 36 \\
                 CHAPTER FOUR \\
                 RAND's Early Computers / 45 \\
                 Mid-20th Century Computation / 45 \\
                 Reeves Electronic Analog Computer / 47 \\
                 Plug-Board Interconnections / 50 \\
                 Chopper-Stabilized Amplifiers / 50 \\
                 Arbitrary Function Input / 51 \\
                 The JOHNNIAC Digital Computer / 53 \\
                 JOHNNIAC's ``Obituary'' / 63 \\
                 IBM Mainframes / 64 \\
                 Other Machinery. / 66 \\
                 CHAPTER FIVE \\
                 A Building for People with Computers / 67 \\
                 A New Building and Campus. / 68 \\
                 The Machine Room. / 72 \\
                 Two-Story Installation / 72 \\
                 REAC Installation. / 73 \\
                 Raised-Floor Installation / 73 \\
                 Air Conditioning. / 74 \\
                 Configurations of the Machine Room / 75 \\
                 Open House. / 75 \\
                 Later Enhancements / 79 \\
                 The Camera / 79 \\
                 Kevershan's Trough / 80 \\
                 Programmer-Alert Lights / 80 \\
                 CHAPTER SIX \\
                 Project Essays / 83 \\
                 Approximations / 83 \\
                 Random Digits and Normal Deviates / 87 \\
                 The Bombing Simulator (aka Pinball Machine) / 90 \\
                 The Air-Combat Room / 94 \\
                 System Research Laboratory / 94 \\
                 The RAND Tablet, Videographics, and Related Projects /
                 98 \\
                 The RAND Tablet / 98 \\
                 Handwriting Recognition / 99 \\
                 Chinese-Character Lookup / 100 \\
                 Map Annotation / 100 \\
                 Videographic System / 103 \\
                 GRAIL / 105 \\
                 BIOMOD / 105 \\
                 CLINFO / 107 \\
                 Time-Shared Computing: JOSS / 109 \\
                 JOSS-1 / 110 \\
                 JOSS-2 / 113 \\
                 Networked Computing: Packet Switching and Distributed
                 Communications / 115 \\
                 The Beginnings of Packet Switching: Some Underlying
                 Concepts / 116 \\
                 Text Editors (NED and e) / 122 \\
                 Word Processing / 126 \\
                 The Mail Handler / 128 \\
                 The Original MH-Proposal Memorandum / 129 \\
                 Implementation / 132 \\
                 Another Perspective / 134 \\
                 A User's Perspective / 135 \\
                 The Developers' Present Views / 137 \\
                 Artificial-Intelligence Research / 138 \\
                 The Beginnings of Artificial Intelligence / 138 \\
                 Newell, Shaw, and Simon: The Development of
                 List-Processing Languages / 138 \\
                 Expert Systems / 140 \\
                 Knowledge-Based Simulation / 142 \\
                 Computational Linguistics / 143 \\
                 The Perfect Buddy / 144 \\
                 Department of Defense Computer Institute / 147 \\
                 Officer Career Paths / 149 \\
                 Software / 150 \\
                 Security and Privacy / 152 \\
                 Security / 152 \\
                 Privacy / 154 \\
                 Fair Information Practices / 155 \\
                 CHAPTER SEVEN \\
                 Lore, Snippets, and Snapshots / 159 \\
                 The Great Machine Fire / 159 \\
                 The Gavel Caper / 159 \\
                 Department-Head-Office Decor / 161 \\
                 Oliver Alfred Gross and JOSS-1 / 162 \\
                 The Soviet ``Threat'' / 163 \\
                 Social Events / 164 \\
                 The One-Way Wire / 166 \\
                 Soviet Cybernetics / 166 \\
                 Inter/Exhume / 167 \\
                 The RAND Computer Symposia / 168 \\
                 Professional Societies / 169 \\
                 Microvignettes / 170 \\
                 The Marchant March / 170 \\
                 Getting Out the Documents / 171 \\
                 Hero of the Week / 171 \\
                 The Chiquita Banana War / 171 \\
                 The Mengel Joint / 171 \\
                 John Williams' Jaguar / 172 \\
                 Programmer Sweepstakes / 173 \\
                 CHAPTER EIGHT \\
                 Epilogue / 175 \\
                 Bibliography / 177 \\
                 Index / 191",
}


