# bibtab

bibtype: inproceedings
filename: elefunt.bib
label: Bradford:2002:TBS
author: Russell Bradford and James H. Davenport
booktitle: Proceedings of the 2002 International Symposium on Symbolic and Algebraic Computation
title: Towards better simplification of elementary functions
publisher: pub-ACM
address: pub-ACM:adr
pages: 16--22 (electronic)
year: 2002
MRclass: 68W30 (33B10)
MRnumber: MR2035228
bibdate: Wed Apr 13 06:46:35 2005
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2005.04.13 06:46:35 ???
acknowledgement: ack-nhfb
rawauthor: Russell Bradford and James H. Davenport
rawbooktitle: Proceedings of the 2002 International Symposium on Symbolic and Algebraic Computation
rawtitle: Towards better simplification of elementary functions
entry:
@InProceedings{Bradford:2002:TBS,
  author =       "Russell Bradford and James H. Davenport",
  booktitle =    "Proceedings of the 2002 International Symposium on
                 Symbolic and Algebraic Computation",
  title =        "Towards better simplification of elementary
                 functions",
  publisher =    pub-ACM,
  address =      pub-ACM:adr,
  pages =        "16--22 (electronic)",
  year =         "2002",
  MRclass =      "68W30 (33B10)",
  MRnumber =     "MR2035228",
  bibdate =      "Wed Apr 13 06:46:35 2005",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: elefunt.bib
label: Bryc:2002:UAR
author: W. Bryc
title: A uniform approximation to the right normal tail integral
journal: j-APPL-MATH-COMP
volume: 127
number: 2--3
pages: 365--374
day: 15
month: apr
monthnumber: 4
year: 2002
CODEN: AMHCBQ
DOI: https://doi.org/10.1016/S0096-3003(01)00015-7
ISSN: 0096-3003 (print), 1873-5649 (electronic) OR 00963003 (print), 18735649 (electronic)
ISSNL: 0096-3003 OR 00963003
bibdate: Wed Feb 27 08:48:29 MST 2002
bibtimestamp: 2002.02.27 08:48:29 MST
fjournal: Applied Mathematics and Computation
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00963003/
rawauthor: W. Bryc
rawtitle: A uniform approximation to the right normal tail integral
bibsource:
http://www.elsevier.com/locate/issn/00963003; https://www.math.utah.edu/pub/tex/bib/applmathcomput2000.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

URL:
http://www.elsevier.com/gej-ng/10/9/12/123/27/44/abstract.html; http://www.sciencedirect.com/science/article/pii/S0096300301000157

entry:
@Article{Bryc:2002:UAR,
  author =       "W. Bryc",
  title =        "A uniform approximation to the right normal tail
                 integral",
  journal =      j-APPL-MATH-COMP,
  volume =       "127",
  number =       "2--3",
  pages =        "365--374",
  day =          "15",
  month =        apr,
  year =         "2002",
  CODEN =        "AMHCBQ",
  DOI =          "https://doi.org/10.1016/S0096-3003(01)00015-7",
  ISSN =         "0096-3003 (print), 1873-5649 (electronic)",
  ISSN-L =       "0096-3003",
  bibdate =      "Wed Feb 27 08:48:29 MST 2002",
  bibsource =    "http://www.elsevier.com/locate/issn/00963003;
                 https://www.math.utah.edu/pub/tex/bib/applmathcomput2000.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.elsevier.com/gej-ng/10/9/12/123/27/44/abstract.html;
                 http://www.sciencedirect.com/science/article/pii/S0096300301000157",
  acknowledgement = ack-nhfb,
  fjournal =     "Applied Mathematics and Computation",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00963003/",
}


bibtype: article
filename: elefunt.bib
label: Ceberio:2002:HRI
author: M. Ceberio and L. Granvilliers
title: Horner's Rule for Interval Evaluation Revisited
journal: j-COMPUTING
volume: 69
number: 1
pages: 51--81
month: mar
monthnumber: 3
year: 2002
CODEN: CMPTA2
DOI: https://doi.org/10.1007/s00607-002-1448-y
ISSN: 0010-485X (print), 1436-5057 (electronic) OR 0010485X (print), 14365057 (electronic)
ISSNL: 0010-485X OR 0010485X
bibdate: Tue Nov 5 07:12:39 MST 2002
bibtimestamp: 2002.11.05 07:12:39 MST
fjournal: Computing
keywords: interval arithmetic; number of multiplications to evaluate a polynomial
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/607
rawauthor: M. Ceberio and L. Granvilliers
rawtitle: {Horner}'s Rule for Interval Evaluation Revisited
bibsource:
http://link.springer-ny.com/link/service/journals/00607/tocs/t2069001.htm; http://springerlink.metapress.com/openurl.asp?genre=journal&issn=0010-485X; https://www.math.utah.edu/pub/tex/bib/computing.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

URL:
http://link.springer.de/link/service/journals/00607/bibs/2069001/20690051.htm; http://link.springer.de/link/service/journals/00607/papers/2069001/20690051.pdf

entry:
@Article{Ceberio:2002:HRI,
  author =       "M. Ceberio and L. Granvilliers",
  title =        "{Horner}'s Rule for Interval Evaluation Revisited",
  journal =      j-COMPUTING,
  volume =       "69",
  number =       "1",
  pages =        "51--81",
  month =        mar,
  year =         "2002",
  CODEN =        "CMPTA2",
  DOI =          "https://doi.org/10.1007/s00607-002-1448-y",
  ISSN =         "0010-485X (print), 1436-5057 (electronic)",
  ISSN-L =       "0010-485X",
  bibdate =      "Tue Nov 5 07:12:39 MST 2002",
  bibsource =    "http://link.springer-ny.com/link/service/journals/00607/tocs/t2069001.htm;
                 http://springerlink.metapress.com/openurl.asp?genre=journal&issn=0010-485X;
                 https://www.math.utah.edu/pub/tex/bib/computing.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://link.springer.de/link/service/journals/00607/bibs/2069001/20690051.htm;
                 http://link.springer.de/link/service/journals/00607/papers/2069001/20690051.pdf",
  acknowledgement = ack-nhfb,
  fjournal =     "Computing",
  journal-URL =  "http://link.springer.com/journal/607",
  keywords =     "interval arithmetic; number of multiplications to
                 evaluate a polynomial",
}


bibtype: inproceedings
filename: elefunt.bib
label: Chiani:2002:IEB
author: M. Chiani and D. Dardari
booktitle: Global Telecommunications Conference, 2002. GLOBECOM '02. IEEE
publisher: pub-IEEE
address: pub-IEEE:adr
pages: 51--81
year: 2002
DOI: https://doi.org/10.1109/glocom.2002.1188428
bibdate: Sat Dec 16 16:54:47 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2017.12.16 16:54:47 ???
URL: http://ieeexplore.ieee.org/document/1188428/
acknowledgement: ack-nhfb
rawauthor: M. Chiani and D. Dardari
rawbooktitle: Global Telecommunications Conference, 2002. {GLOBECOM '02}. {IEEE}
title:
Improved exponential bounds and approximation for the $Q$-function with application to average error probability computation

rawtitle:
Improved exponential bounds and approximation for the {$Q$}-function with application to average error probability computation

entry:
@InProceedings{Chiani:2002:IEB,
  author =       "M. Chiani and D. Dardari",
  booktitle =    "Global Telecommunications Conference, 2002. {GLOBECOM
                 '02}. {IEEE}",
  title =        "Improved exponential bounds and approximation for the
                 {$Q$}-function with application to average error
                 probability computation",
  publisher =    pub-IEEE,
  address =      pub-IEEE:adr,
  year =         "2002",
  DOI =          "https://doi.org/10.1109/glocom.2002.1188428",
  bibdate =      "Sat Dec 16 16:54:47 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://ieeexplore.ieee.org/document/1188428/",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: elefunt.bib
label: Fabijonas:2002:LMC
author: Bruce R. Fabijonas
journal: j-J-COMPUT-APPL-MATH
volume: 146
number: 2
pages: 323--342
day: 15
month: sep
monthnumber: 9
year: 2002
CODEN: JCAMDI
ISSN: 0377-0427 (print), 1879-1778 (electronic) OR 03770427 (print), 18791778 (electronic)
ISSNL: 0377-0427 OR 03770427
bibdate: Sat Feb 25 12:52:30 MST 2017
bibtimestamp: 2017.02.25 12:52:30 MST
URL: http://www.sciencedirect.com/science/article/pii/S0377042702003643
fjournal: Journal of Computational and Applied Mathematics
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/03770427
rawauthor: Bruce R. Fabijonas
title:
Laplace's method on a computer algebra system with an application to the real valued modified Bessel functions

rawtitle:
{Laplace}'s method on a computer algebra system with an application to the real valued modified {Bessel} functions

entry:
@Article{Fabijonas:2002:LMC,
  author =       "Bruce R. Fabijonas",
  title =        "{Laplace}'s method on a computer algebra system with
                 an application to the real valued modified {Bessel}
                 functions",
  journal =      j-J-COMPUT-APPL-MATH,
  volume =       "146",
  number =       "2",
  pages =        "323--342",
  day =          "15",
  month =        sep,
  year =         "2002",
  CODEN =        "JCAMDI",
  ISSN =         "0377-0427 (print), 1879-1778 (electronic)",
  ISSN-L =       "0377-0427",
  bibdate =      "Sat Feb 25 12:52:30 MST 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2000.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0377042702003643",
  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: Gautschi:2002:GQA
author: W. Gautschi
title: Gauss quadrature approximations to hypergeometric and confluent hypergeometric functions
journal: j-J-COMPUT-APPL-MATH
volume: 139
number: 1
pages: 173--187
day: 1
month: feb
monthnumber: 2
year: 2002
CODEN: JCAMDI
ISSN: 0377-0427 (print), 1879-1778 (electronic) OR 03770427 (print), 18791778 (electronic)
ISSNL: 0377-0427 OR 03770427
MRclass: 33F05 (33C05 33C15 65D20)
MRnumber: MR1876879 (2002m:33029)
bibdate: Thu Dec 01 09:11:13 2011
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2011.12.01 09:11:13 ???
fjournal: Journal of Computational and Applied Mathematics
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/03770427
rawauthor: W. Gautschi
rawtitle: {Gauss} quadrature approximations to hypergeometric and confluent hypergeometric functions
remark:
The paper treats ordinary and confluent hypergeometric functions $_2 F_1$ and $_1 F_1$, using their integral representations to obtain Gaussian quadrature rules.

entry:
@Article{Gautschi:2002:GQA,
  author =       "W. Gautschi",
  title =        "{Gauss} quadrature approximations to hypergeometric
                 and confluent hypergeometric functions",
  journal =      j-J-COMPUT-APPL-MATH,
  volume =       "139",
  number =       "1",
  pages =        "173--187",
  day =          "1",
  month =        feb,
  year =         "2002",
  CODEN =        "JCAMDI",
  ISSN =         "0377-0427 (print), 1879-1778 (electronic)",
  ISSN-L =       "0377-0427",
  MRclass =      "33F05 (33C05 33C15 65D20)",
  MRnumber =     "MR1876879 (2002m:33029)",
  bibdate =      "Thu Dec 01 09:11:13 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "Journal of Computational and Applied Mathematics",
  journal-URL =  "http://www.sciencedirect.com/science/journal/03770427",
  remark =       "The paper treats ordinary and confluent hypergeometric
                 functions $_2 F_1$ and $_1 F_1$, using their integral
                 representations to obtain Gaussian quadrature rules.",
}


bibtype: article
filename: elefunt.bib
label: Gil:2002:AAB
author: Amparo Gil and Javier Segura and Nico M. Temme
title: Algorithm 819: AIZ, BIZ: two Fortran 77 routines for the computation of complex Airy functions
journal: j-TOMS
volume: 28
number: 3
pages: 325--336
month: sep
monthnumber: 9
year: 2002
CODEN: ACMSCU
DOI: https://doi.org/10.1145/569147.569150
ISSN: 0098-3500 (print), 1557-7295 (electronic) OR 00983500 (print), 15577295 (electronic)
ISSNL: 0098-3500 OR 00983500
bibdate: Sat Nov 9 11:16:50 MST 2002
bibtimestamp: 2002.11.09 11:16:50 MST
fjournal: ACM Transactions on Mathematical Software (TOMS)
acknowledgement: ack-nhfb
journalURL: https://dl.acm.org/loi/toms
rawauthor: Amparo Gil and Javier Segura and Nico M. Temme
abstract:
Two Fortran 77 routines for the evaluation of Airy functions of complex arguments $ A i(z) $, $ B i(z) $ and their derivatives are presented. The routines are based on the use of Gaussian quadrature, Maclaurin series and asymptotic expansions. Comparison with a previous code by D. E. Amos (ACM TOMS 12 (1986)) is provided.

rawabstract:
Two Fortran 77 routines for the evaluation of Airy functions of complex arguments $ A i(z) $, $ B i(z) $ and their derivatives are presented. The routines are based on the use of Gaussian quadrature, Maclaurin series and asymptotic expansions. Comparison with a previous code by D. E. Amos (ACM TOMS 12 (1986)) is provided.

rawtitle:
{Algorithm 819}: {AIZ}, {BIZ}: two {Fortran 77} routines for the computation of complex {Airy} functions

entry:
@Article{Gil:2002:AAB,
  author =       "Amparo Gil and Javier Segura and Nico M. Temme",
  title =        "{Algorithm 819}: {AIZ}, {BIZ}: two {Fortran 77}
                 routines for the computation of complex {Airy}
                 functions",
  journal =      j-TOMS,
  volume =       "28",
  number =       "3",
  pages =        "325--336",
  month =        sep,
  year =         "2002",
  CODEN =        "ACMSCU",
  DOI =          "https://doi.org/10.1145/569147.569150",
  ISSN =         "0098-3500 (print), 1557-7295 (electronic)",
  ISSN-L =       "0098-3500",
  bibdate =      "Sat Nov 9 11:16:50 MST 2002",
  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 =     "Two Fortran 77 routines for the evaluation of Airy
                 functions of complex arguments $ A i(z) $, $ B i(z) $
                 and their derivatives are presented. The routines are
                 based on the use of Gaussian quadrature, Maclaurin
                 series and asymptotic expansions. Comparison with a
                 previous code by D. E. Amos (ACM TOMS 12 (1986)) is
                 provided.",
  acknowledgement = ack-nhfb,
  fjournal =     "ACM Transactions on Mathematical Software (TOMS)",
  journal-URL =  "https://dl.acm.org/loi/toms",
}


bibtype: article
filename: elefunt.bib
label: Gil:2002:AGH
author: Amparo Gil and Javier Segura and Nico M. Temme
title: Algorithm 822: GIZ, HIZ: two Fortran 77 routines for the computation of complex Scorer functions
journal: j-TOMS
volume: 28
number: 4
pages: 436--447
month: dec
monthnumber: 12
year: 2002
CODEN: ACMSCU
DOI: https://doi.org/10.1145/592843.592847
ISSN: 0098-3500 (print), 1557-7295 (electronic) OR 00983500 (print), 15577295 (electronic)
ISSNL: 0098-3500 OR 00983500
bibdate: Fri Mar 28 08:17:55 MST 2003
bibtimestamp: 2003.03.28 08:17:55 MST
fjournal: ACM Transactions on Mathematical Software (TOMS)
acknowledgement: ack-nhfb
journalURL: https://dl.acm.org/loi/toms
rawauthor: Amparo Gil and Javier Segura and Nico M. Temme
abstract:
Two Fortran 77 routines for the evaluation of Scorer functions of complex arguments $ G i(z) $, $ H i(z) $ and their derivatives are presented. The routines are based on the use of quadrature, Maclaurin series and asymptotic expansions. For real $z$ comparison with a previous code by A. J. Macleod (J. Comput. Appl. Math. 53 (1994)) is provided.

rawabstract:
Two Fortran 77 routines for the evaluation of Scorer functions of complex arguments $ G i(z) $, $ H i(z) $ and their derivatives are presented. The routines are based on the use of quadrature, Maclaurin series and asymptotic expansions. For real $z$ comparison with a previous code by A. J. Macleod (J. Comput. Appl. Math. 53 (1994)) is provided.

rawtitle:
{Algorithm 822}: {GIZ}, {HIZ}: two {Fortran} 77 routines for the computation of complex {Scorer} functions

entry:
@Article{Gil:2002:AGH,
  author =       "Amparo Gil and Javier Segura and Nico M. Temme",
  title =        "{Algorithm 822}: {GIZ}, {HIZ}: two {Fortran} 77
                 routines for the computation of complex {Scorer}
                 functions",
  journal =      j-TOMS,
  volume =       "28",
  number =       "4",
  pages =        "436--447",
  month =        dec,
  year =         "2002",
  CODEN =        "ACMSCU",
  DOI =          "https://doi.org/10.1145/592843.592847",
  ISSN =         "0098-3500 (print), 1557-7295 (electronic)",
  ISSN-L =       "0098-3500",
  bibdate =      "Fri Mar 28 08:17:55 MST 2003",
  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 =     "Two Fortran 77 routines for the evaluation of Scorer
                 functions of complex arguments $ G i(z) $, $ H i(z) $
                 and their derivatives are presented. The routines are
                 based on the use of quadrature, Maclaurin series and
                 asymptotic expansions. For real $z$ comparison with a
                 previous code by A. J. Macleod (J. Comput. Appl. Math.
                 53 (1994)) is provided.",
  acknowledgement = ack-nhfb,
  fjournal =     "ACM Transactions on Mathematical Software (TOMS)",
  journal-URL =  "https://dl.acm.org/loi/toms",
}


bibtype: techreport
filename: elefunt.bib
label: Gil:2002:CSF
author: A. Gil and J. Segura and N. M. Temme
title: Computing special functions by using quadrature rules
type: Report
number: MAS-R0230
institution: pub-CWI
address: pub-CWI:adr
pages: 11
year: 2002
LCCN: QA9.A1 R426 MAS-R0230
bibdate: Sat Oct 30 19:13:12 2010
bibsource: http://cat.cisti-icist.nrc-cnrc.gc.ca/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2010.10.30 19:13:12 ???
note: Published in cite Gil:2003:CSF.
acknowledgement: ack-nhfb
rawauthor: A. Gil and J. Segura and N. M. Temme
rawnote: Published in \cite{Gil:2003:CSF}.
rawtitle: Computing special functions by using quadrature rules
entry:
@TechReport{Gil:2002:CSF,
  author =       "A. Gil and J. Segura and N. M. Temme",
  title =        "Computing special functions by using quadrature
                 rules",
  type =         "Report",
  number =       "MAS-R0230",
  institution =  pub-CWI,
  address =      pub-CWI:adr,
  pages =        "11",
  year =         "2002",
  LCCN =         "QA9.A1 R426 MAS-R0230",
  bibdate =      "Sat Oct 30 19:13:12 2010",
  bibsource =    "http://cat.cisti-icist.nrc-cnrc.gc.ca/;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  note =         "Published in \cite{Gil:2003:CSF}.",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: elefunt.bib
label: Gil:2002:EMB
author: Amparo Gil and Javier Segura and Nico M. Temme
title: Evaluation of the Modified Bessel Function of the Third Kind of Imaginary Orders
journal: j-J-COMPUT-PHYS
volume: 175
number: 2
pages: 398--411
day: 20
month: jan
monthnumber: 1
year: 2002
CODEN: JCTPAH
DOI: https://doi.org/10.1006/jcph.2001.6894
ISSN: 0021-9991 (print), 1090-2716 (electronic) OR 00219991 (print), 10902716 (electronic)
ISSNL: 0021-9991 OR 00219991
bibdate: Mon Jan 2 22:12:13 MST 2012
bibtimestamp: 2012.01.02 22:12:13 MST
URL: http://www.sciencedirect.com/science/article/pii/S0021999101968949
fjournal: Journal of Computational Physics
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00219991/
rawauthor: Amparo Gil and Javier Segura and Nico M. Temme
rawtitle: Evaluation of the Modified {Bessel} Function of the Third Kind of Imaginary Orders
bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputphys2000.bib

abstract:
The evaluation of the modified Bessel function of the third kind of purely imaginary order $ mathrm K_ia(x) $ is discussed; we also present analogous results for the derivative. The methods are based on the use of Maclaurin series, nonoscillatory integral representations, asymptotic expansions, and a continued fraction method, depending on the ranges of x and a. We discuss the range of applicability of the different approaches considered and conclude that power series, the continued fraction method, and the nonoscillatory integral representation can be used to accurately compute the function $ mathrm K_ia(x) $ in the range $ 0 leq a leq 200 $, $ 0 leq x leq 100 $; using a similar scheme the derivative $ mathrm K'_ia(x) $ can also be computed within these ranges.

rawabstract:
The evaluation of the modified Bessel function of the third kind of purely imaginary order $ \mathrm {K}_{ia}(x) $ is discussed; we also present analogous results for the derivative. The methods are based on the use of Maclaurin series, nonoscillatory integral representations, asymptotic expansions, and a continued fraction method, depending on the ranges of x and a. We discuss the range of applicability of the different approaches considered and conclude that power series, the continued fraction method, and the nonoscillatory integral representation can be used to accurately compute the function $ \mathrm {K}_{ia}(x) $ in the range $ 0 \leq a \leq 200 $, $ 0 \leq x \leq 100 $; using a similar scheme the derivative $ \mathrm {K}'_{ia(x)} $ can also be computed within these ranges.