bibtype: article
filename: elefunt.bib
label: Zhu:2008:SNR
author: Ling Zhu and Jinju Sun
title: Six new Redheffer-type inequalities for circular and hyperbolic functions
journal: j-COMPUT-MATH-APPL
volume: 56
number: 2
pages: 522--529
month: jul
monthnumber: 7
year: 2008
CODEN: CMAPDK
ISSN: 0898-1221 (print), 1873-7668 (electronic) OR 08981221 (print), 18737668 (electronic)
ISSNL: 0898-1221 OR 08981221
bibdate: Wed Mar 1 21:50:15 MST 2017
bibtimestamp: 2017.03.01 21:50:15 MST
URL: http://www.sciencedirect.com/science/article/pii/S0898122108000813
fjournal: Computers and Mathematics with Applications
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/08981221
rawauthor: Ling Zhu and Jinju Sun
rawtitle: Six new {Redheffer}-type inequalities for circular and hyperbolic functions
entry:
@Article{Zhu:2008:SNR,
  author =       "Ling Zhu and Jinju Sun",
  title =        "Six new {Redheffer}-type inequalities for circular and
                 hyperbolic functions",
  journal =      j-COMPUT-MATH-APPL,
  volume =       "56",
  number =       "2",
  pages =        "522--529",
  month =        jul,
  year =         "2008",
  CODEN =        "CMAPDK",
  ISSN =         "0898-1221 (print), 1873-7668 (electronic)",
  ISSN-L =       "0898-1221",
  bibdate =      "Wed Mar 1 21:50:15 MST 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/computmathappl2000.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0898122108000813",
  acknowledgement = ack-nhfb,
  fjournal =     "Computers and Mathematics with Applications",
  journal-URL =  "http://www.sciencedirect.com/science/journal/08981221",
}


bibtype: article
filename: elefunt.bib
label: Anand:2009:OCS
author: C. K. Anand and W. Kahl
title: An Optimized Cell BE Special Function Library Generated by Coconut
journal: j-IEEE-TRANS-COMPUT
volume: 58
number: 8
pages: 1126--1138
month: aug
monthnumber: 8
year: 2009
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2008.223
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Mon Jul 4 11:37:43 MDT 2011
bibtimestamp: 2011.07.04 11:37:43 MDT
URL: https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4731241
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: C. K. Anand and W. Kahl
rawtitle: An Optimized {Cell BE} Special Function Library Generated by {Coconut}
abstract:
Coconut, a tool for developing high-assurance, high-performance kernels for scientific computing, contains an extensible domain-specific language (DSL) embedded in Haskell. The DSL supports interactive prototyping and unit testing, simplifying the process of designing efficient implementations of common patterns. Unscheduled C and scheduled assembly language output are supported. Using the patterns, even nonexpert users can write efficient function implementations, leveraging special hardware features. A production-quality library of elementary functions for the cell BE SPU compute engines has been developed. Coconut-generated and -scheduled vector functions were more than four times faster than commercially distributed functions written in C with intrinsics (a nicer syntax for in-line assembly), wrapped in loops and scheduled by tt spuxlc. All Coconut functions were faster, but the difference was larger for hard-to-approximate functions for which register-level SIMD lookups made a bigger difference. Other helpful features in the language include facilities for translating interval and polynomial descriptions between GHCi, a Haskell interpreter used to prototype in the DSL, and Maple, used for exploration and minimax polynomial generation. This makes it easier to match mathematical properties of the functions with efficient calculational patterns in the SPU ISA. By using single, literate source files, the resulting functions are remarkably readable.