entry:
@Article{Gil:2002:EMB,
  author =       "Amparo Gil and Javier Segura and Nico M. Temme",
  title =        "Evaluation of the Modified {Bessel} Function of the
                 Third Kind of Imaginary Orders",
  journal =      j-J-COMPUT-PHYS,
  volume =       "175",
  number =       "2",
  pages =        "398--411",
  day =          "20",
  month =        jan,
  year =         "2002",
  CODEN =        "JCTPAH",
  DOI =          "https://doi.org/10.1006/jcph.2001.6894",
  ISSN =         "0021-9991 (print), 1090-2716 (electronic)",
  ISSN-L =       "0021-9991",
  bibdate =      "Mon Jan 2 22:12:13 MST 2012",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jcomputphys2000.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0021999101968949",
  abstract =     "The evaluation of the modified Bessel function of the
                 third kind of purely imaginary order $ \mathrm
                 {K}_{ia}(x) $ is discussed; we also present analogous
                 results for the derivative. The methods are based on
                 the use of Maclaurin series, nonoscillatory integral
                 representations, asymptotic expansions, and a continued
                 fraction method, depending on the ranges of x and a. We
                 discuss the range of applicability of the different
                 approaches considered and conclude that power series,
                 the continued fraction method, and the nonoscillatory
                 integral representation can be used to accurately
                 compute the function $ \mathrm {K}_{ia}(x) $ in the
                 range $ 0 \leq a \leq 200 $, $ 0 \leq x \leq 100 $;
                 using a similar scheme the derivative $ \mathrm
                 {K}'_{ia(x)} $ can also be computed within these
                 ranges.",
  acknowledgement = ack-nhfb,
  fjournal =     "Journal of Computational Physics",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00219991/",
}


bibtype: article
filename: elefunt.bib
label: Gray:2002:ARE
author: Norman Gray
title: Automatic reduction of elliptic integrals using Carlson's relations
journal: j-MATH-COMPUT
volume: 71
number: 237
pages: 311--318
month: jan
monthnumber: 1
year: 2002
CODEN: MCMPAF
ISSN: 0025-5718 (print), 1088-6842 (electronic) OR 00255718 (print), 10886842 (electronic)
ISSNL: 0025-5718 OR 00255718
bibdate: Thu Jan 31 06:16:28 MST 2002
bibsource: http://www.ams.org/mcom/2002-71-237; https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2002.01.31 06:16:28 MST
fjournal: Mathematics of Computation
acknowledgement: ack-nhfb
journalURL: http://www.ams.org/mcom/
rawauthor: Norman Gray
rawtitle: Automatic reduction of elliptic integrals using {Carlson}'s relations
URL:
http://www.ams.org/journal-getitem?pii=S0025-5718-01-01333-3; http://www.ams.org/mcom/2002-71-237/S0025-5718-01-01333-3/S0025-5718-01-01333-3.dvi; http://www.ams.org/mcom/2002-71-237/S0025-5718-01-01333-3/S0025-5718-01-01333-3.pdf; http://www.ams.org/mcom/2002-71-237/S0025-5718-01-01333-3/S0025-5718-01-01333-3.ps; http://www.ams.org/mcom/2002-71-237/S0025-5718-01-01333-3/S0025-5718-01-01333-3.tex

entry:
@Article{Gray:2002:ARE,
  author =       "Norman Gray",
  title =        "Automatic reduction of elliptic integrals using
                 {Carlson}'s relations",
  journal =      j-MATH-COMPUT,
  volume =       "71",
  number =       "237",
  pages =        "311--318",
  month =        jan,
  year =         "2002",
  CODEN =        "MCMPAF",
  ISSN =         "0025-5718 (print), 1088-6842 (electronic)",
  ISSN-L =       "0025-5718",
  bibdate =      "Thu Jan 31 06:16:28 MST 2002",
  bibsource =    "http://www.ams.org/mcom/2002-71-237;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.ams.org/journal-getitem?pii=S0025-5718-01-01333-3;
                 http://www.ams.org/mcom/2002-71-237/S0025-5718-01-01333-3/S0025-5718-01-01333-3.dvi;
                 http://www.ams.org/mcom/2002-71-237/S0025-5718-01-01333-3/S0025-5718-01-01333-3.pdf;
                 http://www.ams.org/mcom/2002-71-237/S0025-5718-01-01333-3/S0025-5718-01-01333-3.ps;
                 http://www.ams.org/mcom/2002-71-237/S0025-5718-01-01333-3/S0025-5718-01-01333-3.tex",
  acknowledgement = ack-nhfb,
  fjournal =     "Mathematics of Computation",
  journal-URL =  "http://www.ams.org/mcom/",
}


bibtype: article
filename: elefunt.bib
label: Hassenpflug:2002:EAS
author: W. C. Hassenpflug
title: Error analysis in the series evaluation of the exponential type integral $ e^z E_1 (z) $
journal: j-COMPUT-MATH-APPL
volume: 43
number: 1--2
pages: 207--266
month: jan
monthnumber: 1
year: 2002
CODEN: CMAPDK
ISSN: 0898-1221 (print), 1873-7668 (electronic) OR 08981221 (print), 18737668 (electronic)
ISSNL: 0898-1221 OR 08981221
bibdate: Wed Mar 1 21:49:20 MST 2017
bibtimestamp: 2017.03.01 21:49:20 MST
URL: http://www.sciencedirect.com/science/article/pii/S0898122101002838
fjournal: Computers and Mathematics with Applications
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/08981221
rawauthor: W. C. Hassenpflug
rawtitle: Error analysis in the series evaluation of the exponential type integral {$ e^z E_1 (z) $}
bibsource:
https://www.math.utah.edu/pub/tex/bib/computmathappl2000.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

entry:
@Article{Hassenpflug:2002:EAS,
  author =       "W. C. Hassenpflug",
  title =        "Error analysis in the series evaluation of the
                 exponential type integral {$ e^z E_1 (z) $}",
  journal =      j-COMPUT-MATH-APPL,
  volume =       "43",
  number =       "1--2",
  pages =        "207--266",
  month =        jan,
  year =         "2002",
  CODEN =        "CMAPDK",
  ISSN =         "0898-1221 (print), 1873-7668 (electronic)",
  ISSN-L =       "0898-1221",
  bibdate =      "Wed Mar 1 21:49:20 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/S0898122101002838",
  acknowledgement = ack-nhfb,
  fjournal =     "Computers and Mathematics with Applications",
  journal-URL =  "http://www.sciencedirect.com/science/journal/08981221",
}


bibtype: article
filename: elefunt.bib
label: Kilbas:2002:ARH
author: Anatoly A. Kilbas and Luis Rodriguez and Juan J. Trujillo
title: Asymptotic representations for hypergeometric-Bessel type function and fractional integrals
journal: j-J-COMPUT-APPL-MATH
volume: 149
number: 2
pages: 469--487
day: 15
month: dec
monthnumber: 12
year: 2002
CODEN: JCAMDI
ISSN: 0377-0427 (print), 1879-1778 (electronic) OR 03770427 (print), 18791778 (electronic)
ISSNL: 0377-0427 OR 03770427
bibdate: Sat Feb 25 12:52:32 MST 2017
bibtimestamp: 2017.02.25 12:52:32 MST
URL: http://www.sciencedirect.com/science/article/pii/S0377042702005629
fjournal: Journal of Computational and Applied Mathematics
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/03770427
rawauthor: Anatoly A. Kilbas and Luis Rodr{\'\i}guez and Juan J. Trujillo
rawtitle: Asymptotic representations for hypergeometric-{Bessel} type function and fractional integrals
entry:
@Article{Kilbas:2002:ARH,
  author =       "Anatoly A. Kilbas and Luis Rodr{\'\i}guez and Juan J.
                 Trujillo",
  title =        "Asymptotic representations for hypergeometric-{Bessel}
                 type function and fractional integrals",
  journal =      j-J-COMPUT-APPL-MATH,
  volume =       "149",
  number =       "2",
  pages =        "469--487",
  day =          "15",
  month =        dec,
  year =         "2002",
  CODEN =        "JCAMDI",
  ISSN =         "0377-0427 (print), 1879-1778 (electronic)",
  ISSN-L =       "0377-0427",
  bibdate =      "Sat Feb 25 12:52:32 MST 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2000.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0377042702005629",
  acknowledgement = ack-nhfb,
  fjournal =     "Journal of Computational and Applied Mathematics",
  journal-URL =  "http://www.sciencedirect.com/science/journal/03770427",
}


bibtype: book
filename: elefunt.bib
label: Korenev:2002:BFT
author: B. G. (Boris Grigorevich) Korenev
title: Bessel Functions and Their Applications
publisher: pub-TAYLOR-FRANCIS
address: pub-TAYLOR-FRANCIS:adr
pages: ix + 276
year: 2002
ISBN: 0-415-28130-X (hardcover) OR 041528130X (hardcover)
ISBN13: 978-0-415-28130-0 (hardcover) OR 9780415281300 (hardcover)
LCCN: QA408 .K67 2002
bibdate: Sat Oct 30 17:01:51 MDT 2010
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib; prodorbis.library.yale.edu:7090/voyager
bibtimestamp: 2010.10.30 17:01:51 MDT
series: Analytical methods and special functions
subject: Bessel functions
acknowledgement: ack-nhfb
rawauthor: B. G. (Boris Grigorevich) Korenev
rawtitle: {Bessel} Functions and Their Applications
TOC:
Part 1. Foundation of the theory of Bessel functions 1. The Bessel equation Properties of Bessel functions 2. Definite and improper integrals Series in Bessel functions Part 2. Applications of Bessel functions 3. Problems of the theory of plates and shells 4. Problems of the theory of oscillations, hydrodynamics and heat transfer Appendix A. Brief information on gamma functions

rawTOC:
Part 1. Foundation of the theory of Bessel functions \\ 1. The Bessel equation \\ Properties of Bessel functions \\ 2. Definite and improper integrals \\ Series in Bessel functions \\ Part 2. Applications of Bessel functions \\ 3. Problems of the theory of plates and shells \\ 4. Problems of the theory of oscillations, hydrodynamics and heat transfer \\ Appendix A. Brief information on gamma functions

entry:
@Book{Korenev:2002:BFT,
  author =       "B. G. (Boris Grigorevich) Korenev",
  title =        "{Bessel} Functions and Their Applications",
  publisher =    pub-TAYLOR-FRANCIS,
  address =      pub-TAYLOR-FRANCIS:adr,
  pages =        "ix + 276",
  year =         "2002",
  ISBN =         "0-415-28130-X (hardcover)",
  ISBN-13 =      "978-0-415-28130-0 (hardcover)",
  LCCN =         "QA408 .K67 2002",
  bibdate =      "Sat Oct 30 17:01:51 MDT 2010",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 prodorbis.library.yale.edu:7090/voyager",
  series =       "Analytical methods and special functions",
  acknowledgement = ack-nhfb,
  subject =      "Bessel functions",
  tableofcontents = "Part 1. Foundation of the theory of Bessel
                 functions \\
                 1. The Bessel equation \\
                 Properties of Bessel functions \\
                 2. Definite and improper integrals \\
                 Series in Bessel functions \\
                 Part 2. Applications of Bessel functions \\
                 3. Problems of the theory of plates and shells \\
                 4. Problems of the theory of oscillations,
                 hydrodynamics and heat transfer \\
                 Appendix A. Brief information on gamma functions",
  xxaddress =    pub-CRC:adr,
  xxpublisher =  pub-CRC,
}


bibtype: book
filename: elefunt.bib
label: Li:2002:SWF
author: Le-Wei Li and Xiao-Kang Kang and Mook-Seng Leong
title: Spheroidal Wave Functions in Electromagnetic Theory
publisher: pub-WILEY
address: pub-WILEY:adr
pages: xiii + 295
year: 2002
ISBN: 0-471-03170-4 (hardcover) OR 0471031704 (hardcover)
ISBN13: 978-0-471-03170-3 (hardcover) OR 9780471031703 (hardcover)
LCCN: QC670 .L49 2002
bibdate: Sat Apr 1 14:32:29 MDT 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager
bibtimestamp: 2017.04.01 14:32:29 MDT
series: Wiley series in microwave and optical engineering
subject: Electromagnetic theory; Spheroidal functions
acknowledgement: ack-nhfb
rawauthor: Le-Wei Li and Xiao-Kang Kang and Mook-Seng Leong
rawtitle: Spheroidal Wave Functions in Electromagnetic Theory
URL:
http://www.loc.gov/catdir/bios/wiley043/2001045399.html; http://www.loc.gov/catdir/description/wiley036/2001045399.html; http://www.loc.gov/catdir/toc/onix07/2001045399.html

TOC:
Preface Acknowledgments Introduction Spheroidal Coordinates and Wave Functions Dyadic Green's Functions in Spheroidal Systems EM Scattering by a Conducting Spheroid EM Scattering by a Coated Dielectric Spheroid Spheroidal Antennas SAR Distributions in a Spheroidal Head Model Analysis of Rainfall Attenuation Using Oblate Raindrops EM Eigenfrequencies in a Spheroidal Cavity Appendix A: Expressions of Spheroidal Vector Wave Functions Appendix B: Intermediates $I_t,ell ^mn(c)$ in Closed Form Appendix C: $cal U^q(i),t$ and $cal V^(i),t$ Used in the Matrix Equation System References Index

rawTOC:
Preface \\ Acknowledgments \\ Introduction \\ Spheroidal Coordinates and Wave Functions \\ Dyadic Green's Functions in Spheroidal Systems \\ EM Scattering by a Conducting Spheroid \\ EM Scattering by a Coated Dielectric Spheroid \\ Spheroidal Antennas \\ SAR Distributions in a Spheroidal Head Model \\ Analysis of Rainfall Attenuation Using Oblate Raindrops \\ EM Eigenfrequencies in a Spheroidal Cavity \\ Appendix A: Expressions of Spheroidal Vector Wave Functions \\ Appendix B: Intermediates $I_{t,\ell}^{mn}(c)$ in Closed Form \\ Appendix C: ${\cal U}^{q(i),t}$ and ${\cal V}^{(i),t}$ Used in the Matrix Equation System \\ References \\ Index

entry:
@Book{Li:2002:SWF,
  author =       "Le-Wei Li and Xiao-Kang Kang and Mook-Seng Leong",
  title =        "Spheroidal Wave Functions in Electromagnetic Theory",
  publisher =    pub-WILEY,
  address =      pub-WILEY:adr,
  pages =        "xiii + 295",
  year =         "2002",
  ISBN =         "0-471-03170-4 (hardcover)",
  ISBN-13 =      "978-0-471-03170-3 (hardcover)",
  LCCN =         "QC670 .L49 2002",
  bibdate =      "Sat Apr 1 14:32:29 MDT 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 z3950.loc.gov:7090/Voyager",
  series =       "Wiley series in microwave and optical engineering",
  URL =          "http://www.loc.gov/catdir/bios/wiley043/2001045399.html;
                 http://www.loc.gov/catdir/description/wiley036/2001045399.html;
                 http://www.loc.gov/catdir/toc/onix07/2001045399.html",
  acknowledgement = ack-nhfb,
  subject =      "Electromagnetic theory; Spheroidal functions",
  tableofcontents = "Preface \\
                 Acknowledgments \\
                 Introduction \\
                 Spheroidal Coordinates and Wave Functions \\
                 Dyadic Green's Functions in Spheroidal Systems \\
                 EM Scattering by a Conducting Spheroid \\
                 EM Scattering by a Coated Dielectric Spheroid \\
                 Spheroidal Antennas \\
                 SAR Distributions in a Spheroidal Head Model \\
                 Analysis of Rainfall Attenuation Using Oblate Raindrops
                 \\
                 EM Eigenfrequencies in a Spheroidal Cavity \\
                 Appendix A: Expressions of Spheroidal Vector Wave
                 Functions \\
                 Appendix B: Intermediates $I_{t,\ell}^{mn}(c)$ in
                 Closed Form \\
                 Appendix C: ${\cal U}^{q(i),t}$ and ${\cal V}^{(i),t}$
                 Used in the Matrix Equation System \\
                 References \\
                 Index",
}


bibtype: article
filename: elefunt.bib
label: McCluskey:2002:MLF
author: Glen McCluskey
title: Math Library Functions in C9X
journal: j-LOGIN
volume: 27
number: 2
pages: 8--13
month: apr
monthnumber: 4
year: 2002
CODEN: LOGNEM
ISSN: 1044-6397 OR 10446397
bibdate: Tue Apr 11 10:52:14 MDT 2006
bibtimestamp: 2006.04.11 10:52:14 MDT
URL: http://www.usenix.org/publications/login/2002-04/pdfs/mccluskey.pdf
fjournal: ;login: the USENIX Association newsletter
remark: This is a short tutorial on some of the new math library functions in C99.
acknowledgement: ack-nhfb
rawauthor: Glen McCluskey
rawtitle: Math Library Functions in {C9X}
bibsource:
http://www.usenix.org/publications/login/2002-04/index.html; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

entry:
@Article{McCluskey:2002:MLF,
  author =       "Glen McCluskey",
  title =        "Math Library Functions in {C9X}",
  journal =      j-LOGIN,
  volume =       "27",
  number =       "2",
  pages =        "8--13",
  month =        apr,
  year =         "2002",
  CODEN =        "LOGNEM",
  ISSN =         "1044-6397",
  bibdate =      "Tue Apr 11 10:52:14 MDT 2006",
  bibsource =    "http://www.usenix.org/publications/login/2002-04/index.html;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.usenix.org/publications/login/2002-04/pdfs/mccluskey.pdf",
  acknowledgement = ack-nhfb,
  fjournal =     ";login: the USENIX Association newsletter",
  remark =       "This is a short tutorial on some of the new math
                 library functions in C99.",
}


bibtype: article
filename: elefunt.bib
label: Paris:2002:EBU
author: R. B. Paris
title: Error bounds for the uniform asymptotic expansion of the incomplete gamma function
journal: j-J-COMPUT-APPL-MATH
volume: 147
number: 1
pages: 215--231
day: 1
month: oct
monthnumber: 10
year: 2002
CODEN: JCAMDI
ISSN: 0377-0427 (print), 1879-1778 (electronic) OR 03770427 (print), 18791778 (electronic)
ISSNL: 0377-0427 OR 03770427
bibdate: Sat Feb 25 12:52:30 MST 2017
bibtimestamp: 2017.02.25 12:52:30 MST
URL: http://www.sciencedirect.com/science/article/pii/S037704270200434X
fjournal: Journal of Computational and Applied Mathematics
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/03770427
rawauthor: R. B. Paris
rawtitle: Error bounds for the uniform asymptotic expansion of the incomplete gamma function
entry:
@Article{Paris:2002:EBU,
  author =       "R. B. Paris",
  title =        "Error bounds for the uniform asymptotic expansion of
                 the incomplete gamma function",
  journal =      j-J-COMPUT-APPL-MATH,
  volume =       "147",
  number =       "1",
  pages =        "215--231",
  day =          "1",
  month =        oct,
  year =         "2002",
  CODEN =        "JCAMDI",
  ISSN =         "0377-0427 (print), 1879-1778 (electronic)",
  ISSN-L =       "0377-0427",
  bibdate =      "Sat Feb 25 12:52:30 MST 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2000.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S037704270200434X",
  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: Paris:2002:UAE
author: R. B. Paris
title: A uniform asymptotic expansion for the incomplete gamma function
journal: j-J-COMPUT-APPL-MATH
volume: 148
number: 2
pages: 323--339
month: nov
monthnumber: 11
year: 2002
CODEN: JCAMDI
DOI: https://doi.org/10.1016/s0377-0427(02)00553-8
ISSN: 0377-0427 (print), 1879-1778 (electronic) OR 03770427 (print), 18791778 (electronic)
ISSNL: 0377-0427 OR 03770427
bibdate: Sat Feb 18 09:18:08 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2017.02.18 09:18:08 ???
note: See related work cite Paris:2016:UAE.
fjournal: Journal of Computational and Applied Mathematics
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/03770427
rawauthor: R. B. Paris
rawnote: See related work \cite{Paris:2016:UAE}.
rawtitle: A uniform asymptotic expansion for the incomplete gamma function
entry:
@Article{Paris:2002:UAE,
  author =       "R. B. Paris",
  title =        "A uniform asymptotic expansion for the incomplete
                 gamma function",
  journal =      j-J-COMPUT-APPL-MATH,
  volume =       "148",
  number =       "2",
  pages =        "323--339",
  month =        nov,
  year =         "2002",
  CODEN =        "JCAMDI",
  DOI =          "https://doi.org/10.1016/s0377-0427(02)00553-8",
  ISSN =         "0377-0427 (print), 1879-1778 (electronic)",
  ISSN-L =       "0377-0427",
  bibdate =      "Sat Feb 18 09:18:08 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  note =         "See related work \cite{Paris:2016:UAE}.",
  acknowledgement = ack-nhfb,
  fjournal =     "Journal of Computational and Applied Mathematics",
  journal-URL =  "http://www.sciencedirect.com/science/journal/03770427",
}


bibtype: inproceedings
filename: elefunt.bib
label: Pineiro:2002:HRL
author: J.-A. Pineiro and M. D. Ercegovac and J. D. Bruguera
title: High-radix logarithm with selection by rounding
publisher: pub-IEEE
address: pub-IEEE:adr
pages: 101--110
year: 2002
bibdate: Sat Jul 16 11:25:05 MDT 2005
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2005.07.16 11:25:05 MDT
acknowledgement: ack-nhfb
rawauthor: J.-A. Pineiro and M. D. Ercegovac and J. D. Bruguera
rawtitle: High-radix logarithm with selection by rounding
booktitle:
The IEEE International Conference on Application-Specific Systems, Architectures and Processors, 2002. Proceedings. 17--19 July 2002

rawbooktitle:
{The IEEE International Conference on Application-Specific Systems, Architectures and Processors, 2002. Proceedings. 17--19 July 2002}

entry:
@InProceedings{Pineiro:2002:HRL,
  author =       "J.-A. Pineiro and M. D. Ercegovac and J. D. Bruguera",
  booktitle =    "{The IEEE International Conference on
                 Application-Specific Systems, Architectures and
                 Processors, 2002. Proceedings. 17--19 July 2002}",
  title =        "High-radix logarithm with selection by rounding",
  publisher =    pub-IEEE,
  address =      pub-IEEE:adr,
  pages =        "101--110",
  year =         "2002",
  CODEN =        "????",
  ISSN =         "????",
  bibdate =      "Sat Jul 16 11:25:05 MDT 2005",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
  summary =      "A high-radix digit-recurrence algorithm or the
                 computation of the logarithm is presented in this
                 paper. Selection by rounding is used in iterations
                 j/spl ges/2, and selection by table in the first
                 iteration is combined with a restricted digit-set
                 \ldots{}",
}


bibtype: article
filename: elefunt.bib
label: Pineiro:2002:HSD
author: J. A. Pineiro and J. D. Bruguera
journal: j-IEEE-TRANS-COMPUT
volume: 51
number: 12
pages: 1377--1388
month: dec
monthnumber: 12
year: 2002
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2002.1146704
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Sat Jul 16 15:28:13 MDT 2005
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2005.07.16 15:28:13 MDT
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: J. A. Pi{\~n}eiro and J. D. Bruguera
title:
High-Speed Double Precision Computation of Reciprocal, Division, Square Root, and Inverse Square Root

rawtitle:
High-Speed Double Precision Computation of Reciprocal, Division, Square Root, and Inverse Square Root

entry:
@Article{Pineiro:2002:HSD,
  author =       "J. A. Pi{\~n}eiro and J. D. Bruguera",
  title =        "High-Speed Double Precision Computation of Reciprocal,
                 Division, Square Root, and Inverse Square Root",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "51",
  number =       "12",
  pages =        "1377--1388",
  month =        dec,
  year =         "2002",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2002.1146704",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Sat Jul 16 15:28:13 MDT 2005",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
  ajournal =     "IEEE Trans. Comput.",
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
  summary =      "A new method for the high-speed computation of
                 double-precision floating-point reciprocal, division,
                 square root, and inverse square root operations is
                 presented in this paper. This method employs a
                 second-degree minimax polynomial approximation to
                 \ldots{}",
}


bibtype: book
filename: elefunt.bib
label: Samko:2002:HIT
author: S. G. (Stefan Grigorevich) Samko
title: Hypersingular Integrals and Their Applications
volume: 5
publisher: pub-TAYLOR-FRANCIS
address: pub-TAYLOR-FRANCIS:adr
pages: xvii + 359
year: 2002
DOI: https://doi.org/10.1201/9781482264968
ISBN: 0-415-27268-8 OR 0415272688
ISBN13: 978-0-415-27268-1 OR 9780415272681
LCCN: QA403.5 .S26 2002
bibdate: Sat Oct 30 17:22:10 MDT 2010
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib; prodorbis.library.yale.edu:7090/voyager
bibtimestamp: 2010.10.30 17:22:10 MDT
series: Analytical methods and special functions
subject: singular integrals
acknowledgement: ack-nhfb
rawauthor: S. G. (Stefan Grigorevich) Samko
rawtitle: Hypersingular Integrals and Their Applications
TOC:
Part 1: Hypersingular integrals 1: Some basics from the theory of special functions and operator theory 2: The Riesz potential operator and Lizorkin type invariant spaces $Phi _v$ 3: Hypersingular integrals with constant characteristics 4: Potentials and hypersingular integrals with homogeneous characteristics 5: Hypersingular integrals with non-homogeneous characteristics 6: Hypersingular integrals on the unit sphere Part 2: Applications of hypersingular integrals 7: Characterization of some function spaces in terms of hypersingular integrals 8: Solution of multidimensional integral equations of the first kind with a potential type kernel 9: Hypersingular operators as positive fractional powers of some operators of mathematical physics 10: Regularization of multidimensional integral equations of the first kind with a potential type kernel 11: Some modifications of hypersingular integrals and their applications