rawabstract:
Coconut, a tool for developing high-assurance, high-performance kernels for scientific computing, contains an extensible domain-specific language (DSL) embedded in Haskell. The DSL supports interactive prototyping and unit testing, simplifying the process of designing efficient implementations of common patterns. Unscheduled C and scheduled assembly language output are supported. Using the patterns, even nonexpert users can write efficient function implementations, leveraging special hardware features. A production-quality library of elementary functions for the cell BE SPU compute engines has been developed. Coconut-generated and -scheduled vector functions were more than four times faster than commercially distributed functions written in C with intrinsics (a nicer syntax for in-line assembly), wrapped in loops and scheduled by {\tt spuxlc}. All Coconut functions were faster, but the difference was larger for hard-to-approximate functions for which register-level SIMD lookups made a bigger difference. Other helpful features in the language include facilities for translating interval and polynomial descriptions between GHCi, a Haskell interpreter used to prototype in the DSL, and Maple, used for exploration and minimax polynomial generation. This makes it easier to match mathematical properties of the functions with efficient calculational patterns in the SPU ISA. By using single, literate source files, the resulting functions are remarkably readable.

entry:
@Article{Anand:2009:OCS,
  author =       "C. K. Anand and W. Kahl",
  title =        "An Optimized {Cell BE} Special Function Library
                 Generated by {Coconut}",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "58",
  number =       "8",
  pages =        "1126--1138",
  month =        aug,
  year =         "2009",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2008.223",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Mon Jul 4 11:37:43 MDT 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib",
  URL =          "https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4731241",
  abstract =     "Coconut, a tool for developing high-assurance,
                 high-performance kernels for scientific computing,
                 contains an extensible domain-specific language (DSL)
                 embedded in Haskell. The DSL supports interactive
                 prototyping and unit testing, simplifying the process
                 of designing efficient implementations of common
                 patterns. Unscheduled C and scheduled assembly language
                 output are supported. Using the patterns, even
                 nonexpert users can write efficient function
                 implementations, leveraging special hardware features.
                 A production-quality library of elementary functions
                 for the cell BE SPU compute engines has been developed.
                 Coconut-generated and -scheduled vector functions were
                 more than four times faster than commercially
                 distributed functions written in C with intrinsics (a
                 nicer syntax for in-line assembly), wrapped in loops
                 and scheduled by {\tt spuxlc}. All Coconut functions
                 were faster, but the difference was larger for
                 hard-to-approximate functions for which register-level
                 SIMD lookups made a bigger difference. Other helpful
                 features in the language include facilities for
                 translating interval and polynomial descriptions
                 between GHCi, a Haskell interpreter used to prototype
                 in the DSL, and Maple, used for exploration and minimax
                 polynomial generation. This makes it easier to match
                 mathematical properties of the functions with efficient
                 calculational patterns in the SPU ISA. By using single,
                 literate source files, the resulting functions are
                 remarkably readable.",
  acknowledgement = ack-nhfb,
  ajournal =     "IEEE Trans. Comput.",
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
}


bibtype: article
filename: elefunt.bib
label: Backeljauw:2009:ACF
author: Franky Backeljauw and Annie Cuyt
title: Algorithm 895: a continued fractions package for special functions
journal: j-TOMS
volume: 36
number: 3
pages: 15:1--15:20
month: jul
monthnumber: 7
year: 2009
CODEN: ACMSCU
DOI: https://doi.org/10.1145/1527286.1527289
ISSN: 0098-3500 (print), 1557-7295 (electronic) OR 00983500 (print), 15577295 (electronic)
ISSNL: 0098-3500 OR 00983500
bibdate: Tue Jul 21 14:09:07 MDT 2009
bibtimestamp: 2009.07.21 14:09:07 MDT
fjournal: ACM Transactions on Mathematical Software (TOMS)
keywords: CAS software; continued fractions; Maple; special functions
acknowledgement: ack-nhfb
articleno: 15
journalURL: https://dl.acm.org/loi/toms
rawauthor: Franky Backeljauw and Annie Cuyt
rawtitle: {Algorithm 895}: a continued fractions package for special functions
abstract:
The continued fractions for special functions package (in the sequel abbreviated as CFSF package) complements a systematic study of continued fraction representations for special functions. It provides all the functionality to create continued fractions, in particular $k$-periodic or limit $k$-periodic fractions, to compute approximants, make use of continued fraction tails, perform equivalence transformations and contractions, and much more. The package, developed in Maple, includes a library of more than 200 representations of special functions, of which only 10% can be found in the 1964 NBS em Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables by M. Abramowitz and I. Stegun.

rawabstract:
The continued fractions for special functions package (in the sequel abbreviated as CFSF package) complements a systematic study of continued fraction representations for special functions. It provides all the functionality to create continued fractions, in particular $k$-periodic or limit $k$-periodic fractions, to compute approximants, make use of continued fraction tails, perform equivalence transformations and contractions, and much more. The package, developed in Maple, includes a library of more than 200 representations of special functions, of which only 10\% can be found in the 1964 NBS {\em Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables\/} by M. Abramowitz and I. Stegun.

entry:
@Article{Backeljauw:2009:ACF,
  author =       "Franky Backeljauw and Annie Cuyt",
  title =        "{Algorithm 895}: a continued fractions package for
                 special functions",
  journal =      j-TOMS,
  volume =       "36",
  number =       "3",
  pages =        "15:1--15:20",
  month =        jul,
  year =         "2009",
  CODEN =        "ACMSCU",
  DOI =          "https://doi.org/10.1145/1527286.1527289",
  ISSN =         "0098-3500 (print), 1557-7295 (electronic)",
  ISSN-L =       "0098-3500",
  bibdate =      "Tue Jul 21 14:09:07 MDT 2009",
  bibsource =    "http://www.acm.org/pubs/contents/journals/toms/;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/toms.bib",
  abstract =     "The continued fractions for special functions package
                 (in the sequel abbreviated as CFSF package) complements
                 a systematic study of continued fraction
                 representations for special functions. It provides all
                 the functionality to create continued fractions, in
                 particular $k$-periodic or limit $k$-periodic
                 fractions, to compute approximants, make use of
                 continued fraction tails, perform equivalence
                 transformations and contractions, and much more. The
                 package, developed in Maple, includes a library of more
                 than 200 representations of special functions, of which
                 only 10\% can be found in the 1964 NBS {\em Handbook of
                 Mathematical Functions with Formulas, Graphs and
                 Mathematical Tables\/} by M. Abramowitz and I.
                 Stegun.",
  acknowledgement = ack-nhfb,
  articleno =    "15",
  fjournal =     "ACM Transactions on Mathematical Software (TOMS)",
  journal-URL =  "https://dl.acm.org/loi/toms",
  keywords =     "CAS software; continued fractions; Maple; special
                 functions",
}


bibtype: article
filename: elefunt.bib
label: Blomquist:2009:MSC
author: Frithjof Blomquist and Werner Hofschuster and Walter Kramer
title: A Modified Staggered Correction Arithmetic with Enhanced Accuracy and Very Wide Exponent Range
journal: j-LECT-NOTES-COMP-SCI
volume: 5492
pages: 41--67
year: 2009
CODEN: LNCSD9
DOI: https://doi.org/10.1007/978-3-642-01591-5_4
ISSN: 0302-9743 (print), 1611-3349 (electronic) OR 03029743 (print), 16113349 (electronic)
ISSNL: 0302-9743 OR 03029743
bibdate: Tue Apr 10 08:32:19 2012
bibtimestamp: 2012.04.10 08:32:19 ???
URL: http://www.springerlink.com/content/k038294004403504/
fjournal: Lecture Notes in Computer Science
remark: Conference on Numerical Validation in Current Hardware Architectures
acknowledgement: ack-nhfb
authordates: 1952--2014 (WK)
journalURL: http://link.springer.com/bookseries/558
rawauthor: Frithjof Blomquist and Werner Hofschuster and Walter Kr{\"a}mer
rawtitle: A Modified Staggered Correction Arithmetic with Enhanced Accuracy and Very Wide Exponent Range
keywords:
C-XSC; complex interval functions; interval computation; multiple precision; reliable numerical computations; staggered correction; wide exponent range