rawTOC:
Part 1: Hypersingular integrals \\ 1: Some basics from the theory of special functions and operator theory \\ 2: The Riesz potential operator and Lizorkin type invariant spaces $\Phi_v$ \\ 3: Hypersingular integrals with constant characteristics \\ 4: Potentials and hypersingular integrals with homogeneous characteristics \\ 5: Hypersingular integrals with non-homogeneous characteristics \\ 6: Hypersingular integrals on the unit sphere \\ Part 2: Applications of hypersingular integrals \\ 7: Characterization of some function spaces in terms of hypersingular integrals \\ 8: Solution of multidimensional integral equations of the first kind with a potential type kernel \\ 9: Hypersingular operators as positive fractional powers of some operators of mathematical physics \\ 10: Regularization of multidimensional integral equations of the first kind with a potential type kernel \\ 11: Some modifications of hypersingular integrals and their applications

entry:
@Book{Samko:2002:HIT,
  author =       "S. G. (Stefan Grigorevich) Samko",
  title =        "Hypersingular Integrals and Their Applications",
  volume =       "5",
  publisher =    pub-TAYLOR-FRANCIS,
  address =      pub-TAYLOR-FRANCIS:adr,
  pages =        "xvii + 359",
  year =         "2002",
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                 6: Hypersingular integrals on the unit sphere \\
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                 7: Characterization of some function spaces in terms of
                 hypersingular integrals \\
                 8: Solution of multidimensional integral equations of
                 the first kind with a potential type kernel \\
                 9: Hypersingular operators as positive fractional
                 powers of some operators of mathematical physics \\
                 10: Regularization of multidimensional integral
                 equations of the first kind with a potential type
                 kernel \\
                 11: Some modifications of hypersingular integrals and
                 their applications",
}


bibtype: inproceedings
filename: elefunt.bib
label: Sawada:2002:FVD
author: J. Sawada
title: Formal verification of divide and square root algorithms using series calculation
crossref: Borrione:2002:TIW
pages: 31--49
year: 2002
bibdate: Fri Jun 24 15:14:00 2005
bibtimestamp: 2005.06.24 15:14:00 ???
acknowledgement: ack-nhfb
rawauthor: J. Sawada
rawtitle: Formal verification of divide and square root algorithms using series calculation
entry:
@InProceedings{Sawada:2002:FVD,
  author =       "J. Sawada",
  title =        "Formal verification of divide and square root
                 algorithms using series calculation",
  crossref =     "Borrione:2002:TIW",
  pages =        "31--49",
  year =         "2002",
  bibdate =      "Fri Jun 24 15:14:00 2005",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
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bibtype: article
filename: elefunt.bib
label: Sawada:2002:MVS
author: Jun Sawada and Ruben Gamboa
title: Mechanical Verification of a Square Root Algorithm Using Taylor's Theorem
journal: j-LECT-NOTES-COMP-SCI
volume: 2517
pages: 274--??
year: 2002
CODEN: LNCSD9
ISSN: 0302-9743 (print), 1611-3349 (electronic) OR 03029743 (print), 16113349 (electronic)
ISSNL: 0302-9743 OR 03029743
bibdate: Sat Nov 30 20:58:00 MST 2002
bibtimestamp: 2002.11.30 20:58:00 MST
fjournal: Lecture Notes in Computer Science
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/bookseries/558
rawauthor: Jun Sawada and Ruben Gamboa
rawtitle: Mechanical Verification of a Square Root Algorithm Using {Taylor}'s Theorem
bibsource:
http://link.springer-ny.com/link/service/series/0558/tocs/t2517.htm; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

URL:
http://link.springer.de/link/service/series/0558/bibs/2517/25170274.htm; http://link.springer.de/link/service/series/0558/papers/2517/25170274.pdf

entry:
@Article{Sawada:2002:MVS,
  author =       "Jun Sawada and Ruben Gamboa",
  title =        "Mechanical Verification of a Square Root Algorithm
                 Using {Taylor}'s Theorem",
  journal =      j-LECT-NOTES-COMP-SCI,
  volume =       "2517",
  pages =        "274--??",
  year =         "2002",
  CODEN =        "LNCSD9",
  ISSN =         "0302-9743 (print), 1611-3349 (electronic)",
  ISSN-L =       "0302-9743",
  bibdate =      "Sat Nov 30 20:58:00 MST 2002",
  bibsource =    "http://link.springer-ny.com/link/service/series/0558/tocs/t2517.htm;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://link.springer.de/link/service/series/0558/bibs/2517/25170274.htm;
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  acknowledgement = ack-nhfb,
  fjournal =     "Lecture Notes in Computer Science",
  journal-URL =  "http://link.springer.com/bookseries/558",
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bibtype: misc
filename: elefunt.bib
label: Sebah:2002:IGF
author: Pascal Sebah and Xavier Gourdon
title: Introduction to the Gamma Function
pages: 274--??
day: 4
month: feb
monthnumber: 2
year: 2002
bibdate: Sat May 01 16:07:51 2010
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2010.05.01 16:07:51 ???
acknowledgement: ack-nhfb
howpublished: World-Wide Web document
rawauthor: Pascal Sebah and Xavier Gourdon
rawtitle: Introduction to the Gamma Function
URL:
http://numbers.computation.free.fr/Constants/constants.html; http://numbers.computation.free.fr/Constants/Miscellaneous/gammaFunction.ps

entry:
@Misc{Sebah:2002:IGF,
  author =       "Pascal Sebah and Xavier Gourdon",
  title =        "Introduction to the Gamma Function",
  howpublished = "World-Wide Web document",
  day =          "4",
  month =        feb,
  year =         "2002",
  bibdate =      "Sat May 01 16:07:51 2010",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://numbers.computation.free.fr/Constants/constants.html;
                 http://numbers.computation.free.fr/Constants/Miscellaneous/gammaFunction.ps",
  acknowledgement = ack-nhfb,
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bibtype: article
filename: elefunt.bib
label: Shore:2002:RMM
author: Haim Shore
title: Response Modeling Methodology (RMM)-Exploring the Properties of the Implied Error Distribution
journal: j-COMMUN-STAT-THEORY-METH
volume: 31
number: 12
pages: 2225--2249
year: 2002
CODEN: CSTMDC
DOI: https://doi.org/10.1081/STA-120017223
ISSN: 0361-0926 (print), 1532-415X (electronic) OR 03610926 (print), 1532415X (electronic)
ISSNL: 0361-0926 OR 03610926
bibdate: Wed Jan 27 05:41:30 MST 2016
bibtimestamp: 2016.01.27 05:41:30 MST
fjournal: Communications in Statistics: Theory and Methods
acknowledgement: ack-nhfb
journalURL: http://www.tandfonline.com/loi/lsta20
rawauthor: Haim Shore
rawtitle: Response Modeling Methodology ({RMM})-Exploring the Properties of the Implied Error Distribution
bibsource:
https://www.math.utah.edu/pub/tex/bib/communstattheorymeth2000.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

entry:
@Article{Shore:2002:RMM,
  author =       "Haim Shore",
  title =        "Response Modeling Methodology ({RMM})-Exploring the
                 Properties of the Implied Error Distribution",
  journal =      j-COMMUN-STAT-THEORY-METH,
  volume =       "31",
  number =       "12",
  pages =        "2225--2249",
  year =         "2002",
  CODEN =        "CSTMDC",
  DOI =          "https://doi.org/10.1081/STA-120017223",
  ISSN =         "0361-0926 (print), 1532-415X (electronic)",
  ISSN-L =       "0361-0926",
  bibdate =      "Wed Jan 27 05:41:30 MST 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/communstattheorymeth2000.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "Communications in Statistics: Theory and Methods",
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bibtype: article
filename: elefunt.bib
label: Tornaria:2002:SRM
author: Gonzalo Tornaria
title: Square Roots Modulo $p$
journal: j-LECT-NOTES-COMP-SCI
volume: 2286
pages: 430--??
year: 2002
CODEN: LNCSD9
ISSN: 0302-9743 (print), 1611-3349 (electronic) OR 03029743 (print), 16113349 (electronic)
ISSNL: 0302-9743 OR 03029743
bibdate: Tue Sep 10 19:09:12 MDT 2002
bibtimestamp: 2002.09.10 19:09:12 MDT
fjournal: Lecture Notes in Computer Science
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/bookseries/558
rawauthor: Gonzalo Tornar{\'\i}a
rawtitle: Square Roots Modulo $p$
bibsource:
http://link.springer-ny.com/link/service/series/0558/tocs/t2286.htm; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib

URL:
http://link.springer-ny.com/link/service/series/0558/bibs/2286/22860430.htm; http://link.springer-ny.com/link/service/series/0558/papers/2286/22860430.pdf

entry:
@Article{Tornaria:2002:SRM,
  author =       "Gonzalo Tornar{\'\i}a",
  title =        "Square Roots Modulo $p$",
  journal =      j-LECT-NOTES-COMP-SCI,
  volume =       "2286",
  pages =        "430--??",
  year =         "2002",
  CODEN =        "LNCSD9",
  ISSN =         "0302-9743 (print), 1611-3349 (electronic)",
  ISSN-L =       "0302-9743",
  bibdate =      "Tue Sep 10 19:09:12 MDT 2002",
  bibsource =    "http://link.springer-ny.com/link/service/series/0558/tocs/t2286.htm;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://link.springer-ny.com/link/service/series/0558/bibs/2286/22860430.htm;
                 http://link.springer-ny.com/link/service/series/0558/papers/2286/22860430.pdf",
  acknowledgement = ack-nhfb,
  fjournal =     "Lecture Notes in Computer Science",
  journal-URL =  "http://link.springer.com/bookseries/558",
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bibtype: book
filename: elefunt.bib
label: Vladimirov:2002:MTG
author: V. S. (Vasilii Sergeevich) Vladimirov
title: Methods of the Theory of Generalized Functions
volume: 6
publisher: pub-TAYLOR-FRANCIS
address: pub-TAYLOR-FRANCIS:adr
pages: xiv + 311
year: 2002
DOI: https://doi.org/10.1201/9781482288162
ISBN: 0-415-27356-0 OR 0415273560
ISBN13: 978-0-415-27356-5 OR 9780415273565
LCCN: QC20.7.T45 V53 2002
bibdate: Sat Oct 30 17:22:15 MDT 2010
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib; prodorbis.library.yale.edu:7090/voyager
bibtimestamp: 2010.10.30 17:22:15 MDT
series: Analytical methods and special functions
subject: Theory of distributions (Functional analysis); Integral transforms; Mathematical physics
acknowledgement: ack-nhfb
rawauthor: V. S. (Vasilii Sergeevich) Vladimirov
rawtitle: Methods of the Theory of Generalized Functions
TOC:
Preface / xi Symbols and Definitions / 1 1. Generalized Functions and their Properties / 5 1. Test and Generalized Functions / 5 1.1. Introduction / 5 1.2. The space of test functions $mathcal D(mathcal O)$ / 6 1.3. The space of generalized functions $mathcal D'(mathcal O)$ / 10 1.4. The completeness of the space of generalized functions $mathcal D'(mathcal O)$ / 12 1.5. The support of a generalized function / 13 1.6. Regular generalized functions / 15 1.7. Measures / 16 1.8. Sochozki formulae / 19 1.9. Change of variables in generalized functions / 21 1.10. Multiplication of generalized functions / 23 2. Differentiation of Generalized Functions / 25 2.1. Derivatives of generalized functions / 25 2.2. The antiderivative (primitive) of a generalized function / 27 2.3. Examples / 29 2.4. The local structure of generalized functions / 35 2.5. Generalized functions with compact support / 36 2.6. Generalized functions with point support / 37 2.7. Generalized functions $mathcal P(pi _nu|x|^alpha - 1)$ / 39 3. Direct Product of Generalized Functions / 41 3.1. The definition of a direct product / 41 3.2. The properties of a direct product / 43 3.3. Some applications / 56 3.4. Generalized functions that are smooth with respect to some of the variables / 48 4. The Convolution of Generalized Functions / 50 4.1. The definition of convolution / 50 4.2. The properties of a convolution / 53 4.3. The existence of a convolution / 57 4.4. Cones in $mathcal R^n$ / 59 4.5. Convolution algebras $mathcal D'(Gamma +)$ and $mathcal D'(Gamma )$ / 63 4.6. Mean functions of generalized functions / 64 4.7. Multiplication of generalized functions / 66 4.8. Convolution as a continuous linear translation invariant operator / 66 4.9. Some applications / 68 5. Tempered Generalized Functions / 74 5.1. The space $S$ of test (rapidly decreasing) functions / 74 5.2. The space $S'$ of tempered generalized functions / 77 5.3. Examples of tempered generalized functions and elementary operations in $S$ /' 78 5.4. The structure of tempered generalized functions / 80 5.5. The direct product of tempered generalized functions / 81 5.6. The convolution of tempered generalized functions / 82 5.7. Homogeneous generalized functions / 85 2. Integral Transformations of Generalized Functions / 89 6. The Fourier Transform of Tempered Generalized Functions / 89 6.1. The Fourier transform of test functions in $S$ / 89 6.2. The Fourier transform of tempered generalized functions / 90 6.3. Properties of the Fourier transform / 92 6.4. The Fourier transform of generalized functions with compact support / 93 6.5. The Fourier transform of a convolution / 94 6.6. Examples / 96 6.7. The Mellin transform / 109 7. Fourier Series of Periodic Generalized Functions / 113 7.1. The definition and elementary properties of periodic generalized functions / 113 7.2. Fourier series of periodic generalized functions / 116 7.3. The convolution algebra $mathcal D'_T$ / 117 7.4. Examples / 119 8. Positive Definite Generalized Functions / 121 8.1. The definition and elementary properties of positive definite generalized functions / 121 8.2. The Bochner--Schwartz theorem / 123 8.3. Examples / 125 9. The Laplace Transform of Tempered Generalized Functions / 126 9.1. Definition of the Laplace transform / 126 9.2. Properties of the Laplace transform / 128 9.3. Examples / 130 10. The Cauchy Kernel and the Transforms of Cauchy--Bochner and Hilbert / 133 10.1. The space $mathcal H_s$ / 133 10.2. The Cauchy kernel $mathcal K_(z)$ / 138 10.3. The Cauchy--Bochner transform / 144 10.4. The Hilbert transform / 146 10.5. Holomorphic functions of the class $mathcal H_a^(s)(C)$ / 147 10.6. The generalized Cauchy--Bochner representation / 151 11. Poisson Kernel and Poisson Transform / 152 11.1. The definition and properties of the Poisson kernel / 152 11.2. The Poisson transform and Poisson representation / 155 11.3. Boundary values of the Poisson integral / 157 12. Algebras of Holomorphic Functions / 159 12.1. The definition of the $H_+(C)$ and $H(C)$ algebras / 160 12.2. Isomorphism of the algebras $S'(C*+) sim H_+(C)$ and $S'(C*) sim H(C)$ / 160 12.3. The Paley--Wiener--Schwartz theorem and its generalizations / 165 12.4. The space $H_a(C)$ is the projective limit of the spaces $H_a'(C)$ / 166 12.5. The Schwartz representation / 168 12.6. A generalization of the Phragmen--Lindelof theorem / 171 13. Equations in Convolution Algebras / 171 13.1. Divisors of unity in the $H_+(C)$ and $H(C)$ algebras / 171 13.2. On division by a polynomial in the $H(C)$ algebra / 172 13.3. Estimates for holomorphic functions with nonnegative imaginary part in $T^C$ / 174 13.4. Divisors of unity in the algebra $W(C)$ / 177 13.5. Example / 177 14. Tauberian Theorems for Generalized Functions / 179 14.1. Preliminary results / 179 14.2. General Tauberian theorem / 183 14.3. One-dimensional Tauberian theorems / 186 14.4. Tauberian and Abelian theorems for nonnegative measures / 187 14.5. Tauberian theorems for holomorphic functions of bounded argument / 188 3. Some Applications in Mathematical Physics / 191 15. Differential Operators with Constant Coefficients / 191 15.1. Fundamental solutions in $mathcal D'$ / 191 15.2. Tempered fundamental solutions / 194 15.3. A descent method / 196 15.4. Examples / 199 15.5. A comparison of differential operators / 207 15.6. Elliptic and hypoelliptic operators / 210 15.7. Hyperbolic operators / 212 15.8. The sweeping principle / 212 16. The Cauchy Problem / 213 16.1. The generalized Cauchy problem for a hyperbolic equation / 213 16.2. Wave potential / 216 16.3. Surface wave potentials / 220 16.4. The Cauchy problem for the wave equation / 222 16.5. A statement of the generalized Cauchy problem for the heat equation / 224 16.6. Heat potential / 224 16.7. Solution of the Cauchy problem for the heat equation / 228 17. Holomorphic Functions with Nonnegative Imaginary Part in $T^C$ / 229 17.1. Preliminary remarks / 229 17.2. Properties of functions of the class $mathcal P_+(T^C)$ / 231 17.3. Estimates of the growth of functions of the class $H_+(T^C)$ / 238 17.4. Smoothness of the spectral function / 240 17.5. Indicator of growth of functions of the class $mathcal P_+T^C$ / 242 17.6. An integral representation of functions of the class $H_+(T^C)$ / 245 18. Holomorphic Functions with Nonnegative Imaginary Part in $T^n$ / 249 18.1. Lemmas / 249 18.2. Functions of the classes $H_+(T^1)$ and $mathcal P_+(T^1)$ / 254 18.3. Functions of the class $mathcal P_+(T^n)$ / 258 18.4. Functions of the class $H_+(T^n)$ / 263 19. Positive Real Matrix Functions in $T^C$ / 266 19.1. Positive real functions in $T^C$ / 267 19.2. Positive real matrix functions in $T^C$ / 269 20. Linear Passive Systems / 271 20.1. Introduction / 271 20.2. Corollaries to the condition of passivity / 273 20.3. The necessary and sufficient conditions for passivity / 277 20.4. Multidimensional dispersion relations / 282 20.5. The fundamental solution and the Cauchy problem / 285 20.6. What differential and difference operators are passive operators? / 287 20.7. Examples / 290 20.8. Quasiasymptotics of the solutions of systems of equations in convolutions / 294 21. Abstract Scattering Operator / 295 21.1. The definition and properties of an abstract scattering matrix / 295 21.2. A description of abstract scattering matrices / 298 21.3. The relationship between passive operators and scattering operators / 299 Bibliography / 303 Index / 309

rawTOC:
Preface / xi \\ Symbols and Definitions / 1 \\ \\ 1. Generalized Functions and their Properties / 5 \\ 1. Test and Generalized Functions / 5 \\ 1.1. Introduction / 5 \\ 1.2. The space of test functions $\mathcal{D}(\mathcal{O})$ / 6 \\ 1.3. The space of generalized functions $\mathcal{D}'(\mathcal{O})$ / 10 \\ 1.4. The completeness of the space of generalized functions $\mathcal{D}'(\mathcal{O})$ / 12 \\ 1.5. The support of a generalized function / 13 \\ 1.6. Regular generalized functions / 15 \\ 1.7. Measures / 16 \\ 1.8. Sochozki formulae / 19 \\ 1.9. Change of variables in generalized functions / 21 \\ 1.10. Multiplication of generalized functions / 23 \\ \\ 2. Differentiation of Generalized Functions / 25 \\ 2.1. Derivatives of generalized functions / 25 \\ 2.2. The antiderivative (primitive) of a generalized function / 27 \\ 2.3. Examples / 29 \\ 2.4. The local structure of generalized functions / 35 \\ 2.5. Generalized functions with compact support / 36 \\ 2.6. Generalized functions with point support / 37 \\ 2.7. Generalized functions $\mathcal{P}(\pi_nu|x|^{\alpha - 1})$ / 39 \\ \\ 3. Direct Product of Generalized Functions / 41 \\ 3.1. The definition of a direct product / 41 \\ 3.2. The properties of a direct product / 43 \\ 3.3. Some applications / 56 \\ 3.4. Generalized functions that are smooth with respect to some of the variables / 48 \\ \\ 4. The Convolution of Generalized Functions / 50 \\ 4.1. The definition of convolution / 50 \\ 4.2. The properties of a convolution / 53 \\ 4.3. The existence of a convolution / 57 \\ 4.4. Cones in $\mathcal{R}^n$ / 59 \\ 4.5. Convolution algebras $\mathcal{D}'(\Gamma+)$ and $\mathcal{D}'(\Gamma)$ / 63 \\ 4.6. Mean functions of generalized functions / 64 \\ 4.7. Multiplication of generalized functions / 66 \\ 4.8. Convolution as a continuous linear translation invariant operator / 66 \\ 4.9. Some applications / 68 \\ \\ 5. Tempered Generalized Functions / 74 \\ 5.1. The space $S$ of test (rapidly decreasing) functions / 74 \\ 5.2. The space $S'$ of tempered generalized functions / 77 \\ 5.3. Examples of tempered generalized functions and elementary operations in $S$ /' 78 \\ 5.4. The structure of tempered generalized functions / 80 \\ 5.5. The direct product of tempered generalized functions / 81 \\ 5.6. The convolution of tempered generalized functions / 82 \\ 5.7. Homogeneous generalized functions / 85 \\ 2. Integral Transformations of Generalized Functions / 89 \\ \\ 6. The Fourier Transform of Tempered Generalized Functions / 89 \\ 6.1. The Fourier transform of test functions in $S$ / 89 \\ 6.2. The Fourier transform of tempered generalized functions / 90 \\ 6.3. Properties of the Fourier transform / 92 \\ 6.4. The Fourier transform of generalized functions with compact support / 93 \\ 6.5. The Fourier transform of a convolution / 94 \\ 6.6. Examples / 96 \\ 6.7. The Mellin transform / 109 \\ \\ 7. Fourier Series of Periodic Generalized Functions / 113 \\ 7.1. The definition and elementary properties of periodic generalized functions / 113 \\ 7.2. Fourier series of periodic generalized functions / 116 \\ 7.3. The convolution algebra $\mathcal{D}'_T$ / 117 \\ 7.4. Examples / 119 \\ \\ 8. Positive Definite Generalized Functions / 121 \\ 8.1. The definition and elementary properties of positive definite generalized functions / 121 \\ 8.2. The Bochner--Schwartz theorem / 123 \\ 8.3. Examples / 125 \\ \\ 9. The Laplace Transform of Tempered Generalized Functions / 126 \\ 9.1. Definition of the Laplace transform / 126 \\ 9.2. Properties of the Laplace transform / 128 \\ 9.3. Examples / 130 \\ \\ 10. The Cauchy Kernel and the Transforms of Cauchy--Bochner and Hilbert / 133 \\ 10.1. The space $\mathcal{H}_s$ / 133 \\ 10.2. The Cauchy kernel $\mathcal{K}_(z)$ / 138 \\ 10.3. The Cauchy--Bochner transform / 144 \\ 10.4. The Hilbert transform / 146 \\ 10.5. Holomorphic functions of the class $\mathcal{H}_a^{(s)}(C)$ / 147 \\ 10.6. The generalized Cauchy--Bochner representation / 151 \\ \\ 11. Poisson Kernel and Poisson Transform / 152 \\ 11.1. The definition and properties of the Poisson kernel / 152 \\ 11.2. The Poisson transform and Poisson representation / 155 \\ 11.3. Boundary values of the Poisson integral / 157 \\ \\ 12. Algebras of Holomorphic Functions / 159 \\ 12.1. The definition of the $H_+(C)$ and $H(C)$ algebras / 160 \\ 12.2. Isomorphism of the algebras $S'(C*+) \sim H_+(C)$ and $S'(C*) \sim H(C)$ / 160 \\ 12.3. The Paley--Wiener--Schwartz theorem and its generalizations / 165 \\ 12.4. The space $H_a(C)$ is the projective limit of the spaces $H_{a'}(C)$ / 166 \\ 12.5. The Schwartz representation / 168 \\ 12.6. A generalization of the Phragmen--Lindelof theorem / 171 \\ \\ 13. Equations in Convolution Algebras / 171 \\ 13.1. Divisors of unity in the $H_+(C)$ and $H(C)$ algebras / 171 \\ 13.2. On division by a polynomial in the $H(C)$ algebra / 172 \\ 13.3. Estimates for holomorphic functions with nonnegative imaginary part in $T^C$ / 174 \\ 13.4. Divisors of unity in the algebra $W(C)$ / 177 \\ 13.5. Example / 177 \\ \\ 14. Tauberian Theorems for Generalized Functions / 179 \\ 14.1. Preliminary results / 179 \\ 14.2. General Tauberian theorem / 183 \\ 14.3. One-dimensional Tauberian theorems / 186 \\ 14.4. Tauberian and Abelian theorems for nonnegative measures / 187 \\ 14.5. Tauberian theorems for holomorphic functions of bounded argument / 188 \\ 3. Some Applications in Mathematical Physics / 191 \\ \\ 15. Differential Operators with Constant Coefficients / 191 \\ 15.1. Fundamental solutions in $\mathcal{D}'$ / 191 \\ 15.2. Tempered fundamental solutions / 194 \\ 15.3. A descent method / 196 \\ 15.4. Examples / 199 \\ 15.5. A comparison of differential operators / 207 \\ 15.6. Elliptic and hypoelliptic operators / 210 \\ 15.7. Hyperbolic operators / 212 \\ 15.8. The sweeping principle / 212 \\ \\ 16. The Cauchy Problem / 213 \\ 16.1. The generalized Cauchy problem for a hyperbolic equation / 213 \\ 16.2. Wave potential / 216 \\ 16.3. Surface wave potentials / 220 \\ 16.4. The Cauchy problem for the wave equation / 222 \\ 16.5. A statement of the generalized Cauchy problem for the heat equation / 224 \\ 16.6. Heat potential / 224 \\ 16.7. Solution of the Cauchy problem for the heat equation / 228 \\ \\ 17. Holomorphic Functions with Nonnegative Imaginary Part in $T^C$ / 229 \\ 17.1. Preliminary remarks / 229 \\ 17.2. Properties of functions of the class $\mathcal{P}_+(T^C)$ / 231 \\ 17.3. Estimates of the growth of functions of the class $H_+(T^C)$ / 238 \\ 17.4. Smoothness of the spectral function / 240 \\ 17.5. Indicator of growth of functions of the class $\mathcal{P}_+T^C$ / 242 \\ 17.6. An integral representation of functions of the class $H_+(T^C)$ / 245 \\ \\ 18. Holomorphic Functions with Nonnegative Imaginary Part in $T^n$ / 249 \\ 18.1. Lemmas / 249 \\ 18.2. Functions of the classes $H_+(T^1)$ and $\mathcal{P}_+(T^1)$ / 254 \\ 18.3. Functions of the class $\mathcal{P}_+(T^n)$ / 258 \\ 18.4. Functions of the class $H_+(T^n)$ / 263 \\ \\ 19. Positive Real Matrix Functions in $T^C$ / 266 \\ 19.1. Positive real functions in $T^C$ / 267 \\ 19.2. Positive real matrix functions in $T^C$ / 269 \\ \\ 20. Linear Passive Systems / 271 \\ 20.1. Introduction / 271 \\ 20.2. Corollaries to the condition of passivity / 273 \\ 20.3. The necessary and sufficient conditions for passivity / 277 \\ 20.4. Multidimensional dispersion relations / 282 \\ 20.5. The fundamental solution and the Cauchy problem / 285 \\ 20.6. What differential and difference operators are passive operators? / 287 \\ 20.7. Examples / 290 \\ 20.8. Quasiasymptotics of the solutions of systems of equations in convolutions / 294 \\ \\ 21. Abstract Scattering Operator / 295 \\ 21.1. The definition and properties of an abstract scattering matrix / 295 \\ 21.2. A description of abstract scattering matrices / 298 \\ 21.3. The relationship between passive operators and scattering operators / 299 \\ \\ Bibliography / 303 \\ Index / 309

entry:
@Book{Vladimirov:2002:MTG,
  author =       "V. S. (Vasilii Sergeevich) Vladimirov",
  title =        "Methods of the Theory of Generalized Functions",
  volume =       "6",
  publisher =    pub-TAYLOR-FRANCIS,
  address =      pub-TAYLOR-FRANCIS:adr,
  pages =        "xiv + 311",
  year =         "2002",
  DOI =          "https://doi.org/10.1201/9781482288162",
  ISBN =         "0-415-27356-0",
  ISBN-13 =      "978-0-415-27356-5",
  LCCN =         "QC20.7.T45 V53 2002",
  bibdate =      "Sat Oct 30 17:22:15 MDT 2010",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 prodorbis.library.yale.edu:7090/voyager",
  series =       "Analytical methods and special functions",
  acknowledgement = ack-nhfb,
  subject =      "Theory of distributions (Functional analysis);
                 Integral transforms; Mathematical physics",
  tableofcontents = "Preface / xi \\
                 Symbols and Definitions / 1 \\
                 \\
                 1. Generalized Functions and their Properties / 5 \\
                 1. Test and Generalized Functions / 5 \\
                 1.1. Introduction / 5 \\
                 1.2. The space of test functions
                 $\mathcal{D}(\mathcal{O})$ / 6 \\
                 1.3. The space of generalized functions
                 $\mathcal{D}'(\mathcal{O})$ / 10 \\
                 1.4. The completeness of the space of generalized
                 functions $\mathcal{D}'(\mathcal{O})$ / 12 \\
                 1.5. The support of a generalized function / 13 \\
                 1.6. Regular generalized functions / 15 \\
                 1.7. Measures / 16 \\
                 1.8. Sochozki formulae / 19 \\
                 1.9. Change of variables in generalized functions / 21
                 \\
                 1.10. Multiplication of generalized functions / 23 \\
                 \\
                 2. Differentiation of Generalized Functions / 25 \\
                 2.1. Derivatives of generalized functions / 25 \\
                 2.2. The antiderivative (primitive) of a generalized
                 function / 27 \\
                 2.3. Examples / 29 \\
                 2.4. The local structure of generalized functions / 35
                 \\
                 2.5. Generalized functions with compact support / 36
                 \\
                 2.6. Generalized functions with point support / 37 \\
                 2.7. Generalized functions
                 $\mathcal{P}(\pi_nu|x|^{\alpha - 1})$ / 39 \\
                 \\
                 3. Direct Product of Generalized Functions / 41 \\
                 3.1. The definition of a direct product / 41 \\
                 3.2. The properties of a direct product / 43 \\
                 3.3. Some applications / 56 \\
                 3.4. Generalized functions that are smooth with respect
                 to some of the variables / 48 \\
                 \\
                 4. The Convolution of Generalized Functions / 50 \\
                 4.1. The definition of convolution / 50 \\
                 4.2. The properties of a convolution / 53 \\
                 4.3. The existence of a convolution / 57 \\
                 4.4. Cones in $\mathcal{R}^n$ / 59 \\
                 4.5. Convolution algebras $\mathcal{D}'(\Gamma+)$ and
                 $\mathcal{D}'(\Gamma)$ / 63 \\
                 4.6. Mean functions of generalized functions / 64 \\
                 4.7. Multiplication of generalized functions / 66 \\
                 4.8. Convolution as a continuous linear translation
                 invariant operator / 66 \\
                 4.9. Some applications / 68 \\
                 \\
                 5. Tempered Generalized Functions / 74 \\
                 5.1. The space $S$ of test (rapidly decreasing)
                 functions / 74 \\
                 5.2. The space $S'$ of tempered generalized functions /
                 77 \\
                 5.3. Examples of tempered generalized functions and
                 elementary operations in $S$ /' 78 \\
                 5.4. The structure of tempered generalized functions /
                 80 \\
                 5.5. The direct product of tempered generalized
                 functions / 81 \\
                 5.6. The convolution of tempered generalized functions
                 / 82 \\
                 5.7. Homogeneous generalized functions / 85 \\
                 2. Integral Transformations of Generalized Functions /
                 89 \\
                 \\
                 6. The Fourier Transform of Tempered Generalized
                 Functions / 89 \\
                 6.1. The Fourier transform of test functions in $S$ /
                 89 \\
                 6.2. The Fourier transform of tempered generalized
                 functions / 90 \\
                 6.3. Properties of the Fourier transform / 92 \\
                 6.4. The Fourier transform of generalized functions
                 with compact support / 93 \\
                 6.5. The Fourier transform of a convolution / 94 \\
                 6.6. Examples / 96 \\
                 6.7. The Mellin transform / 109 \\
                 \\
                 7. Fourier Series of Periodic Generalized Functions /
                 113 \\
                 7.1. The definition and elementary properties of
                 periodic generalized functions / 113 \\
                 7.2. Fourier series of periodic generalized functions /
                 116 \\
                 7.3. The convolution algebra $\mathcal{D}'_T$ / 117 \\
                 7.4. Examples / 119 \\
                 \\
                 8. Positive Definite Generalized Functions / 121 \\
                 8.1. The definition and elementary properties of
                 positive definite generalized functions / 121 \\
                 8.2. The Bochner--Schwartz theorem / 123 \\
                 8.3. Examples / 125 \\
                 \\
                 9. The Laplace Transform of Tempered Generalized
                 Functions / 126 \\
                 9.1. Definition of the Laplace transform / 126 \\
                 9.2. Properties of the Laplace transform / 128 \\
                 9.3. Examples / 130 \\
                 \\
                 10. The Cauchy Kernel and the Transforms of
                 Cauchy--Bochner and Hilbert / 133 \\
                 10.1. The space $\mathcal{H}_s$ / 133 \\
                 10.2. The Cauchy kernel $\mathcal{K}_(z)$ / 138 \\
                 10.3. The Cauchy--Bochner transform / 144 \\
                 10.4. The Hilbert transform / 146 \\
                 10.5. Holomorphic functions of the class
                 $\mathcal{H}_a^{(s)}(C)$ / 147 \\
                 10.6. The generalized Cauchy--Bochner representation /
                 151 \\
                 \\
                 11. Poisson Kernel and Poisson Transform / 152 \\
                 11.1. The definition and properties of the Poisson
                 kernel / 152 \\
                 11.2. The Poisson transform and Poisson representation
                 / 155 \\
                 11.3. Boundary values of the Poisson integral / 157 \\
                 \\
                 12. Algebras of Holomorphic Functions / 159 \\
                 12.1. The definition of the $H_+(C)$ and $H(C)$
                 algebras / 160 \\
                 12.2. Isomorphism of the algebras $S'(C*+) \sim H_+(C)$
                 and $S'(C*) \sim H(C)$ / 160 \\
                 12.3. The Paley--Wiener--Schwartz theorem and its
                 generalizations / 165 \\
                 12.4. The space $H_a(C)$ is the projective limit of the
                 spaces $H_{a'}(C)$ / 166 \\
                 12.5. The Schwartz representation / 168 \\
                 12.6. A generalization of the Phragmen--Lindelof
                 theorem / 171 \\
                 \\
                 13. Equations in Convolution Algebras / 171 \\
                 13.1. Divisors of unity in the $H_+(C)$ and $H(C)$
                 algebras / 171 \\
                 13.2. On division by a polynomial in the $H(C)$ algebra
                 / 172 \\
                 13.3. Estimates for holomorphic functions with
                 nonnegative imaginary part in $T^C$ / 174 \\
                 13.4. Divisors of unity in the algebra $W(C)$ / 177 \\
                 13.5. Example / 177 \\
                 \\
                 14. Tauberian Theorems for Generalized Functions / 179
                 \\
                 14.1. Preliminary results / 179 \\
                 14.2. General Tauberian theorem / 183 \\
                 14.3. One-dimensional Tauberian theorems / 186 \\
                 14.4. Tauberian and Abelian theorems for nonnegative
                 measures / 187 \\
                 14.5. Tauberian theorems for holomorphic functions of
                 bounded argument / 188 \\
                 3. Some Applications in Mathematical Physics / 191 \\
                 \\
                 15. Differential Operators with Constant Coefficients /
                 191 \\
                 15.1. Fundamental solutions in $\mathcal{D}'$ / 191 \\
                 15.2. Tempered fundamental solutions / 194 \\
                 15.3. A descent method / 196 \\
                 15.4. Examples / 199 \\
                 15.5. A comparison of differential operators / 207 \\
                 15.6. Elliptic and hypoelliptic operators / 210 \\
                 15.7. Hyperbolic operators / 212 \\
                 15.8. The sweeping principle / 212 \\
                 \\
                 16. The Cauchy Problem / 213 \\
                 16.1. The generalized Cauchy problem for a hyperbolic
                 equation / 213 \\
                 16.2. Wave potential / 216 \\
                 16.3. Surface wave potentials / 220 \\
                 16.4. The Cauchy problem for the wave equation / 222
                 \\
                 16.5. A statement of the generalized Cauchy problem for
                 the heat equation / 224 \\
                 16.6. Heat potential / 224 \\
                 16.7. Solution of the Cauchy problem for the heat
                 equation / 228 \\
                 \\
                 17. Holomorphic Functions with Nonnegative Imaginary
                 Part in $T^C$ / 229 \\
                 17.1. Preliminary remarks / 229 \\
                 17.2. Properties of functions of the class
                 $\mathcal{P}_+(T^C)$ / 231 \\
                 17.3. Estimates of the growth of functions of the class
                 $H_+(T^C)$ / 238 \\
                 17.4. Smoothness of the spectral function / 240 \\
                 17.5. Indicator of growth of functions of the class
                 $\mathcal{P}_+T^C$ / 242 \\
                 17.6. An integral representation of functions of the
                 class $H_+(T^C)$ / 245 \\
                 \\
                 18. Holomorphic Functions with Nonnegative Imaginary
                 Part in $T^n$ / 249 \\
                 18.1. Lemmas / 249 \\
                 18.2. Functions of the classes $H_+(T^1)$ and
                 $\mathcal{P}_+(T^1)$ / 254 \\
                 18.3. Functions of the class $\mathcal{P}_+(T^n)$ / 258
                 \\
                 18.4. Functions of the class $H_+(T^n)$ / 263 \\
                 \\
                 19. Positive Real Matrix Functions in $T^C$ / 266 \\
                 19.1. Positive real functions in $T^C$ / 267 \\
                 19.2. Positive real matrix functions in $T^C$ / 269 \\
                 \\
                 20. Linear Passive Systems / 271 \\
                 20.1. Introduction / 271 \\
                 20.2. Corollaries to the condition of passivity / 273
                 \\
                 20.3. The necessary and sufficient conditions for
                 passivity / 277 \\
                 20.4. Multidimensional dispersion relations / 282 \\
                 20.5. The fundamental solution and the Cauchy problem /
                 285 \\
                 20.6. What differential and difference operators are
                 passive operators? / 287 \\
                 20.7. Examples / 290 \\
                 20.8. Quasiasymptotics of the solutions of systems of
                 equations in convolutions / 294 \\
                 \\
                 21. Abstract Scattering Operator / 295 \\
                 21.1. The definition and properties of an abstract
                 scattering matrix / 295 \\
                 21.2. A description of abstract scattering matrices /
                 298 \\
                 21.3. The relationship between passive operators and
                 scattering operators / 299 \\
                 \\
                 Bibliography / 303 \\
                 Index / 309",
}


bibtype: article
filename: elefunt.bib
label: Aarts:2003:ASF
author: Ronald M. Aarts and Augustus J. E. M. Janssen
title: Approximation of the Struve function $ H_1 $ occurring in impedance calculations
journal: j-J-ACOUST-SOC-AM
volume: 113
number: 5
pages: 2635--2637
month: may
monthnumber: 5
year: 2003
CODEN: JASMAN
DOI: https://doi.org/10.1121/1.1564019
ISSN: 0001-4966 OR 00014966
ISSNL: 0001-4966 OR 00014966
bibdate: Tue Mar 28 07:23:10 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2017.03.28 07:23:10 ???
fjournal: Journal of the Acoustical Society of America
acknowledgement: ack-nhfb
journalURL: http://scitation.aip.org/content/asa/journal/jasa
rawauthor: Ronald M. Aarts and Augustus J. E. M. Janssen
rawtitle: Approximation of the {Struve} function {$ H_1 $} occurring in impedance calculations
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bibtype: article
filename: elefunt.bib
label: Abad:2003:AEQ
author: J. Abad and J. Sesma
title: Asymptotic expansion of the quasiconfluent hypergeometric function
journal: j-J-MATH-PHYS
volume: 44
number: 4
pages: 1723--1729
month: apr
monthnumber: 4
year: 2003
CODEN: JMAPAQ
ISSNL: 0022-2488 OR 00222488
bibdate: Tue Dec 16 11:36:01 MST 2003
bibtimestamp: 2003.12.16 11:36:01 MST
fjournal: Journal of Mathematical Physics
acknowledgement: ack-nhfb
journalURL: http://jmp.aip.org/
rawauthor: J. Abad and J. Sesma
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bibtype: article
filename: elefunt.bib
label: Agou:2003:SPR
author: Simon Joseph Agou and Marc Deleglise and Jean-Louis Nicolas
title: Short Polynomial Representations for Square Roots Modulo $p$
journal: j-DESIGNS-CODES-CRYPTOGR
volume: 28
number: 1
pages: 33--44
month: jan
monthnumber: 1
year: 2003
CODEN: DCCREC
ISSN: 0925-1022 (print), 1573-7586 (electronic) OR 09251022 (print), 15737586 (electronic)
ISSNL: 0925-1022 OR 09251022
bibdate: Thu Dec 11 06:27:20 MST 2003
bibtimestamp: 2003.12.11 06:27:20 MST
fjournal: Designs, codes, and cryptography
acknowledgement: ack-nhfb
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bibtype: article
filename: elefunt.bib
label: Alzer:2003:GHM
author: Horst Alzer
title: On Gautschi's harmonic mean inequality for the gamma function
journal: j-J-COMPUT-APPL-MATH
volume: 157
number: 1
pages: 243--249
day: 1
month: aug
monthnumber: 8
year: 2003
CODEN: JCAMDI
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ISSNL: 0377-0427 OR 03770427
bibdate: Sat Feb 25 12:52:37 MST 2017
bibtimestamp: 2017.02.25 12:52:37 MST
URL: http://www.sciencedirect.com/science/article/pii/S0377042703004564
fjournal: Journal of Computational and Applied Mathematics
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/03770427
rawauthor: Horst Alzer
rawtitle: On {Gautschi}'s harmonic mean inequality for the gamma function
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  number =       "1",
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bibtype: inproceedings
filename: elefunt.bib
label: Beaumont:2003:BSE
author: James Beaumont and Russell Bradford and James H. Davenport
booktitle: Proceedings of the 2003 International Symposium on Symbolic and Algebraic Computation
title: Better simplification of elementary functions through power series
publisher: pub-ACM
address: pub-ACM:adr
pages: 30--36 (electronic)
year: 2003
MRclass: 33F10 (68W30)
MRnumber: MR2035192 (2005e:33018)
MRreviewer: Ekatherina A. Karatsuba
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bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2005.04.13 06:46:35 ???
acknowledgement: ack-nhfb
rawauthor: James Beaumont and Russell Bradford and James H. Davenport
rawbooktitle: Proceedings of the 2003 International Symposium on Symbolic and Algebraic Computation
rawtitle: Better simplification of elementary functions through power series
remark:
A new algorithm testing the correctness of simplifications of elementary functions in the presence of branch cuts is proposed.