entry:
@Article{Blomquist:2009:MSC,
  author =       "Frithjof Blomquist and Werner Hofschuster and Walter
                 Kr{\"a}mer",
  title =        "A Modified Staggered Correction Arithmetic with
                 Enhanced Accuracy and Very Wide Exponent Range",
  journal =      j-LECT-NOTES-COMP-SCI,
  volume =       "5492",
  pages =        "41--67",
  year =         "2009",
  CODEN =        "LNCSD9",
  DOI =          "https://doi.org/10.1007/978-3-642-01591-5_4",
  ISSN =         "0302-9743 (print), 1611-3349 (electronic)",
  ISSN-L =       "0302-9743",
  bibdate =      "Tue Apr 10 08:32:19 2012",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://www.springerlink.com/content/k038294004403504/",
  acknowledgement = ack-nhfb,
  author-dates = "1952--2014 (WK)",
  fjournal =     "Lecture Notes in Computer Science",
  journal-URL =  "http://link.springer.com/bookseries/558",
  keywords =     "C-XSC; complex interval functions; interval
                 computation; multiple precision; reliable numerical
                 computations; staggered correction; wide exponent
                 range",
  remark =       "Conference on Numerical Validation in Current Hardware
                 Architectures",
  remark-2 =     "Includes algorithms for division, $\exp(x)$, $(1 +
                 x)^n$, $\log(x)$, $\log(1 + x)$, and $\sqrt{x}$.
                 Staggered arithmetic represents numbers with tuples
                 $(e, x_1, x_2, \ldots{}, x_n)$ where $e$ is either
                 integer or a floating-point whole number, the $x_k$ are
                 floating-point, and a number has the value $2^e \sum_{k
                 = 1}^n x_k$. For interval arithmetic, the last element
                 is a pair of lower and upper bounds.",
}


bibtype: article
filename: elefunt.bib
label: Boldo:2009:FVA
author: S. Boldo and M. Daumas and Ren-Cang Li
title: Formally Verified Argument Reduction with a Fused Multiply-Add
journal: j-IEEE-TRANS-COMPUT
volume: 58
number: 8
pages: 1139--1145
month: aug
monthnumber: 8
year: 2009
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2008.216
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Mon Jul 4 11:37:43 MDT 2011
bibtimestamp: 2011.07.04 11:37:43 MDT
URL: https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4711042
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: S. Boldo and M. Daumas and Ren-Cang Li
rawtitle: Formally Verified Argument Reduction with a Fused Multiply-Add
abstract:
The Cody and Waite argument reduction technique works perfectly for reasonably large arguments, but as the input grows, there are no bits left to approximate the constant with enough accuracy. Under mild assumptions, we show that the result computed with a fused multiply-add provides a fully accurate result for many possible values of the input with a constant almost accurate to the full working precision. We also present an algorithm for a fully accurate second reduction step to reach full double accuracy (all the significand bits of two numbers are accurate) even in the worst cases of argument reduction. Our work recalls the common algorithms and presents proofs of correctness. All the proofs are formally verified using the Coq automatic proof checker.

rawabstract:
The Cody and Waite argument reduction technique works perfectly for reasonably large arguments, but as the input grows, there are no bits left to approximate the constant with enough accuracy. Under mild assumptions, we show that the result computed with a fused multiply-add provides a fully accurate result for many possible values of the input with a constant almost accurate to the full working precision. We also present an algorithm for a fully accurate second reduction step to reach full double accuracy (all the significand bits of two numbers are accurate) even in the worst cases of argument reduction. Our work recalls the common algorithms and presents proofs of correctness. All the proofs are formally verified using the Coq automatic proof checker.

entry:
@Article{Boldo:2009:FVA,
  author =       "S. Boldo and M. Daumas and Ren-Cang Li",
  title =        "Formally Verified Argument Reduction with a Fused
                 Multiply-Add",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "58",
  number =       "8",
  pages =        "1139--1145",
  month =        aug,
  year =         "2009",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2008.216",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Mon Jul 4 11:37:43 MDT 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib",
  URL =          "https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4711042",
  abstract =     "The Cody and Waite argument reduction technique works
                 perfectly for reasonably large arguments, but as the
                 input grows, there are no bits left to approximate the
                 constant with enough accuracy. Under mild assumptions,
                 we show that the result computed with a fused
                 multiply-add provides a fully accurate result for many
                 possible values of the input with a constant almost
                 accurate to the full working precision. We also present
                 an algorithm for a fully accurate second reduction step
                 to reach full double accuracy (all the significand bits
                 of two numbers are accurate) even in the worst cases of
                 argument reduction. Our work recalls the common
                 algorithms and presents proofs of correctness. All the
                 proofs are formally verified using the Coq automatic
                 proof checker.",
  acknowledgement = ack-nhfb,
  ajournal =     "IEEE Trans. Comput.",
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
}