entry:
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  author =       "James Beaumont and Russell Bradford and James H.
                 Davenport",
  booktitle =    "Proceedings of the 2003 International Symposium on
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                 of branch cuts is proposed.",
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bibtype: article
filename: elefunt.bib
label: Buhring:2003:PSH
author: Wolfgang Buhring
title: Partial sums of hypergeometric functions of unit argument
journal: j-PROC-AM-MATH-SOC
volume: 132
number: 2
pages: 407--415
year: 2003
CODEN: PAMYAR
ISSN: 0002-9939 (print), 1088-6826 (electronic) OR 00029939 (print), 10886826 (electronic)
ISSNL: 0002-9939 OR 00029939
MRclass: 33C20
MRnumber: MR2022363 (2005f:33011)
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fjournal: Proceedings of the American Mathematical Society
acknowledgement: ack-nhfb
journalURL: http://www.ams.org/journals/proc
rawauthor: Wolfgang B{\"u}hring
rawtitle: Partial sums of hypergeometric functions of unit argument
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  MRclass =      "33C20",
  MRnumber =     "MR2022363 (2005f:33011)",
  bibdate =      "Thu Dec 01 09:53:54 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "Proceedings of the American Mathematical Society",
  journal-URL =  "http://www.ams.org/journals/proc",
}


bibtype: article
filename: elefunt.bib
label: Chiani:2003:NEB
author: M. Chiani and D. Dardari and M. K. Simon
journal: j-IEEE-TRANS-WIREL-COMMUN
volume: 24
number: 5
pages: 840--845
month: may
monthnumber: 5
year: 2003
CODEN: ITWCAX
DOI: https://doi.org/10.1109/twc.2003.814350
ISSN: 1536-1276 (print), 1558-2248 (electronic) OR 15361276 (print), 15582248 (electronic)
ISSNL: 1536-1276 OR 15361276
bibdate: Sat Dec 16 15:47:42 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2017.12.16 15:47:42 ???
URL: http://ieeexplore.ieee.org/document/1210748/
fjournal: IEEE Transactions on Wireless Communications
acknowledgement: ack-nhfb
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=7693
rawauthor: M. Chiani and D. Dardari and M. K. Simon
title:
New exponential bounds and approximations for the computation of error probability in fading channels

rawtitle:
New exponential bounds and approximations for the computation of error probability in fading channels

entry:
@Article{Chiani:2003:NEB,
  author =       "M. Chiani and D. Dardari and M. K. Simon",
  title =        "New exponential bounds and approximations for the
                 computation of error probability in fading channels",
  journal =      j-IEEE-TRANS-WIREL-COMMUN,
  volume =       "24",
  number =       "5",
  pages =        "840--845",
  month =        may,
  year =         "2003",
  CODEN =        "ITWCAX",
  DOI =          "https://doi.org/10.1109/twc.2003.814350",
  ISSN =         "1536-1276 (print), 1558-2248 (electronic)",
  ISSN-L =       "1536-1276",
  bibdate =      "Sat Dec 16 15:47:42 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://ieeexplore.ieee.org/document/1210748/",
  acknowledgement = ack-nhfb,
  fjournal =     "IEEE Transactions on Wireless Communications",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=7693",
}


bibtype: techreport
filename: elefunt.bib
label: Cornea:2003:DSR
author: M. Cornea and J. Harrison and C. Iordache and B. Norin and S. Story
title: Division, Square Root and Remainder Algorithms for the Intel Itanium Architecture
type: Report
institution: pub-INTEL
address: pub-INTEL:adr
pages: 840--845
month: nov
monthnumber: 11
year: 2003
bibdate: Fri Jun 24 12:05:58 2005
bibtimestamp: 2005.06.24 12:05:58 ???
acknowledgement: ack-nhfb
rawauthor: M. Cornea and J. Harrison and C. Iordache and B. Norin and S. Story
rawtitle: Division, Square Root and Remainder Algorithms for the {Intel Itanium} Architecture
entry:
@TechReport{Cornea:2003:DSR,
  author =       "M. Cornea and J. Harrison and C. Iordache and B. Norin
                 and S. Story",
  title =        "Division, Square Root and Remainder Algorithms for the
                 {Intel Itanium} Architecture",
  type =         "Report",
  institution =  pub-INTEL,
  address =      pub-INTEL:adr,
  month =        nov,
  year =         "2003",
  bibdate =      "Fri Jun 24 12:05:58 2005",
  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: Coussement:2003:AMO
author: Els Coussement and Walter Van Assche
journal: j-J-COMPUT-APPL-MATH
volume: 153
number: 1--2
pages: 141--149
day: 1
month: apr
monthnumber: 4
year: 2003
CODEN: JCAMDI
ISSN: 0377-0427 (print), 1879-1778 (electronic) OR 03770427 (print), 18791778 (electronic)
ISSNL: 0377-0427 OR 03770427
bibdate: Sat Feb 25 12:52:34 MST 2017
bibtimestamp: 2017.02.25 12:52:34 MST
URL: http://www.sciencedirect.com/science/article/pii/S0377042702005964
fjournal: Journal of Computational and Applied Mathematics
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/03770427
rawauthor: Els Coussement and Walter {Van Assche}
title:
Asymptotics of multiple orthogonal polynomials associated with the modified Bessel functions of the first kind

rawtitle:
Asymptotics of multiple orthogonal polynomials associated with the modified {Bessel} functions of the first kind

entry:
@Article{Coussement:2003:AMO,
  author =       "Els Coussement and Walter {Van Assche}",
  title =        "Asymptotics of multiple orthogonal polynomials
                 associated with the modified {Bessel} functions of the
                 first kind",
  journal =      j-J-COMPUT-APPL-MATH,
  volume =       "153",
  number =       "1--2",
  pages =        "141--149",
  day =          "1",
  month =        apr,
  year =         "2003",
  CODEN =        "JCAMDI",
  ISSN =         "0377-0427 (print), 1879-1778 (electronic)",
  ISSN-L =       "0377-0427",
  bibdate =      "Sat Feb 25 12:52:34 MST 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2000.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0377042702005964",
  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: Dominici:2003:NDS
author: Diego Dominici
title: Nested derivatives: a simple method for computing series expansions of inverse functions
journal: j-INT-J-MATH-MATH-SCI
volume: 58
pages: 3699--3715
year: 2003
ISSN: 0161-1712 (print), 1687-0425 (electronic) OR 01611712 (print), 16870425 (electronic)
ISSNL: 0161-1712 OR 01611712
MRclass: 41A58 (33F10)
MRnumber: MR2031140 (2005f:41079)
MRreviewer: Tord H. Ganelius
bibdate: Mon Oct 24 11:37:20 2011
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2011.10.24 11:37:20 ???
URL: http://www2.newpaltz.edu/~dominicd/NESTED14.pdf
fjournal: International Journal of Mathematics and Mathematical Sciences
acknowledgement: ack-nhfb
journalURL: https://www.hindawi.com/journals/ijmms/
rawauthor: Diego Dominici
rawtitle: Nested derivatives: a simple method for computing series expansions of inverse functions
abstract:
We give an algorithm to compute the series expansion for the inverse of a given function. The algorithm is extremely easy to implement and gives the first $N$ terms of the series. We show several examples of its application in calculating the inverses of some special functions.

keywords:
error function, erf(x); incomplete beta function, B(nu,mu,x); incomplete gamma function, gamma(nu,x); logarithm integral, li(x); Maple; sine integral, Si(x)

rawabstract:
We give an algorithm to compute the series expansion for the inverse of a given function. The algorithm is extremely easy to implement and gives the first $N$ terms of the series. We show several examples of its application in calculating the inverses of some special functions.

entry:
@Article{Dominici:2003:NDS,
  author =       "Diego Dominici",
  title =        "Nested derivatives: a simple method for computing
                 series expansions of inverse functions",
  journal =      j-INT-J-MATH-MATH-SCI,
  volume =       "58",
  pages =        "3699--3715",
  year =         "2003",
  CODEN =        "????",
  ISSN =         "0161-1712 (print), 1687-0425 (electronic)",
  ISSN-L =       "0161-1712",
  MRclass =      "41A58 (33F10)",
  MRnumber =     "MR2031140 (2005f:41079)",
  MRreviewer =   "Tord H. Ganelius",
  bibdate =      "Mon Oct 24 11:37:20 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www2.newpaltz.edu/~dominicd/NESTED14.pdf",
  abstract =     "We give an algorithm to compute the series expansion
                 for the inverse of a given function. The algorithm is
                 extremely easy to implement and gives the first $N$
                 terms of the series. We show several examples of its
                 application in calculating the inverses of some special
                 functions.",
  acknowledgement = ack-nhfb,
  fjournal =     "International Journal of Mathematics and Mathematical
                 Sciences",
  journal-URL =  "https://www.hindawi.com/journals/ijmms/",
  keywords =     "error function, erf(x); incomplete beta function,
                 B(nu,mu,x); incomplete gamma function, gamma(nu,x);
                 logarithm integral, li(x); Maple; sine integral,
                 Si(x)",
}


bibtype: inproceedings
filename: elefunt.bib
label: Ercegovac:2003:DRA
author: M. D. Ercegovac and J.-M. Muller
booktitle: Conference Record of the Thirty-Seventh Asilomar Conference on Signals, Systems and Computers, 2003
title: Digit-recurrence algorithms for division and square root with limited precision primitives
volume: 2
publisher: pub-IEEE
address: pub-IEEE:adr
pages: 1440--1444
year: 2003
bibdate: Sat Jul 16 15:28:13 MDT 2005
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2005.07.16 15:28:13 MDT
acknowledgement: ack-nhfb
rawauthor: M. D. Ercegovac and J.-M. Muller
rawbooktitle: Conference Record of the Thirty-Seventh Asilomar Conference on Signals, Systems and Computers, 2003
rawtitle: Digit-recurrence algorithms for division and square root with limited precision primitives
entry:
@InProceedings{Ercegovac:2003:DRA,
  author =       "M. D. Ercegovac and J.-M. Muller",
  booktitle =    "Conference Record of the Thirty-Seventh Asilomar
                 Conference on Signals, Systems and Computers, 2003",
  title =        "Digit-recurrence algorithms for division and square
                 root with limited precision primitives",
  volume =       "2",
  publisher =    pub-IEEE,
  address =      pub-IEEE:adr,
  pages =        "1440--1444",
  year =         "2003",
  CODEN =        "????",
  ISSN =         "????",
  bibdate =      "Sat Jul 16 15:28:13 MDT 2005",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
  summary =      "We propose a digit-recurrence algorithm for square
                 root using limited-precision multipliers, adders, and
                 table-lookups. The algorithm, except in the
                 initialization, uses the digit-recurrence algorithm for
                 division with limited-precision primitives \ldots{}",
}


bibtype: article
filename: elefunt.bib
label: Fabijonas:2003:ACM
author: B. R. Fabijonas and Daniel W. Lozier and J. M. Rappoport
title: Algorithms and Codes for the Macdonald Function: Recent Progress and Comparisons
journal: j-J-COMPUT-APPL-MATH
volume: 161
number: 1
pages: 179--192
year: 2003
CODEN: JCAMDI
ISSN: 0377-0427 (print), 1879-1778 (electronic) OR 03770427 (print), 18791778 (electronic)
ISSNL: 0377-0427 OR 03770427
MRclass: 33F05 (33C10 65D20)
MRnumber: MR2018582
bibdate: Fri Jul 09 06:21:51 2004
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2004.07.09 06:21:51 ???
URL: http://math.nist.gov/acmd/Staff/DLozier/publications/nistir6596.ps
fjournal: Journal of Computational and Applied Mathematics
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/03770427
rawauthor: B. R. Fabijonas and Daniel W. Lozier and J. M. Rappoport
rawtitle: Algorithms and Codes for the {Macdonald} Function: Recent Progress and Comparisons
entry:
@Article{Fabijonas:2003:ACM,
  author =       "B. R. Fabijonas and Daniel W. Lozier and J. M.
                 Rappoport",
  title =        "Algorithms and Codes for the {Macdonald} Function:
                 Recent Progress and Comparisons",
  journal =      j-J-COMPUT-APPL-MATH,
  volume =       "161",
  number =       "1",
  pages =        "179--192",
  month =        "????",
  year =         "2003",
  CODEN =        "JCAMDI",
  ISSN =         "0377-0427 (print), 1879-1778 (electronic)",
  ISSN-L =       "0377-0427",
  MRclass =      "33F05 (33C10 65D20)",
  MRnumber =     "MR2018582",
  bibdate =      "Fri Jul 09 06:21:51 2004",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://math.nist.gov/acmd/Staff/DLozier/publications/nistir6596.ps",
  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: Galue:2003:FRG
author: L. Galue and A. Al-Zamel and Shyam L. Kalla
title: Further results on generalized hypergeometric functions
journal: j-APPL-MATH-COMP
volume: 136
number: 1
pages: 17--25
day: 25
month: mar
monthnumber: 3
year: 2003
CODEN: AMHCBQ
ISSN: 0096-3003 (print), 1873-5649 (electronic) OR 00963003 (print), 18735649 (electronic)
ISSNL: 0096-3003 OR 00963003
bibdate: Fri Jan 9 08:40:52 MST 2004
bibtimestamp: 2004.01.09 08:40:52 MST
fjournal: Applied Mathematics and Computation
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00963003
rawauthor: L. Galu{\'e} and A. Al-Zamel and Shyam L. Kalla
rawtitle: Further results on generalized hypergeometric functions
bibsource:
https://www.math.utah.edu/pub/tex/bib/applmathcomput2000.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

entry:
@Article{Galue:2003:FRG,
  author =       "L. Galu{\'e} and A. Al-Zamel and Shyam L. Kalla",
  title =        "Further results on generalized hypergeometric
                 functions",
  journal =      j-APPL-MATH-COMP,
  volume =       "136",
  number =       "1",
  pages =        "17--25",
  day =          "25",
  month =        mar,
  year =         "2003",
  CODEN =        "AMHCBQ",
  ISSN =         "0096-3003 (print), 1873-5649 (electronic)",
  ISSN-L =       "0096-3003",
  bibdate =      "Fri Jan 9 08:40:52 MST 2004",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/applmathcomput2000.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "Applied Mathematics and Computation",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00963003",
}


bibtype: article
filename: elefunt.bib
label: Gautschi:2003:EEI
author: W. Gautschi and F. E. Harris and N. M. Temme
title: Expansions of the exponential integral in incomplete gamma functions
journal: j-APPL-MATH-LETT
volume: 16
number: 7
pages: 1095--1099
month: oct
monthnumber: 10
year: 2003
CODEN: AMLEEL
DOI: https://doi.org/10.1016/S0893-9659(03)90100-5
ISSN: 0893-9659 (print), 1873-5452 (electronic) OR 08939659 (print), 18735452 (electronic)
ISSNL: 0893-9659 OR 08939659
MRclass: 33B20
bibdate: Wed Dec 4 10:29:43 2013
bibtimestamp: 2013.12.04 10:29:43 ???
fjournal: Applied Mathematics Letters
ZMnumber: 1058.33002
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/08939659
rawauthor: W. Gautschi and F. E. Harris and N. M. Temme
rawtitle: Expansions of the exponential integral in incomplete gamma 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{Gautschi:2003:EEI,
  author =       "W. Gautschi and F. E. Harris and N. M. Temme",
  title =        "Expansions of the exponential integral in incomplete
                 gamma functions",
  journal =      j-APPL-MATH-LETT,
  volume =       "16",
  number =       "7",
  pages =        "1095--1099",
  month =        oct,
  year =         "2003",
  CODEN =        "AMLEEL",
  DOI =          "https://doi.org/10.1016/S0893-9659(03)90100-5",
  ISSN =         "0893-9659 (print), 1873-5452 (electronic)",
  ISSN-L =       "0893-9659",
  MRclass =      "33B20",
  bibdate =      "Wed Dec 4 10:29:43 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",
  ZMnumber =     "1058.33002",
  acknowledgement = ack-nhfb,
  fjournal =     "Applied Mathematics Letters",
  journal-URL =  "http://www.sciencedirect.com/science/journal/08939659",
}


bibtype: article
filename: elefunt.bib
label: Gil:2003:CMB
author: Amparo Gil and Javier Segura and Nico M. Temme
journal: j-J-COMPUT-APPL-MATH
volume: 153
number: 1--2
pages: 225--234
day: 1
month: apr
monthnumber: 4
year: 2003
CODEN: JCAMDI
ISSN: 0377-0427 (print), 1879-1778 (electronic) OR 03770427 (print), 18791778 (electronic)
ISSNL: 0377-0427 OR 03770427
bibdate: Sat Feb 25 12:52:34 MST 2017
bibtimestamp: 2017.02.25 12:52:34 MST
URL: http://www.sciencedirect.com/science/article/pii/S0377042702006088
fjournal: Journal of Computational and Applied Mathematics
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/03770427
rawauthor: Amparo Gil and Javier Segura and Nico M. Temme
title:
Computation of the modified Bessel function of the third kind of imaginary orders: uniform Airy-type asymptotic expansion

rawtitle:
Computation of the modified {Bessel} function of the third kind of imaginary orders: uniform {Airy}-type asymptotic expansion

entry:
@Article{Gil:2003:CMB,
  author =       "Amparo Gil and Javier Segura and Nico M. Temme",
  title =        "Computation of the modified {Bessel} function of the
                 third kind of imaginary orders: uniform {Airy}-type
                 asymptotic expansion",
  journal =      j-J-COMPUT-APPL-MATH,
  volume =       "153",
  number =       "1--2",
  pages =        "225--234",
  day =          "1",
  month =        apr,
  year =         "2003",
  CODEN =        "JCAMDI",
  ISSN =         "0377-0427 (print), 1879-1778 (electronic)",
  ISSN-L =       "0377-0427",
  bibdate =      "Sat Feb 25 12:52:34 MST 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2000.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0377042702006088",
  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: Gil:2003:CSF
author: Amparo Gil and Javier Segura and Nico M. Temme
title: Computing Special Functions by Using Quadrature Rules
journal: j-NUMER-ALGORITHMS
volume: 33
number: 1--4
pages: 265--275
month: aug
monthnumber: 8
year: 2003
CODEN: NUALEG
ISSN: 1017-1398 (print), 1572-9265 (electronic) OR 10171398 (print), 15729265 (electronic)
ISSNL: 1017-1398 OR 10171398
bibdate: Mon Sep 29 08:37:11 MDT 2003
bibsource: http://www.kluweronline.com/issn/1017-1398; https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2003.09.29 08:37:11 MDT
fjournal: Numerical Algorithms
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/11075
rawauthor: Amparo Gil and Javier Segura and Nico M. Temme
rawtitle: Computing Special Functions by Using Quadrature Rules
URL:
http://ipsapp007.kluweronline.com/content/getfile/5058/46/22/abstract.htm; http://ipsapp007.kluweronline.com/content/getfile/5058/46/22/fulltext.pdf

entry:
@Article{Gil:2003:CSF,
  author =       "Amparo Gil and Javier Segura and Nico M. Temme",
  title =        "Computing Special Functions by Using Quadrature
                 Rules",
  journal =      j-NUMER-ALGORITHMS,
  volume =       "33",
  number =       "1--4",
  pages =        "265--275",
  month =        aug,
  year =         "2003",
  CODEN =        "NUALEG",
  ISSN =         "1017-1398 (print), 1572-9265 (electronic)",
  ISSN-L =       "1017-1398",
  bibdate =      "Mon Sep 29 08:37:11 MDT 2003",
  bibsource =    "http://www.kluweronline.com/issn/1017-1398;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://ipsapp007.kluweronline.com/content/getfile/5058/46/22/abstract.htm;
                 http://ipsapp007.kluweronline.com/content/getfile/5058/46/22/fulltext.pdf",
  acknowledgement = ack-nhfb,
  fjournal =     "Numerical Algorithms",
  journal-URL =  "http://link.springer.com/journal/11075",
}


bibtype: article
filename: elefunt.bib
label: Harrison:2003:FVS
author: John Harrison
title: Formal verification of square root algorithms
journal: j-FORM-METHODS-SYST-DES
volume: 22
number: 2
pages: 143--153
month: mar
monthnumber: 3
year: 2003
CODEN: FMSDE6
DOI: https://doi.org/10.1023/A:1022973506233
ISSN: 0925-9856 (print), 1572-8102 (electronic) OR 09259856 (print), 15728102 (electronic)
ISSNL: 0925-9856 OR 09259856
bibdate: Sat Feb 08 08:47:21 2020
bibtimestamp: 2020.02.08 08:47:21 ???
URL: https://dl.acm.org/doi/abs/10.1023/A:1022973506233
fjournal: Formal Methods in System Design
acknowledgement: ack-nhfb
journalURL: https://dl.acm.org/loi/fmsd
rawauthor: John Harrison
rawtitle: Formal verification of square root algorithms
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/intel-ia-64.bib

abstract:
We discuss the formal verification of some low-level mathematical software for the Intel Itanium architecture. A number of important algorithms have been proven correct using the HOL Light theorem prover. After briefly surveying some of our formal verification work, we discuss in more detail the verification of a square root algorithm, which helps to illustrate why some features of HOL Light, in particular programmability, make it especially suitable for these applications.

rawabstract:
We discuss the formal verification of some low-level mathematical software for the Intel Itanium architecture. A number of important algorithms have been proven correct using the HOL Light theorem prover. After briefly surveying some of our formal verification work, we discuss in more detail the verification of a square root algorithm, which helps to illustrate why some features of HOL Light, in particular programmability, make it especially suitable for these applications.

entry:
@Article{Harrison:2003:FVS,
  author =       "John Harrison",
  title =        "Formal verification of square root algorithms",
  journal =      j-FORM-METHODS-SYST-DES,
  volume =       "22",
  number =       "2",
  pages =        "143--153",
  month =        mar,
  year =         "2003",
  CODEN =        "FMSDE6",
  DOI =          "https://doi.org/10.1023/A:1022973506233",
  ISSN =         "0925-9856 (print), 1572-8102 (electronic)",
  ISSN-L =       "0925-9856",
  bibdate =      "Sat Feb 08 08:47:21 2020",
  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/intel-ia-64.bib",
  URL =          "https://dl.acm.org/doi/abs/10.1023/A:1022973506233",
  abstract =     "We discuss the formal verification of some low-level
                 mathematical software for the Intel Itanium
                 architecture. A number of important algorithms have
                 been proven correct using the HOL Light theorem prover.
                 After briefly surveying some of our formal verification
                 work, we discuss in more detail the verification of a
                 square root algorithm, which helps to illustrate why
                 some features of HOL Light, in particular
                 programmability, make it especially suitable for these
                 applications.",
  acknowledgement = ack-nhfb,
  fjournal =     "Formal Methods in System Design",
  journal-URL =  "https://dl.acm.org/loi/fmsd",
}


bibtype: misc
filename: elefunt.bib
label: Intel:2003:DSR
author: Intel
title: Divide, Square Root, and Remainder Algorithms for the Itanium Architecture
pages: 143--153
day: 18
month: dec
monthnumber: 12
year: 2003
bibdate: Tue Nov 18 16:23:36 2008
bibtimestamp: 2008.11.18 16:23:36 ???
URL: http://www.intel.com/cd/software/products/asmo-na/eng/219863.htm
acknowledgement: ack-nhfb
howpublished: Intel Software Development Products
rawauthor: {Intel}
rawtitle: Divide, Square Root, and Remainder Algorithms for the {Itanium} Architecture
entry:
@Misc{Intel:2003:DSR,
  author =       "{Intel}",
  title =        "Divide, Square Root, and Remainder Algorithms for the
                 {Itanium} Architecture",
  howpublished = "Intel Software Development Products",
  day =          "18",
  month =        dec,
  year =         "2003",
  bibdate =      "Tue Nov 18 16:23:36 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.intel.com/cd/software/products/asmo-na/eng/219863.htm",
  acknowledgement = ack-nhfb,
}


bibtype: misc
filename: elefunt.bib
label: Intel:2003:NID
author: Intel
pages: 143--153
day: 18
month: dec
monthnumber: 12
year: 2003
bibdate: Tue Nov 18 16:23:36 2008
bibtimestamp: 2008.11.18 16:23:36 ???
URL: http://www.intel.com/cd/software/products/asmo-na/eng/219864.htm
acknowledgement: ack-nhfb
howpublished: Intel Software Development Products
rawauthor: {Intel}
title:
Non-IEEE Division, Square Root, Reciprocal, and Reciprocal Square Root Algorithms for the Intel Itanium Architecture

rawtitle:
Non-{IEEE} Division, Square Root, Reciprocal, and Reciprocal Square Root Algorithms for the {Intel Itanium} Architecture

entry:
@Misc{Intel:2003:NID,
  author =       "{Intel}",
  title =        "Non-{IEEE} Division, Square Root, Reciprocal, and
                 Reciprocal Square Root Algorithms for the {Intel
                 Itanium} Architecture",
  howpublished = "Intel Software Development Products",
  day =          "18",
  month =        dec,
  year =         "2003",
  bibdate =      "Tue Nov 18 16:23:36 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.intel.com/cd/software/products/asmo-na/eng/219864.htm",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: elefunt.bib
label: Kzaz:2003:CAG
author: M. Kzaz and M. Prevost
title: Convergence Acceleration of Gauss--Chebyshev Quadrature Formulae
journal: j-NUMER-ALGORITHMS
volume: 34
number: 2--4
pages: 379--391
month: dec
monthnumber: 12
year: 2003
CODEN: NUALEG
ISSN: 1017-1398 (print), 1572-9265 (electronic) OR 10171398 (print), 15729265 (electronic)
ISSNL: 1017-1398 OR 10171398
bibdate: Tue Jan 13 17:32:50 MST 2004
bibsource: http://www.kluweronline.com/issn/1017-1398; https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2004.01.13 17:32:50 MST
fjournal: Numerical Algorithms
keywords: convergence acceleration
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/11075
rawauthor: M. Kzaz and M. Pr{\'e}vost
rawtitle: Convergence Acceleration of {Gauss--Chebyshev} Quadrature Formulae
URL:
http://ipsapp007.kluweronline.com/content/getfile/5058/48/6/abstract.htm; http://ipsapp007.kluweronline.com/content/getfile/5058/48/6/fulltext.pdf

entry:
@Article{Kzaz:2003:CAG,
  author =       "M. Kzaz and M. Pr{\'e}vost",
  title =        "Convergence Acceleration of {Gauss--Chebyshev}
                 Quadrature Formulae",
  journal =      j-NUMER-ALGORITHMS,
  volume =       "34",
  number =       "2--4",
  pages =        "379--391",
  month =        dec,
  year =         "2003",
  CODEN =        "NUALEG",
  ISSN =         "1017-1398 (print), 1572-9265 (electronic)",
  ISSN-L =       "1017-1398",
  bibdate =      "Tue Jan 13 17:32:50 MST 2004",
  bibsource =    "http://www.kluweronline.com/issn/1017-1398;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://ipsapp007.kluweronline.com/content/getfile/5058/48/6/abstract.htm;
                 http://ipsapp007.kluweronline.com/content/getfile/5058/48/6/fulltext.pdf",
  acknowledgement = ack-nhfb,
  fjournal =     "Numerical Algorithms",
  journal-URL =  "http://link.springer.com/journal/11075",
  keywords =     "convergence acceleration",
}


bibtype: article
filename: elefunt.bib
label: Lang:2003:RRS
author: Tomas Lang and Elisardo Antelo
title: Radix-$4$ Reciprocal Square-root and Its Combination with Division and Square Root
journal: j-IEEE-TRANS-COMPUT
volume: 52
number: 9
pages: 1100--1114
month: sep
monthnumber: 9
year: 2003
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2003.1228508
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Sat Jul 16 15:28:13 MDT 2005
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2005.07.16 15:28:13 MDT
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: Tom{\'a}s Lang and Elisardo Antelo
rawtitle: Radix-$4$ Reciprocal Square-root and Its Combination with Division and Square Root
abstract:
In this work, we present a reciprocal square root algorithm by digit recurrence and selection by a staircase function and the radix-$4$ implementation. As in similar algorithms for division and square root, the results are obtained correctly rounded in a straightforward manner (in contrast to existing methods to compute the reciprocal square root). Although, apparently, a single selection function can only be used for $ j geq 2 $ (the selection constants are different for $ j = 0 $, $ j = 1 $, and $ j geq 2 $ ), we show that it is possible to use a single selection function for all iterations. We perform a rough comparison with existing methods and we conclude that our implementation is a low hardware complexity solution with moderate latency, especially for exactly rounded results. We also extend the unit to support division and square root with the same selection function and with slight modifications in the initialization of the reciprocal square root unit.

rawabstract:
In this work, we present a reciprocal square root algorithm by digit recurrence and selection by a staircase function and the radix-$4$ implementation. As in similar algorithms for division and square root, the results are obtained correctly rounded in a straightforward manner (in contrast to existing methods to compute the reciprocal square root). Although, apparently, a single selection function can only be used for $ j \geq 2 $ (the selection constants are different for $ j = 0 $, $ j = 1 $, and $ j \geq 2 $ ), we show that it is possible to use a single selection function for all iterations. We perform a rough comparison with existing methods and we conclude that our implementation is a low hardware complexity solution with moderate latency, especially for exactly rounded results. We also extend the unit to support division and square root with the same selection function and with slight modifications in the initialization of the reciprocal square root unit.

entry:
@Article{Lang:2003:RRS,
  author =       "Tom{\'a}s Lang and Elisardo Antelo",
  title =        "Radix-$4$ Reciprocal Square-root and Its Combination
                 with Division and Square Root",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "52",
  number =       "9",
  pages =        "1100--1114",
  month =        sep,
  year =         "2003",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2003.1228508",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Sat Jul 16 15:28:13 MDT 2005",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  abstract =     "In this work, we present a reciprocal square root
                 algorithm by digit recurrence and selection by a
                 staircase function and the radix-$4$ implementation. As
                 in similar algorithms for division and square root, the
                 results are obtained correctly rounded in a
                 straightforward manner (in contrast to existing methods
                 to compute the reciprocal square root). Although,
                 apparently, a single selection function can only be
                 used for $ j \geq 2 $ (the selection constants are
                 different for $ j = 0 $, $ j = 1 $, and $ j \geq 2 $ ),
                 we show that it is possible to use a single selection
                 function for all iterations. We perform a rough
                 comparison with existing methods and we conclude that
                 our implementation is a low hardware complexity
                 solution with moderate latency, especially for exactly
                 rounded results. We also extend the unit to support
                 division and square root with the same selection
                 function and with slight modifications in the
                 initialization of the reciprocal square root unit.",
  acknowledgement = ack-nhfb,
  ajournal =     "IEEE Trans. Comput.",
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
}


bibtype: techreport
filename: elefunt.bib
label: Lefevre:2003:WCC
author: Vincent Lefevre and Jean-Michel Muller
title: Worst Cases for Correct Rounding for the Elementary Functions in Double Precision
type: Technical report
institution: INRIA, Projet Spaces, LORIA, Campus Scientifique
address: B.P. 239, 54506 Vandoeuvre-les-Nancy Cedex, France
pages: 1100--1114
day: 14
month: aug
monthnumber: 8
year: 2003
bibdate: Thu Jul 08 08:27:53 2004
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2004.07.08 08:27:53 ???
URL: http://perso.ens-lyon.fr/jean-michel.muller/TMDworstcases.pdf
keywords: computer arithmetic; elementary functions; floating-point arithmetic; Table Maker's Dilemma
acknowledgement: ack-nhfb
rawauthor: Vincent Lef{\`e}vre and Jean-Michel Muller
rawtitle: Worst Cases for Correct Rounding for the Elementary Functions in Double Precision
abstract:
We give the results of our search for the worst cases for correct rounding of the major elementary functions in double precision floating-point arithmetic. These results allow the design of reasonably fast routines that will compute these functions with correct rounding, at least in some interval, for any of the four rounding modes specified by the IEEE-754 standard. They will also allow one to easily test libraries that are claimed to provide correctly rounded functions.

rawabstract:
We give the results of our search for the worst cases for correct rounding of the major elementary functions in double precision floating-point arithmetic. These results allow the design of reasonably fast routines that will compute these functions with correct rounding, at least in some interval, for any of the four rounding modes specified by the IEEE-754 standard. They will also allow one to easily test libraries that are claimed to provide correctly rounded functions.

entry:
@TechReport{Lefevre:2003:WCC,
  author =       "Vincent Lef{\`e}vre and Jean-Michel Muller",
  title =        "Worst Cases for Correct Rounding for the Elementary
                 Functions in Double Precision",
  type =         "Technical report",
  institution =  "INRIA, Projet Spaces, LORIA, Campus Scientifique",
  address =      "B.P. 239, 54506 Vandoeuvre-l{\`e}s-Nancy Cedex,
                 France",
  day =          "14",
  month =        aug,
  year =         "2003",
  bibdate =      "Thu Jul 08 08:27:53 2004",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://perso.ens-lyon.fr/jean-michel.muller/TMDworstcases.pdf",
  abstract =     "We give the results of our search for the worst cases
                 for correct rounding of the major elementary functions
                 in double precision floating-point arithmetic. These
                 results allow the design of reasonably fast routines
                 that will compute these functions with correct
                 rounding, at least in some interval, for any of the
                 four rounding modes specified by the IEEE-754 standard.
                 They will also allow one to easily test libraries that
                 are claimed to provide correctly rounded functions.",
  acknowledgement = ack-nhfb,
  keywords =     "computer arithmetic; elementary functions;
                 floating-point arithmetic; Table Maker's Dilemma",
}


bibtype: article
filename: elefunt.bib
label: Lozier:2003:NDL
author: Daniel W. Lozier
title: NIST Digital Library of Mathematical Functions
journal: j-ANN-MATH-ARTIF-INTELL
volume: 38
number: 1--3
pages: 105--119
month: may
monthnumber: 5
year: 2003
CODEN: AMAIEC
ISSN: 1012-2443 (print), 1573-7470 (electronic) OR 10122443 (print), 15737470 (electronic)
ISSNL: 1012-2443 OR 10122443
bibdate: Fri Jul 09 06:23:08 2004
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2004.07.09 06:23:08 ???
URL: http://math.nist.gov/acmd/Staff/DLozier/publications/Linz01.ps
fjournal: Annals of Mathematics and Artificial Intelligence
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/10472
rawauthor: Daniel W. Lozier
rawtitle: {NIST Digital Library of Mathematical Functions}
entry:
@Article{Lozier:2003:NDL,
  author =       "Daniel W. Lozier",
  title =        "{NIST Digital Library of Mathematical Functions}",
  journal =      j-ANN-MATH-ARTIF-INTELL,
  volume =       "38",
  number =       "1--3",
  pages =        "105--119",
  month =        may,
  year =         "2003",
  CODEN =        "AMAIEC",
  ISSN =         "1012-2443 (print), 1573-7470 (electronic)",
  ISSN-L =       "1012-2443",
  bibdate =      "Fri Jul 09 06:23:08 2004",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://math.nist.gov/acmd/Staff/DLozier/publications/Linz01.ps",
  acknowledgement = ack-nhfb,
  fjournal =     "Annals of Mathematics and Artificial Intelligence",
  journal-URL =  "http://link.springer.com/journal/10472",
}


bibtype: inproceedings
filename: elefunt.bib
label: Markstein:2003:FQP
author: Peter Markstein
title: A fast quad precision elementary function library for Itanium
crossref: Anonymous:2003:CRN
pages: 5--12
year: 2003
bibdate: Fri Jun 24 20:14:39 2005
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2005.06.24 20:14:39 ???
acknowledgement: ack-nhfb
rawauthor: Peter Markstein
rawtitle: A fast quad precision elementary function library for {Itanium}
abstract:
This talk will describe Itanium's floating point architecture and how it has been used to produce a high performance, highly accurate quad precision elementary function library.par Itanium's floating-point features will first be described, from the point of view of a computer architect. Many conflicting requirements vie for consideration during the design of a new computer architecture. These include instruction word size, number of registers, the set of operations, arithmetic precisions supported, and memory access. Some of the trade-offs during the design phase will be discussed.par One of the objectives of the original Itanium design was to accelerate quad precision arithmetic. The talk will describe how the Itanium elementary function library was constructed, with attention to performance and accuracy. Because a pair of double-extended floating point words are used for internal operations involving quad precision numbers, intermediate results, holding 128 bits, provide 15 guard bits during intermediate calculations, resulting in a very low percentage of misrounded results.

rawabstract:
This talk will describe Itanium's floating point architecture and how it has been used to produce a high performance, highly accurate quad precision elementary function library.\par Itanium's floating-point features will first be described, from the point of view of a computer architect. Many conflicting requirements vie for consideration during the design of a new computer architecture. These include instruction word size, number of registers, the set of operations, arithmetic precisions supported, and memory access. Some of the trade-offs during the design phase will be discussed.\par One of the objectives of the original Itanium design was to accelerate quad precision arithmetic. The talk will describe how the Itanium elementary function library was constructed, with attention to performance and accuracy. Because a pair of double-extended floating point words are used for internal operations involving quad precision numbers, intermediate results, holding 128 bits, provide 15 guard bits during intermediate calculations, resulting in a very low percentage of misrounded results.

entry:
@InProceedings{Markstein:2003:FQP,
  author =       "Peter Markstein",
  title =        "A fast quad precision elementary function library for
                 {Itanium}",
  crossref =     "Anonymous:2003:CRN",
  pages =        "5--12",
  year =         "2003",
  bibdate =      "Fri Jun 24 20:14:39 2005",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  abstract =     "This talk will describe Itanium's floating point
                 architecture and how it has been used to produce a high
                 performance, highly accurate quad precision elementary
                 function library.\par

                 Itanium's floating-point features will first be
                 described, from the point of view of a computer
                 architect. Many conflicting requirements vie for
                 consideration during the design of a new computer
                 architecture. These include instruction word size,
                 number of registers, the set of operations, arithmetic
                 precisions supported, and memory access. Some of the
                 trade-offs during the design phase will be
                 discussed.\par

                 One of the objectives of the original Itanium design
                 was to accelerate quad precision arithmetic. The talk
                 will describe how the Itanium elementary function
                 library was constructed, with attention to performance
                 and accuracy. Because a pair of double-extended
                 floating point words are used for internal operations
                 involving quad precision numbers, intermediate results,
                 holding 128 bits, provide 15 guard bits during
                 intermediate calculations, resulting in a very low
                 percentage of misrounded results.",
  acknowledgement = ack-nhfb,
}


bibtype: book
filename: elefunt.bib
label: Mason:2003:CP
author: J. C. Mason and D. C. Handscomb
title: Chebyshev Polynomials
publisher: pub-CHAPMAN-HALL-CRC
address: pub-CHAPMAN-HALL-CRC:adr
pages: xiii + 341
year: 2003
ISBN: 0-8493-0355-9 OR 0849303559
ISBN13: 978-0-8493-0355-5 OR 9780849303555
LCCN: QA404.5 .M37 2003
bibdate: Fri Apr 17 09:45:35 MDT 2009
bibtimestamp: 2009.04.17 09:45:35 MDT
subject: Chebyshev polynomials
acknowledgement: ack-nhfb
rawauthor: J. C. Mason and D. C. Handscomb
rawtitle: {Chebyshev} Polynomials
TOC:
1 Definitions / 1 1.1 Preliminary remarks / 1 1.2 Trigonometric definitions and recurrences / 1 1.2.1 The first-kind polynomial $T_n$ / 2 1.2.2 The second-kind polynomial $U_n$ / 3 1.2.3 The third- and fourth-kind polynomials $V_n$ and $W_n$ (the airfoil polynomials) / 5 1.2.4 Connections between the four kinds of polynomial / 7 1.3 Shifted Chebyshev polynomials / 9 1.3.1 The shifted polynomials $T_n^*$, $U_n^*$, $V_n^*$, $W_n^*$ / 9 1.3.2 Chebyshev polynomials for the general range $[a, b]$ / 11 1.4 Chebyshev polynomials of a complex variable / 11 1.4.1 Conformal mapping of a circle to and from an ellipse / 12 1.4.2 Chebyshev polynomials in $z$ / 14 1.4.3 Shabat polynomials / 17 1.5 Problems for Chapter 1 / 17 2 Basic Properties and Formulae / 19 2.1 Introduction / 19 2.2 Chebyshev polynomial zeros and extrema / 19 2.3 Relations between Chebyshev polynomials and powers of x / 22 2.3.1 Powers of $x$ in terms of $T_n(x)$ / 22 2.3.2 $T_n(x)$ in terms of powers of $x$ / 23 2.3.3 Ratios of coefficients in $T_n(x)$ / 25 2.4 Evaluation of Chebyshev sums, products, integrals and derivatives / 25 2.4.1 Evaluation of a Chebyshev sum / 25 2.4.2 Stability of the evaluation of a Chebyshev sum / 29 2.4.3 Evaluation of a product / 31 2.4.4 Evaluation of an integral / 32 2.4.5 Evaluation of a derivative / 34 6.3.1 Aliasing / 152 6.3.2 Second-kind interpolation / 155 6.3.3 Third- and fourth-kind interpolation / 156 6.3.4 Conditioning / 158 6.4 Best $mathcal L_1$ approximation by Chebyshev interpolation / 158 6.5 Near-minimax approximation by Chebyshev interpolation / 160 6.6 Problems for Chapter 6 / 162 7 Near-Best $mathcal L_infty $, $mathcal L_1$ and $mathcal L_p$ Approximations / 165 7.1 Near-best $mathcal L_infty $ (near-minimax) approximations / 165 7.1.1 Second-kind expansions in $mathcal L_infty $ / 165 7.1.2 Third-kind expansions in $mathcal L_infty $ / 167 7.2 Near-best $mathcal L_1$ approximations / 169 7.3 Best and near-best $mathcal L_p$ approximations / 170 7.3.1 Complex variable results for elliptic-type regions / 172 7.4 Problems for Chapter 7 / 173 8 Integration Using Chebyshev Polynomials / 177 8.1 Indefinite integration with Chebyshev series / 177 8.2 Gauss--Chebyshev quadrature / 180 8.3 Quadrature methods of Clenshaw--Curtis type / 186 8.3.1 Introduction / 186 8.3.2 First-kind formulae / 187 8.3.3 Second-kind formulae / 189 8.3.4 Third-kind formulae / 191 8.3.5 General remark on methods of Clenshaw--Curtis type / 192 8.4 Error estimation for Clenshaw--Curtis methods / 192 8.4.1 First-kind polynomials / 193 8.4.2 Fitting an exponential curve / 195 8.4.3 Other abscissae and polynomials / 196 8.5 Some other work on Clenshaw--Curtis methods / 200 8.6 Problems for Chapter 8 / 201 9 Solution of Integral Equations / 203 9.1 Introduction / 203 9.2 Fredholm equations of the second kind / 204 9.3 Fredholm equations of the third kind / 206 9.4 Fredholm equations of the first kind / 207 9.5 Singular kernels / 209 9.5.1 Hilbert-type kernels and related kernels / 209 9.5.2 Symm's integral equation / 212 9.6 Regularisation of integral equations / 214 9.6.1 Discrete data with second derivative regularisation / 214 9.6.2 Details of a smoothing algorithm (second derivative regularisation) / 215 9.6.3 A smoothing algorithm with weighted function regularisation / 217 9.6.4 Evaluation of $V(lambda )$ / 220 9.6.5 Other basis functions / 221 9.7 Partial differential equations and boundary integral equation methods / 222 9.7.1 A hypersingular integral equation derived from a mixed boundary value problem for Laplace's equation / 222 9.8 Problems for Chapter 9 / 227 10 Solution of Ordinary Differential Equations / 231 10.1 Introduction / 231 10.2 A simple example / 232 10.2.1 Collocation methods / 234 10.2.2 Error of the collocation method / 237 10.2.3 Projection (tau) methods / 239 10.2.4 Error of the preceding projection method / 241 10.3 The original Lanczos tau ($tau $) method / 242 10.4 A more general linear equation / 244 10.4.1 Collocation method / 244 10.4.2 Projection method / 245 10.5 Pseudospectral methods --- another form of collocation / 245 10.5.1 Differentiation matrices / 246 10.5.2 Differentiation matrix for Chebyshev points / 247 10.5.3 Collocation using differentiation matrices / 249 10.6 Nonlinear equations / 251 10.7 Eigenvalue problems / 252 10.7.1 Collocation methods / 252 10.7.2 Collocation using the differentiation matrix / 254 10.8 Differential equations in one space and one time dimension / 256 10.8.1 Collocation methods / 257 10.8.2 Collocation using the differentiation matrix / 258 10.9 Problems for Chapter 10 / 259 11 Chebyshev and Spectral Methods for Partial Differential Equations / 261 11.1 Introduction / 261 11.2 Interior, boundary and mixed methods / 262 11.2.1 Interior methods / 262 11.2.2 Boundary methods / 263 11.2.3 Mixed methods / 265 11.3 Differentiation matrices and nodal representation / 265 11.4 Method of weighted residuals / 265 11.4.1 Continuous MWR / 265 11.4.2 Discrete MWR --- a new nomenclature / 266 11.5 Chebyshev series and Galerkin methods / 267 11.6 Collocation/interpolation and related methods / 269 11.7 PDE methods / 271 11.7.1 Error analysis / 272 11.8 Some PDE problems and various methods / 272 11.8.1 Power basis: collocation for Poisson problem / 273 11.8.2 Power basis: interior collocation for the L-membrane / 275 11.8.3 Chebyshev basis and discrete orthogonalisation / 278 11.8.4 Differentiation matrix approach: Poisson problem / 281 11.8.5 Explicit collocation for the quasilinear Dirichlet problem: Chebyshev basis / 283 11.9 Computational fluid dynamics / 295 11.10 Particular issues in spectral methods / 296 11.11 More advanced problems / 297 11.12 Problems for Chapter 11 / 298 12 Conclusion / 303 Bibliography / 305 Appendices: A Biographical Note / 321 B Summary of Notations, Definitions and Important Properties / 323 B.I Miscellaneous notations / 323 B.2 The four kinds of Chebyshev polynomial / 325 C Tables of Coefficients / 329 Index / 335

rawTOC:
1 Definitions / 1 \\ 1.1 Preliminary remarks / 1 \\ 1.2 Trigonometric definitions and recurrences / 1 \\ 1.2.1 The first-kind polynomial $T_n$ / 2 \\ 1.2.2 The second-kind polynomial $U_n$ / 3 \\ 1.2.3 The third- and fourth-kind polynomials $V_n$ and $W_n$ (the airfoil polynomials) / 5 \\ 1.2.4 Connections between the four kinds of polynomial / 7 \\ 1.3 Shifted Chebyshev polynomials / 9 \\ 1.3.1 The shifted polynomials $T_n^*$, $U_n^*$, $V_n^*$, $W_n^*$ / 9 \\ 1.3.2 Chebyshev polynomials for the general range $[a, b]$ / 11 \\ 1.4 Chebyshev polynomials of a complex variable / 11 \\ 1.4.1 Conformal mapping of a circle to and from an ellipse / 12 \\ 1.4.2 Chebyshev polynomials in $z$ / 14 \\ 1.4.3 Shabat polynomials / 17 \\ 1.5 Problems for Chapter 1 / 17 \\ 2 Basic Properties and Formulae / 19 \\ 2.1 Introduction / 19 \\ 2.2 Chebyshev polynomial zeros and extrema / 19 \\ 2.3 Relations between Chebyshev polynomials and powers of x / 22 \\ 2.3.1 Powers of $x$ in terms of $\{T_n(x)\}$ / 22 \\ 2.3.2 $T_n(x)$ in terms of powers of $x$ / 23 \\ 2.3.3 Ratios of coefficients in $T_n(x)$ / 25 \\ 2.4 Evaluation of Chebyshev sums, products, integrals and derivatives / 25 \\ 2.4.1 Evaluation of a Chebyshev sum / 25 \\ 2.4.2 Stability of the evaluation of a Chebyshev sum / 29 \\ 2.4.3 Evaluation of a product / 31 \\ 2.4.4 Evaluation of an integral / 32 \\ 2.4.5 Evaluation of a derivative / 34 \\ 6.3.1 Aliasing / 152 \\ 6.3.2 Second-kind interpolation / 155 \\ 6.3.3 Third- and fourth-kind interpolation / 156 \\ 6.3.4 Conditioning / 158 \\ 6.4 Best $\mathcal{L}_1$ approximation by Chebyshev interpolation / 158 \\ 6.5 Near-minimax approximation by Chebyshev interpolation / 160 \\ 6.6 Problems for Chapter 6 / 162 \\ 7 Near-Best $\mathcal{L}_\infty$, $\mathcal{L}_1$ and $\mathcal{L}_p$ Approximations / 165 \\ 7.1 Near-best $\mathcal{L}_\infty$ (near-minimax) approximations / 165 \\ 7.1.1 Second-kind expansions in $\mathcal{L}_\infty$ / 165 \\ 7.1.2 Third-kind expansions in $\mathcal{L}_\infty$ / 167 \\ 7.2 Near-best $\mathcal{L}_1$ approximations / 169 \\ 7.3 Best and near-best $\mathcal{L}_p$ approximations / 170 \\ 7.3.1 Complex variable results for elliptic-type regions / 172 \\ 7.4 Problems for Chapter 7 / 173 \\ 8 Integration Using Chebyshev Polynomials / 177 \\ 8.1 Indefinite integration with Chebyshev series / 177 \\ 8.2 Gauss--Chebyshev quadrature / 180 \\ 8.3 Quadrature methods of Clenshaw--Curtis type / 186 \\ 8.3.1 Introduction / 186 \\ 8.3.2 First-kind formulae / 187 \\ 8.3.3 Second-kind formulae / 189 \\ 8.3.4 Third-kind formulae / 191 \\ 8.3.5 General remark on methods of Clenshaw--Curtis type / 192 \\ 8.4 Error estimation for Clenshaw--Curtis methods / 192 \\ 8.4.1 First-kind polynomials / 193 \\ 8.4.2 Fitting an exponential curve / 195 \\ 8.4.3 Other abscissae and polynomials / 196 \\ 8.5 Some other work on Clenshaw--Curtis methods / 200 \\ 8.6 Problems for Chapter 8 / 201 \\ 9 Solution of Integral Equations / 203 \\ 9.1 Introduction / 203 \\ 9.2 Fredholm equations of the second kind / 204 \\ 9.3 Fredholm equations of the third kind / 206 \\ 9.4 Fredholm equations of the first kind / 207 \\ 9.5 Singular kernels / 209 \\ 9.5.1 Hilbert-type kernels and related kernels / 209 \\ 9.5.2 Symm's integral equation / 212 \\ 9.6 Regularisation of integral equations / 214 \\ 9.6.1 Discrete data with second derivative regularisation / 214 \\ 9.6.2 Details of a smoothing algorithm (second derivative regularisation) / 215 \\ 9.6.3 A smoothing algorithm with weighted function regularisation / 217 \\ 9.6.4 Evaluation of $V(\lambda)$ / 220 \\ 9.6.5 Other basis functions / 221 \\ 9.7 Partial differential equations and boundary integral equation methods / 222 \\ 9.7.1 A hypersingular integral equation derived from a mixed boundary value problem for Laplace's equation / 222 \\ 9.8 Problems for Chapter 9 / 227 \\ 10 Solution of Ordinary Differential Equations / 231 \\ 10.1 Introduction / 231 \\ 10.2 A simple example / 232 \\ 10.2.1 Collocation methods / 234 \\ 10.2.2 Error of the collocation method / 237 \\ 10.2.3 Projection (tau) methods / 239 \\ 10.2.4 Error of the preceding projection method / 241 \\ 10.3 The original Lanczos tau ($\tau$) method / 242 \\ 10.4 A more general linear equation / 244 \\ 10.4.1 Collocation method / 244 \\ 10.4.2 Projection method / 245 \\ 10.5 Pseudospectral methods --- another form of collocation / 245 \\ 10.5.1 Differentiation matrices / 246 \\ 10.5.2 Differentiation matrix for Chebyshev points / 247 \\ 10.5.3 Collocation using differentiation matrices / 249 \\ 10.6 Nonlinear equations / 251 \\ 10.7 Eigenvalue problems / 252 \\ 10.7.1 Collocation methods / 252 \\ 10.7.2 Collocation using the differentiation matrix / 254 \\ 10.8 Differential equations in one space and one time dimension / 256 \\ 10.8.1 Collocation methods / 257 \\ 10.8.2 Collocation using the differentiation matrix / 258 \\ 10.9 Problems for Chapter 10 / 259 \\ 11 Chebyshev and Spectral Methods for Partial Differential Equations / 261 \\ 11.1 Introduction / 261 \\ 11.2 Interior, boundary and mixed methods / 262 \\ 11.2.1 Interior methods / 262 \\ 11.2.2 Boundary methods / 263 \\ 11.2.3 Mixed methods / 265 \\ 11.3 Differentiation matrices and nodal representation / 265 \\ 11.4 Method of weighted residuals / 265 \\ 11.4.1 Continuous MWR / 265 \\ 11.4.2 Discrete MWR --- a new nomenclature / 266 \\ 11.5 Chebyshev series and Galerkin methods / 267 \\ 11.6 Collocation/interpolation and related methods / 269 \\ 11.7 PDE methods / 271 \\ 11.7.1 Error analysis / 272 \\ 11.8 Some PDE problems and various methods / 272 \\ 11.8.1 Power basis: collocation for Poisson problem / 273 \\ 11.8.2 Power basis: interior collocation for the L-membrane / 275 \\ 11.8.3 Chebyshev basis and discrete orthogonalisation / 278 \\ 11.8.4 Differentiation matrix approach: Poisson problem / 281 \\ 11.8.5 Explicit collocation for the quasilinear Dirichlet problem: Chebyshev basis / 283 \\ 11.9 Computational fluid dynamics / 295 \\ 11.10 Particular issues in spectral methods / 296 \\ 11.11 More advanced problems / 297 \\ 11.12 Problems for Chapter 11 / 298 \\ 12 Conclusion / 303 \\ Bibliography / 305 \\ Appendices: \\ A Biographical Note / 321 \\ B Summary of Notations, Definitions and Important Properties / 323 \\ B.I Miscellaneous notations / 323 \\ B.2 The four kinds of Chebyshev polynomial / 325 \\ C Tables of Coefficients / 329 \\ Index / 335

entry:
@Book{Mason:2003:CP,
  author =       "J. C. Mason and D. C. Handscomb",
  title =        "{Chebyshev} Polynomials",
  publisher =    pub-CHAPMAN-HALL-CRC,
  address =      pub-CHAPMAN-HALL-CRC:adr,
  pages =        "xiii + 341",
  year =         "2003",
  ISBN =         "0-8493-0355-9",
  ISBN-13 =      "978-0-8493-0355-5",
  LCCN =         "QA404.5 .M37 2003",
  bibdate =      "Fri Apr 17 09:45:35 MDT 2009",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/mathcw.bib;
                 z3950.loc.gov:7090/Voyager",
  acknowledgement = ack-nhfb,
  subject =      "Chebyshev polynomials",
  tableofcontents = "1 Definitions / 1 \\
                 1.1 Preliminary remarks / 1 \\
                 1.2 Trigonometric definitions and recurrences / 1 \\
                 1.2.1 The first-kind polynomial $T_n$ / 2 \\
                 1.2.2 The second-kind polynomial $U_n$ / 3 \\
                 1.2.3 The third- and fourth-kind polynomials $V_n$ and
                 $W_n$ (the airfoil polynomials) / 5 \\
                 1.2.4 Connections between the four kinds of polynomial
                 / 7 \\
                 1.3 Shifted Chebyshev polynomials / 9 \\
                 1.3.1 The shifted polynomials $T_n^*$, $U_n^*$,
                 $V_n^*$, $W_n^*$ / 9 \\
                 1.3.2 Chebyshev polynomials for the general range $[a,
                 b]$ / 11 \\
                 1.4 Chebyshev polynomials of a complex variable / 11
                 \\
                 1.4.1 Conformal mapping of a circle to and from an
                 ellipse / 12 \\
                 1.4.2 Chebyshev polynomials in $z$ / 14 \\
                 1.4.3 Shabat polynomials / 17 \\
                 1.5 Problems for Chapter 1 / 17 \\
                 2 Basic Properties and Formulae / 19 \\
                 2.1 Introduction / 19 \\
                 2.2 Chebyshev polynomial zeros and extrema / 19 \\
                 2.3 Relations between Chebyshev polynomials and powers
                 of x / 22 \\
                 2.3.1 Powers of $x$ in terms of $\{T_n(x)\}$ / 22 \\
                 2.3.2 $T_n(x)$ in terms of powers of $x$ / 23 \\
                 2.3.3 Ratios of coefficients in $T_n(x)$ / 25 \\
                 2.4 Evaluation of Chebyshev sums, products, integrals
                 and derivatives / 25 \\
                 2.4.1 Evaluation of a Chebyshev sum / 25 \\
                 2.4.2 Stability of the evaluation of a Chebyshev sum /
                 29 \\
                 2.4.3 Evaluation of a product / 31 \\
                 2.4.4 Evaluation of an integral / 32 \\
                 2.4.5 Evaluation of a derivative / 34 \\
                 6.3.1 Aliasing / 152 \\
                 6.3.2 Second-kind interpolation / 155 \\
                 6.3.3 Third- and fourth-kind interpolation / 156 \\
                 6.3.4 Conditioning / 158 \\
                 6.4 Best $\mathcal{L}_1$ approximation by Chebyshev
                 interpolation / 158 \\
                 6.5 Near-minimax approximation by Chebyshev
                 interpolation / 160 \\
                 6.6 Problems for Chapter 6 / 162 \\
                 7 Near-Best $\mathcal{L}_\infty$, $\mathcal{L}_1$ and
                 $\mathcal{L}_p$ Approximations / 165 \\
                 7.1 Near-best $\mathcal{L}_\infty$ (near-minimax)
                 approximations / 165 \\
                 7.1.1 Second-kind expansions in $\mathcal{L}_\infty$ /
                 165 \\
                 7.1.2 Third-kind expansions in $\mathcal{L}_\infty$ /
                 167 \\
                 7.2 Near-best $\mathcal{L}_1$ approximations / 169 \\
                 7.3 Best and near-best $\mathcal{L}_p$ approximations /
                 170 \\
                 7.3.1 Complex variable results for elliptic-type
                 regions / 172 \\
                 7.4 Problems for Chapter 7 / 173 \\
                 8 Integration Using Chebyshev Polynomials / 177 \\
                 8.1 Indefinite integration with Chebyshev series / 177
                 \\
                 8.2 Gauss--Chebyshev quadrature / 180 \\
                 8.3 Quadrature methods of Clenshaw--Curtis type / 186
                 \\
                 8.3.1 Introduction / 186 \\
                 8.3.2 First-kind formulae / 187 \\
                 8.3.3 Second-kind formulae / 189 \\
                 8.3.4 Third-kind formulae / 191 \\
                 8.3.5 General remark on methods of Clenshaw--Curtis
                 type / 192 \\
                 8.4 Error estimation for Clenshaw--Curtis methods / 192
                 \\
                 8.4.1 First-kind polynomials / 193 \\
                 8.4.2 Fitting an exponential curve / 195 \\
                 8.4.3 Other abscissae and polynomials / 196 \\
                 8.5 Some other work on Clenshaw--Curtis methods / 200
                 \\
                 8.6 Problems for Chapter 8 / 201 \\
                 9 Solution of Integral Equations / 203 \\
                 9.1 Introduction / 203 \\
                 9.2 Fredholm equations of the second kind / 204 \\
                 9.3 Fredholm equations of the third kind / 206 \\
                 9.4 Fredholm equations of the first kind / 207 \\
                 9.5 Singular kernels / 209 \\
                 9.5.1 Hilbert-type kernels and related kernels / 209
                 \\
                 9.5.2 Symm's integral equation / 212 \\
                 9.6 Regularisation of integral equations / 214 \\
                 9.6.1 Discrete data with second derivative
                 regularisation / 214 \\
                 9.6.2 Details of a smoothing algorithm (second
                 derivative regularisation) / 215 \\
                 9.6.3 A smoothing algorithm with weighted function
                 regularisation / 217 \\
                 9.6.4 Evaluation of $V(\lambda)$ / 220 \\
                 9.6.5 Other basis functions / 221 \\
                 9.7 Partial differential equations and boundary
                 integral equation methods / 222 \\
                 9.7.1 A hypersingular integral equation derived from a
                 mixed boundary value problem for Laplace's equation /
                 222 \\
                 9.8 Problems for Chapter 9 / 227 \\
                 10 Solution of Ordinary Differential Equations / 231
                 \\
                 10.1 Introduction / 231 \\
                 10.2 A simple example / 232 \\
                 10.2.1 Collocation methods / 234 \\
                 10.2.2 Error of the collocation method / 237 \\
                 10.2.3 Projection (tau) methods / 239 \\
                 10.2.4 Error of the preceding projection method / 241
                 \\
                 10.3 The original Lanczos tau ($\tau$) method / 242 \\
                 10.4 A more general linear equation / 244 \\
                 10.4.1 Collocation method / 244 \\
                 10.4.2 Projection method / 245 \\
                 10.5 Pseudospectral methods --- another form of
                 collocation / 245 \\
                 10.5.1 Differentiation matrices / 246 \\
                 10.5.2 Differentiation matrix for Chebyshev points /
                 247 \\
                 10.5.3 Collocation using differentiation matrices / 249
                 \\
                 10.6 Nonlinear equations / 251 \\
                 10.7 Eigenvalue problems / 252 \\
                 10.7.1 Collocation methods / 252 \\
                 10.7.2 Collocation using the differentiation matrix /
                 254 \\
                 10.8 Differential equations in one space and one time
                 dimension / 256 \\
                 10.8.1 Collocation methods / 257 \\
                 10.8.2 Collocation using the differentiation matrix /
                 258 \\
                 10.9 Problems for Chapter 10 / 259 \\
                 11 Chebyshev and Spectral Methods for Partial
                 Differential Equations / 261 \\
                 11.1 Introduction / 261 \\
                 11.2 Interior, boundary and mixed methods / 262 \\
                 11.2.1 Interior methods / 262 \\
                 11.2.2 Boundary methods / 263 \\
                 11.2.3 Mixed methods / 265 \\
                 11.3 Differentiation matrices and nodal representation
                 / 265 \\
                 11.4 Method of weighted residuals / 265 \\
                 11.4.1 Continuous MWR / 265 \\
                 11.4.2 Discrete MWR --- a new nomenclature / 266 \\
                 11.5 Chebyshev series and Galerkin methods / 267 \\
                 11.6 Collocation/interpolation and related methods /
                 269 \\
                 11.7 PDE methods / 271 \\
                 11.7.1 Error analysis / 272 \\
                 11.8 Some PDE problems and various methods / 272 \\
                 11.8.1 Power basis: collocation for Poisson problem /
                 273 \\
                 11.8.2 Power basis: interior collocation for the
                 L-membrane / 275 \\
                 11.8.3 Chebyshev basis and discrete orthogonalisation /
                 278 \\
                 11.8.4 Differentiation matrix approach: Poisson problem
                 / 281 \\
                 11.8.5 Explicit collocation for the quasilinear
                 Dirichlet problem: Chebyshev basis / 283 \\
                 11.9 Computational fluid dynamics / 295 \\
                 11.10 Particular issues in spectral methods / 296 \\
                 11.11 More advanced problems / 297 \\
                 11.12 Problems for Chapter 11 / 298 \\
                 12 Conclusion / 303 \\
                 Bibliography / 305 \\
                 Appendices: \\
                 A Biographical Note / 321 \\
                 B Summary of Notations, Definitions and Important
                 Properties / 323 \\
                 B.I Miscellaneous notations / 323 \\
                 B.2 The four kinds of Chebyshev polynomial / 325 \\
                 C Tables of Coefficients / 329 \\
                 Index / 335",
  xxauthor =     "J. C. Mason and D. C. (David Christopher) Handscomb",
  xxURL =        "http://www.loc.gov/catdir/enhancements/fy0646/2002073578-d.html",
}


bibtype: inproceedings
filename: elefunt.bib
label: Meunier:2003:EAG
author: Ludovic Meunier and Bruno Salvy
editor: Hoon Hong
title: ESF: an automatically generated encyclopedia of special functions
publisher: pub-ACM
address: pub-ACM:adr
pages: xiii + 341
month: aug
monthnumber: 8
year: 2003
DOI: https://doi.org/10.1145/860854.860898
ISBN: 1-58113-641-2 (paperback) OR 1581136412 (paperback)
ISBN13: 978-1-58113-641-8 (paperback) OR 9781581136418 (paperback)
LCCN: QA76.5 S98 2003; QA76.95.I59 2003
bibdate: Sat Nov 11 06:21:45 2023
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2023.11.11 06:21:45 ???
acknowledgement: ack-nhfb
bookDOI: https://doi.org/10.1145/860854
bookURL: https://dl.acm.org/doi/proceedings/10.1145/860854
rawauthor: Ludovic Meunier and Bruno Salvy
raweditor: Hoon Hong
rawtitle: {ESF}: an automatically generated encyclopedia of special functions
booktitle:
Proceedings of the 2003 International Symposium on Symbolic and Algebraic Computation: Philadelphia, PA, USA, August 3--6, 2003

abstract:
We present our on-going work on the automatic generation of an encyclopedia of special functions on the web, called The Encyclopedia of Special Functions (ESF) (url http://algo.inria.fr/esf). All mathematical formulae in the ESF are computed, typeset and displayed without any human intervention. This is achieved by exploiting a collection of computer algebra algorithms in a systematic way, on top of a specially designed data structure for a class of special functions.

rawabstract:
We present our on-going work on the automatic generation of an encyclopedia of special functions on the web, called The Encyclopedia of Special Functions (ESF) (\url{http://algo.inria.fr/esf}). All mathematical formulae in the ESF are computed, typeset and displayed without any human intervention. This is achieved by exploiting a collection of computer algebra algorithms in a systematic way, on top of a specially designed data structure for a class of special functions.

rawbooktitle:
Proceedings of the 2003 International Symposium on Symbolic and Algebraic Computation: {Philadelphia, PA, USA, August 3--6, 2003}

entry:
@InProceedings{Meunier:2003:EAG,
  author =       "Ludovic Meunier and Bruno Salvy",
  editor =       "Hoon Hong",
  booktitle =    "Proceedings of the 2003 International Symposium on
                 Symbolic and Algebraic Computation: {Philadelphia, PA,
                 USA, August 3--6, 2003}",
  title =        "{ESF}: an automatically generated encyclopedia of
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  year =         "2003",
  DOI =          "https://doi.org/10.1145/860854.860898",
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  ISBN-13 =      "978-1-58113-641-8 (paperback)",
  LCCN =         "QA76.5 S98 2003; QA76.95.I59 2003",
  bibdate =      "Sat Nov 11 06:21:45 2023",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  abstract =     "We present our on-going work on the automatic
                 generation of an encyclopedia of special functions on
                 the web, called The Encyclopedia of Special Functions
                 (ESF) (\url{http://algo.inria.fr/esf}). All
                 mathematical formulae in the ESF are computed, typeset
                 and displayed without any human intervention. This is
                 achieved by exploiting a collection of computer algebra
                 algorithms in a systematic way, on top of a specially
                 designed data structure for a class of special
                 functions.",
  acknowledgement = ack-nhfb,
  book-DOI =     "https://doi.org/10.1145/860854",
  book-URL =     "https://dl.acm.org/doi/proceedings/10.1145/860854",
}


bibtype: article
filename: elefunt.bib
label: Ovtchinnikov:2003:CEGb
author: E. Ovtchinnikov
journal: j-SIAM-J-NUMER-ANAL
volume: 41
number: 1
pages: 272--286
month: feb
monthnumber: 2
year: 2003
CODEN: SJNAAM
DOI: https://doi.org/10.1137/S0036142902411768
ISSN: 0036-1429 (print), 1095-7170 (electronic) OR 00361429 (print), 10957170 (electronic)
ISSNL: 0036-1429 OR 00361429
bibdate: Fri Aug 15 05:57:09 MDT 2003
bibsource: http://epubs.siam.org/sam-bin/dbq/toc/SINUM/41/1; https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2003.08.15 05:57:09 MDT
URL: http://epubs.siam.org/sam-bin/dbq/article/41176
fjournal: SIAM Journal on Numerical Analysis
keywords: convergence acceleration
acknowledgement: ack-nhfb
journalURL: http://epubs.siam.org/sinum
rawauthor: E. Ovtchinnikov
title:
Convergence Estimates for the Generalized Davidson Method for Symmetric Eigenvalue Problems II: The Subspace Acceleration

rawtitle:
Convergence Estimates for the Generalized {Davidson} Method for Symmetric Eigenvalue Problems {II}: The Subspace Acceleration

entry:
@Article{Ovtchinnikov:2003:CEGb,
  author =       "E. Ovtchinnikov",
  title =        "Convergence Estimates for the Generalized {Davidson}
                 Method for Symmetric Eigenvalue Problems {II}: The
                 Subspace Acceleration",
  journal =      j-SIAM-J-NUMER-ANAL,
  volume =       "41",
  number =       "1",
  pages =        "272--286",
  month =        feb,
  year =         "2003",
  CODEN =        "SJNAAM",
  DOI =          "https://doi.org/10.1137/S0036142902411768",
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  ISSN-L =       "0036-1429",
  bibdate =      "Fri Aug 15 05:57:09 MDT 2003",
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  acknowledgement = ack-nhfb,
  fjournal =     "SIAM Journal on Numerical Analysis",
  journal-URL =  "http://epubs.siam.org/sinum",
  keywords =     "convergence acceleration",
}


bibtype: article
filename: elefunt.bib
label: Paris:2003:AEG
author: R. B. Paris
title: The asymptotic expansion of a generalised incomplete gamma function
journal: j-J-COMPUT-APPL-MATH
volume: 151
number: 2
pages: 297--306
day: 15
month: feb
monthnumber: 2
year: 2003
CODEN: JCAMDI
ISSN: 0377-0427 (print), 1879-1778 (electronic) OR 03770427 (print), 18791778 (electronic)
ISSNL: 0377-0427 OR 03770427
bibdate: Sat Feb 25 12:52:33 MST 2017
bibtimestamp: 2017.02.25 12:52:33 MST
URL: http://www.sciencedirect.com/science/article/pii/S0377042702008099
fjournal: Journal of Computational and Applied Mathematics
acknowledgement: ack-nhfb
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rawauthor: R. B. Paris
rawtitle: The asymptotic expansion of a generalised incomplete gamma function
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  author =       "R. B. Paris",
  title =        "The asymptotic expansion of a generalised incomplete
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  volume =       "151",
  number =       "2",
  pages =        "297--306",
  day =          "15",
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  bibdate =      "Sat Feb 25 12:52:33 MST 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
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bibtype: article
filename: elefunt.bib
label: Paszkowski:2003:CAS
author: Stefan Paszkowski
title: Convergence Acceleration of Some Continued Fractions
journal: j-NUMER-ALGORITHMS
volume: 32
number: 2--4
pages: 193--247
month: apr
monthnumber: 4
year: 2003
CODEN: NUALEG
ISSN: 1017-1398 (print), 1572-9265 (electronic) OR 10171398 (print), 15729265 (electronic)
ISSNL: 1017-1398 OR 10171398
bibdate: Mon Sep 29 08:37:11 MDT 2003
bibsource: http://www.kluweronline.com/issn/1017-1398; https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2003.09.29 08:37:11 MDT
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rawauthor: Stefan Paszkowski
rawtitle: Convergence Acceleration of Some Continued Fractions
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bibtype: article
filename: elefunt.bib
label: Pedersen:2003:DGF
author: Henrik L. Pedersen
title: The double gamma function and related Pick functions
journal: j-J-COMPUT-APPL-MATH
volume: 153
number: 1--2
pages: 361--369
day: 1
month: apr
monthnumber: 4
year: 2003
CODEN: JCAMDI
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ISSNL: 0377-0427 OR 03770427
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bibtimestamp: 2017.02.25 12:52:34 MST
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fjournal: Journal of Computational and Applied Mathematics
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/03770427
rawauthor: Henrik L. Pedersen
rawtitle: The double gamma function and related {Pick} functions
entry:
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  author =       "Henrik L. Pedersen",
  title =        "The double gamma function and related {Pick}
                 functions",
  journal =      j-J-COMPUT-APPL-MATH,
  volume =       "153",
  number =       "1--2",
  pages =        "361--369",
  day =          "1",
  month =        apr,
  year =         "2003",
  CODEN =        "JCAMDI",
  ISSN =         "0377-0427 (print), 1879-1778 (electronic)",
  ISSN-L =       "0377-0427",
  bibdate =      "Sat Feb 25 12:52:34 MST 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2000.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0377042702006040",
  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: Petropoulou:2003:CZB
author: Eugenia N. Petropoulou and Panayiotis D. Siafarikas and Ioannis D. Stabolas
title: On the common zeros of Bessel functions
journal: j-J-COMPUT-APPL-MATH
volume: 153
number: 1--2
pages: 387--393
day: 1
month: apr
monthnumber: 4
year: 2003
CODEN: JCAMDI
ISSN: 0377-0427 (print), 1879-1778 (electronic) OR 03770427 (print), 18791778 (electronic)
ISSNL: 0377-0427 OR 03770427
bibdate: Sat Feb 25 12:52:34 MST 2017
bibtimestamp: 2017.02.25 12:52:34 MST
URL: http://www.sciencedirect.com/science/article/pii/S0377042702006416
fjournal: Journal of Computational and Applied Mathematics
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/03770427
rawauthor: Eugenia N. Petropoulou and Panayiotis D. Siafarikas and Ioannis D. Stabolas
rawtitle: On the common zeros of {Bessel} functions
entry:
@Article{Petropoulou:2003:CZB,
  author =       "Eugenia N. Petropoulou and Panayiotis D. Siafarikas
                 and Ioannis D. Stabolas",
  title =        "On the common zeros of {Bessel} functions",
  journal =      j-J-COMPUT-APPL-MATH,
  volume =       "153",
  number =       "1--2",
  pages =        "387--393",
  day =          "1",
  month =        apr,
  year =         "2003",
  CODEN =        "JCAMDI",
  ISSN =         "0377-0427 (print), 1879-1778 (electronic)",
  ISSN-L =       "0377-0427",
  bibdate =      "Sat Feb 25 12:52:34 MST 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2000.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0377042702006416",
  acknowledgement = ack-nhfb,
  fjournal =     "Journal of Computational and Applied Mathematics",
  journal-URL =  "http://www.sciencedirect.com/science/journal/03770427",
}


bibtype: inproceedings
filename: elefunt.bib
label: Pineiro:2003:LHR
author: J.-A. Pineiro and J. D. Bruguera and M. D. Ercegovac
booktitle: ISCAS '03. Proceedings of the 2003 International Symposium on Circuits and Systems. 25--28 May 2003
title: On-line high-radix exponential with selection by rounding
volume: 4
publisher: pub-IEEE
address: pub-IEEE:adr
pages: IV-121--IV-124
year: 2003
bibdate: Sat Jul 16 11:25:05 MDT 2005
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2005.07.16 11:25:05 MDT
acknowledgement: ack-nhfb
rawauthor: J.-A. Pineiro and J. D. Bruguera and M. D. Ercegovac
rawtitle: On-line high-radix exponential with selection by rounding
rawbooktitle:
{ISCAS '03. Proceedings of the 2003 International Symposium on Circuits and Systems. 25--28 May 2003}

entry:
@InProceedings{Pineiro:2003:LHR,
  author =       "J.-A. Pineiro and J. D. Bruguera and M. D. Ercegovac",
  booktitle =    "{ISCAS '03. Proceedings of the 2003 International
                 Symposium on Circuits and Systems. 25--28 May 2003}",
  title =        "On-line high-radix exponential with selection by
                 rounding",
  volume =       "4",
  publisher =    pub-IEEE,
  address =      pub-IEEE:adr,
  pages =        "IV-121--IV-124",
  year =         "2003",
  CODEN =        "????",
  ISSN =         "????",
  bibdate =      "Sat Jul 16 11:25:05 MDT 2005",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
  summary =      "An on-line high-radix algorithm for computing the
                 exponential function (e/sup x/) with arbitrary
                 precision n is presented. Selection by rounding and a
                 redundant digit-set for the digits e/sub j/ are used,
                 with selection by table in the first \ldots{}",
}


bibtype: book
filename: elefunt.bib
label: Sidi:2003:PEM
author: Avram Sidi
title: Practical Extrapolation Methods: Theory and Applications
volume: 10
publisher: pub-CAMBRIDGE
address: pub-CAMBRIDGE:adr
pages: xxii + 519
year: 2003
DOI: https://doi.org/10.1017/CBO9780511546815
LCCN: QA281 .S555 2003
bibdate: Mon Jul 5 16:49:09 MDT 2010
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager
bibtimestamp: 2010.07.05 16:49:09 MDT
series: Cambridge monographs on applied and computational mathematics
subject: extrapolation
acknowledgement: ack-nhfb
rawauthor: Avram Sidi
rawtitle: Practical Extrapolation Methods: Theory and Applications
ISBN:
0-521-66159-5 (hardcover), 0-511-54681-5 (e-book), 0-511-06018-1 (Adobe Reader), 0-511-06649-X (e-book) OR 0521661595 (hardcover), 0511546815 (ebook), 0511060181 (Adobe Reader), 051106649X (ebook)

ISBN13:
978-0-521-66159-1 (hardcover), 978-0-511-54681-5 (e-book), 978-0-511-06018-2 (Adobe Reader), 978-0-511-06649-8 (e-book) OR 9780521661591 (hardcover), 9780511546815 (ebook), 9780511060182 (Adobe Reader), 9780511066498 (ebook)

TOC:
Preface / xix--xxii Introduction / 1--18 I: The Richardson Extrapolation Process and Its Generalizations / 19--20 1: The Richardson Extrapolation Process / 21--41 2: Additional Topics in Richardson Extrapolation / 42--56 3: First Generalization of the Richardson Extrapolation Process / 57--80 4: GREP: Further Generalization of the Richardson Extrapolation Process / 81--94 5: The $D$-Transformation: A GREP for Infinite-Range Integrals / 95--120 6: The $d$-Transformation: A GREP for Infinite Series and Sequences / 121--157 7: Recursive Algorithms for GREP / 158--175 8: Analytic Study of GREP(1): Slowly Varying $A(y) in F^(1)$ / 176--202 9: Analytic Study of GREP(1): Quickly Varying $A(y) in F^(1)$ / 203--211 10: Efficient Use of GREP(1): Applications to the $D(1)$-, $d(1)$-, and $d(m)$-Transformations / 212--217 11: Reduction of the $D$-Transformation for Oscillatory Infinite-Range Integrals: The $bar D$-, $D'$-, $W$-, and $mW$-Transformations / 218--237 12: Acceleration of Convergence of Power Series by the $d$-Transformation: Rational $d$-Approximants / 238--252 13: Acceleration of Convergence of Fourier and Generalized Fourier Series by the $d$-Transformation: The Complex Series Approach with APS / 253--262 14: Special Topics in Richardson Extrapolation / 263--276 II: Sequence Transformations / 277--278 15: The Euler Transformation, Aitken 2-Process, and Lubkin W-Transformation / 279--296 16: The Shanks Transformation / 297--322 17: The Pade Table / 323--347 18: Generalizations of Pade Approximants / 348--362 19: The Levin- and Sidi $S$-Transformations / 363--374 20: The Wynn- and Brezinski-Algorithms / 375--383 21: The $G$-Transformation and Its Generalizations / 384--389 22: The Transformations of Overholt and Wimp / 390--395 23: Confluent Transformations / 396--406 24: Formal Theory of Sequence Transformations / 407--412 III: Further Applications / 413--414 25: Further Applications of Extrapolation Methods and Sequence Transformations / 415--456 IV: Appendices / 457--458 A: Review of Basic Asymptotics / 459--462 B: The Laplace Transform and Watson's Lemma / 463--464 C: The Gamma Function / 465--466 D: Bernoulli Numbers and Polynomials and the Euler Maclaurin Formula / 467--476 E: The Riemann Zeta Function and the Generalized Zeta Function / 477--479 F: Some Highlights of Polynomial Approximation Theory / 480--482 G: A Compendium of Sequence Transformations / 483--487 H: Efficient Application of Sequence Transformations: Summary / 488--492 I: FORTRAN 77 Program for the $d(m)$-Transformation / 493--500 Bibliography / 501--514 Index / 515--519

rawTOC:
Preface / xix--xxii \\ Introduction / 1--18 \\ I: The Richardson Extrapolation Process and Its Generalizations / 19--20 \\ 1: The Richardson Extrapolation Process / 21--41 \\ 2: Additional Topics in Richardson Extrapolation / 42--56 \\ 3: First Generalization of the Richardson Extrapolation Process / 57--80 \\ 4: GREP: Further Generalization of the Richardson Extrapolation Process / 81--94 \\ 5: The $D$-Transformation: A GREP for Infinite-Range Integrals / 95--120 \\ 6: The $d$-Transformation: A GREP for Infinite Series and Sequences / 121--157 \\ 7: Recursive Algorithms for GREP / 158--175 \\ 8: Analytic Study of GREP(1): Slowly Varying $A(y) \in F^{(1)}$ / 176--202 \\ 9: Analytic Study of GREP(1): Quickly Varying $A(y) \in F^{(1)}$ / 203--211 \\ 10: Efficient Use of GREP(1): Applications to the $D(1)$-, $d(1)$-, and $d(m)$-Transformations / 212--217 \\ 11: Reduction of the $D$-Transformation for Oscillatory Infinite-Range Integrals: The $\bar{D}$-, $D'$-, $W$-, and $mW$-Transformations / 218--237 \\ 12: Acceleration of Convergence of Power Series by the $d$-Transformation: Rational $d$-Approximants / 238--252 \\ 13: Acceleration of Convergence of Fourier and Generalized Fourier Series by the $d$-Transformation: The Complex Series Approach with APS / 253--262 \\ 14: Special Topics in Richardson Extrapolation / 263--276 \\ II: Sequence Transformations / 277--278 \\ 15: The Euler Transformation, Aitken 2-Process, and Lubkin W-Transformation / 279--296 \\ 16: The Shanks Transformation / 297--322 \\ 17: The Pad{\'e} Table / 323--347 \\ 18: Generalizations of Pad{\'e} Approximants / 348--362 \\ 19: The Levin- and Sidi $S$-Transformations / 363--374 \\ 20: The Wynn- and Brezinski-Algorithms / 375--383 \\ 21: The $G$-Transformation and Its Generalizations / 384--389 \\ 22: The Transformations of Overholt and Wimp / 390--395 \\ 23: Confluent Transformations / 396--406 \\ 24: Formal Theory of Sequence Transformations / 407--412 \\ III: Further Applications / 413--414 \\ 25: Further Applications of Extrapolation Methods and Sequence Transformations / 415--456 \\ IV: Appendices / 457--458 \\ A: Review of Basic Asymptotics / 459--462 \\ B: The Laplace Transform and Watson's Lemma / 463--464 \\ C: The Gamma Function / 465--466 \\ D: Bernoulli Numbers and Polynomials and the Euler Maclaurin Formula / 467--476 \\ E: The Riemann Zeta Function and the Generalized Zeta Function / 477--479 \\ F: Some Highlights of Polynomial Approximation Theory / 480--482 \\ G: A Compendium of Sequence Transformations / 483--487 \\ H: Efficient Application of Sequence Transformations: Summary / 488--492 \\ I: FORTRAN 77 Program for the $d(m)$-Transformation / 493--500 \\ Bibliography / 501--514 \\ Index / 515--519

entry:
@Book{Sidi:2003:PEM,
  author =       "Avram Sidi",
  title =        "Practical Extrapolation Methods: Theory and
                 Applications",
  volume =       "10",
  publisher =    pub-CAMBRIDGE,
  address =      pub-CAMBRIDGE:adr,
  pages =        "xxii + 519",
  year =         "2003",
  DOI =          "https://doi.org/10.1017/CBO9780511546815",
  ISBN =         "0-521-66159-5 (hardcover), 0-511-54681-5 (e-book),
                 0-511-06018-1 (Adobe Reader), 0-511-06649-X (e-book)",
  ISBN-13 =      "978-0-521-66159-1 (hardcover), 978-0-511-54681-5
                 (e-book), 978-0-511-06018-2 (Adobe Reader),
                 978-0-511-06649-8 (e-book)",
  LCCN =         "QA281 .S555 2003",
  bibdate =      "Mon Jul 5 16:49:09 MDT 2010",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 z3950.loc.gov:7090/Voyager",
  series =       "Cambridge monographs on applied and computational
                 mathematics",
  acknowledgement = ack-nhfb,
  subject =      "extrapolation",
  tableofcontents = "Preface / xix--xxii \\
                 Introduction / 1--18 \\
                 I: The Richardson Extrapolation Process and Its
                 Generalizations / 19--20 \\
                 1: The Richardson Extrapolation Process / 21--41 \\
                 2: Additional Topics in Richardson Extrapolation /
                 42--56 \\
                 3: First Generalization of the Richardson Extrapolation
                 Process / 57--80 \\
                 4: GREP: Further Generalization of the Richardson
                 Extrapolation Process / 81--94 \\
                 5: The $D$-Transformation: A GREP for Infinite-Range
                 Integrals / 95--120 \\
                 6: The $d$-Transformation: A GREP for Infinite Series
                 and Sequences / 121--157 \\
                 7: Recursive Algorithms for GREP / 158--175 \\
                 8: Analytic Study of GREP(1): Slowly Varying $A(y) \in
                 F^{(1)}$ / 176--202 \\
                 9: Analytic Study of GREP(1): Quickly Varying $A(y) \in
                 F^{(1)}$ / 203--211 \\
                 10: Efficient Use of GREP(1): Applications to the
                 $D(1)$-, $d(1)$-, and $d(m)$-Transformations / 212--217
                 \\
                 11: Reduction of the $D$-Transformation for Oscillatory
                 Infinite-Range Integrals: The $\bar{D}$-, $D'$-, $W$-,
                 and $mW$-Transformations / 218--237 \\
                 12: Acceleration of Convergence of Power Series by the
                 $d$-Transformation: Rational $d$-Approximants /
                 238--252 \\
                 13: Acceleration of Convergence of Fourier and
                 Generalized Fourier Series by the $d$-Transformation:
                 The Complex Series Approach with APS / 253--262 \\
                 14: Special Topics in Richardson Extrapolation /
                 263--276 \\
                 II: Sequence Transformations / 277--278 \\
                 15: The Euler Transformation, Aitken 2-Process, and
                 Lubkin W-Transformation / 279--296 \\
                 16: The Shanks Transformation / 297--322 \\
                 17: The Pad{\'e} Table / 323--347 \\
                 18: Generalizations of Pad{\'e} Approximants / 348--362
                 \\
                 19: The Levin- and Sidi $S$-Transformations / 363--374
                 \\
                 20: The Wynn- and Brezinski-Algorithms / 375--383 \\
                 21: The $G$-Transformation and Its Generalizations /
                 384--389 \\
                 22: The Transformations of Overholt and Wimp / 390--395
                 \\
                 23: Confluent Transformations / 396--406 \\
                 24: Formal Theory of Sequence Transformations /
                 407--412 \\
                 III: Further Applications / 413--414 \\
                 25: Further Applications of Extrapolation Methods and
                 Sequence Transformations / 415--456 \\
                 IV: Appendices / 457--458 \\
                 A: Review of Basic Asymptotics / 459--462 \\
                 B: The Laplace Transform and Watson's Lemma / 463--464
                 \\
                 C: The Gamma Function / 465--466 \\
                 D: Bernoulli Numbers and Polynomials and the Euler
                 Maclaurin Formula / 467--476 \\
                 E: The Riemann Zeta Function and the Generalized Zeta
                 Function / 477--479 \\
                 F: Some Highlights of Polynomial Approximation Theory /
                 480--482 \\
                 G: A Compendium of Sequence Transformations / 483--487
                 \\
                 H: Efficient Application of Sequence Transformations:
                 Summary / 488--492 \\
                 I: FORTRAN 77 Program for the $d(m)$-Transformation /
                 493--500 \\
                 Bibliography / 501--514 \\
                 Index / 515--519",
  xxURL =        "http://www.loc.gov/catdir/samples/cam033/2002024669.html;
                 http://www.loc.gov/catdir/description/cam022/2002024669.html;
                 http://www.loc.gov/catdir/toc/cam024/2002024669.html",
}


bibtype: misc
filename: elefunt.bib
label: Tkachev:2003:EFI
author: Vladimir G. Tkachev
title: Elliptic functions: Introduction course
pages: xxii + 519
day: 25
month: nov
monthnumber: 11
year: 2003
bibdate: Wed Mar 15 08:43:21 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/agm.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2017.03.15 08:43:21 ???
URL: http://users.mai.liu.se/vlatk48/teaching/lect2-agm.pdf
acknowledgement: ack-nhfb
howpublished: Web lecture notes.
rawauthor: Vladimir G. Tkachev
rawtitle: Elliptic functions: Introduction course
TOC:
Chapter 1. Elliptic integrals and Jacobi's theta functions / 5 1.1. Elliptic integrals and the AGM: real case / 5 1.1.3. The arithmetic-geometric mean iteration / 7 1.2. Lemniscates and elastic curves / 11 1.3. Euler's addition theorem / 18 1.4. Theta functions: preliminaries 5 / 24 Chapter 2. General theory of doubly periodic functions / 31 2.1. Preliminaries / 31 2.2. Periods of analytic functions / 33 2.3. Existence of doubly periodic functions / 36 2.4. Liouville's theorems / 38 2.5. The Weierstrass function $wp (z)$ / 43 2.6. Modular forms / 51 Bibliography / 61

rawTOC:
Chapter 1. Elliptic integrals and Jacobi's theta functions / 5 \\ 1.1. Elliptic integrals and the AGM: real case / 5 \\ 1.1.3. The arithmetic-geometric mean iteration / 7 \\ 1.2. Lemniscates and elastic curves / 11 \\ 1.3. Euler's addition theorem / 18 \\ 1.4. Theta functions: preliminaries 5 / 24 \\ Chapter 2. General theory of doubly periodic functions / 31 \\ 2.1. Preliminaries / 31 \\ 2.2. Periods of analytic functions / 33 \\ 2.3. Existence of doubly periodic functions / 36 \\ 2.4. Liouville's theorems / 38 \\ 2.5. The Weierstrass function $\wp(z)$ / 43 \\ 2.6. Modular forms / 51 \\ Bibliography / 61

entry:
@Misc{Tkachev:2003:EFI,
  author =       "Vladimir G. Tkachev",
  title =        "Elliptic functions: Introduction course",
  howpublished = "Web lecture notes.",
  day =          "25",
  month =        nov,
  year =         "2003",
  bibdate =      "Wed Mar 15 08:43:21 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/agm.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://users.mai.liu.se/vlatk48/teaching/lect2-agm.pdf",
  acknowledgement = ack-nhfb,
  tableofcontents = "Chapter 1. Elliptic integrals and Jacobi's theta
                 functions / 5 \\
                 1.1. Elliptic integrals and the AGM: real case / 5 \\
                 1.1.3. The arithmetic-geometric mean iteration / 7 \\
                 1.2. Lemniscates and elastic curves / 11 \\
                 1.3. Euler's addition theorem / 18 \\
                 1.4. Theta functions: preliminaries 5 / 24 \\
                 Chapter 2. General theory of doubly periodic functions
                 / 31 \\
                 2.1. Preliminaries / 31 \\
                 2.2. Periods of analytic functions / 33 \\
                 2.3. Existence of doubly periodic functions / 36 \\
                 2.4. Liouville's theorems / 38 \\
                 2.5. The Weierstrass function $\wp(z)$ / 43 \\
                 2.6. Modular forms / 51 \\
                 Bibliography / 61",
}


bibtype: inproceedings
filename: elefunt.bib
label: Wang:2003:TDF
author: Xiaojun Wang and B. E. Nelson
title: Tradeoffs of designing floating-point division and square root on Virtex FPGAs
publisher: pub-IEEE
address: pub-IEEE:adr
pages: 195--203
year: 2003
bibdate: Sat Jul 16 15:28:14 MDT 2005
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2005.07.16 15:28:14 MDT
acknowledgement: ack-nhfb
rawauthor: Xiaojun Wang and B. E. Nelson
rawtitle: Tradeoffs of designing floating-point division and square root on {Virtex FPGAs}
booktitle:
FCCM 2003: 11th Annual IEEE Symposium on Field-Programmable Custom Computing Machines, 9--11 April 2003

rawbooktitle:
{FCCM 2003}: 11th Annual {IEEE} Symposium on Field-Programmable Custom Computing Machines, 9--11 April 2003

entry:
@InProceedings{Wang:2003:TDF,
  author =       "Xiaojun Wang and B. E. Nelson",
  booktitle =    "{FCCM 2003}: 11th Annual {IEEE} Symposium on
                 Field-Programmable Custom Computing Machines, 9--11
                 April 2003",
  title =        "Tradeoffs of designing floating-point division and
                 square root on {Virtex FPGAs}",
  publisher =    pub-IEEE,
  address =      pub-IEEE:adr,
  pages =        "195--203",
  year =         "2003",
  CODEN =        "????",
  ISSN =         "????",
  bibdate =      "Sat Jul 16 15:28:14 MDT 2005",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
  summary =      "Low latency, high throughput and small area are three
                 major design considerations of an FPGA (field
                 programmable gate array) design. In this paper, we
                 present a high radix SRT division algorithm and a
                 binary restoring square root algorithm. We \ldots{}",
}


bibtype: article
filename: elefunt.bib
label: Yousif:2003:CBF
author: Hashim A. Yousif and Richard Melka
title: Computing Bessel functions of the second kind in extreme parameter regimes
journal: j-COMP-PHYS-COMM
volume: 151
number: 1
pages: 25--34
day: 1
month: mar
monthnumber: 3
year: 2003
CODEN: CPHCBZ
DOI: https://doi.org/10.1016/S0010-4655(02)00697-5
ISSN: 0010-4655 (print), 1879-2944 (electronic) OR 00104655 (print), 18792944 (electronic)
ISSNL: 0010-4655 OR 00104655
bibdate: Mon Feb 13 23:41:27 MST 2012
bibtimestamp: 2012.02.13 23:41:27 MST
URL: http://www.sciencedirect.com/science/article/pii/S0010465502006975
fjournal: Computer Physics Communications
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00104655
rawauthor: Hashim A. Yousif and Richard Melka
rawtitle: Computing {Bessel} functions of the second kind in extreme parameter regimes
bibsource:
https://www.math.utah.edu/pub/tex/bib/compphyscomm2000.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathematica.bib

abstract:
A useful method of computing the integral order Bessel functions of the second kind $ Y_n(x + i y) $ when either, the absolute value of the real part, or the imaginary part of the argument $ z = x + i y $ is small, is described. This method is based on computing the Bessel functions for extreme parameter regimes when $ x sim 0 $ (or $ y sim 0 $ ) and is useful because a number existing algorithms and methods fail to give correct results for small $x$ or small $y$. The approximating equations are derived by expanding the Bessel function in Taylor series, are tested and discussed. The present work is a continuation of the previous one conducted in regard to the Bessel function of the first kind. The results of our formalism are compared to the available existing numerical methods used in Mathematica, IMSL, MATLAB, and the Amos library. Our numerical method is easy to implement, efficient, and produces reliable results. In addition, this method reduces the computation of the Bessel functions of the second complex argument to that of real argument which simplify the computation considerably.

rawabstract:
A useful method of computing the integral order Bessel functions of the second kind $ Y_n(x + i y) $ when either, the absolute value of the real part, or the imaginary part of the argument $ z = x + i y $ is small, is described. This method is based on computing the Bessel functions for extreme parameter regimes when $ x \sim 0 $ (or $ y \sim 0 $ ) and is useful because a number existing algorithms and methods fail to give correct results for small $x$ or small $y$. The approximating equations are derived by expanding the Bessel function in Taylor series, are tested and discussed. The present work is a continuation of the previous one conducted in regard to the Bessel function of the first kind. The results of our formalism are compared to the available existing numerical methods used in Mathematica, IMSL, MATLAB, and the Amos library. Our numerical method is easy to implement, efficient, and produces reliable results. In addition, this method reduces the computation of the Bessel functions of the second complex argument to that of real argument which simplify the computation considerably.

entry:
@Article{Yousif:2003:CBF,
  author =       "Hashim A. Yousif and Richard Melka",
  title =        "Computing {Bessel} functions of the second kind in
                 extreme parameter regimes",
  journal =      j-COMP-PHYS-COMM,
  volume =       "151",
  number =       "1",
  pages =        "25--34",
  day =          "1",
  month =        mar,
  year =         "2003",
  CODEN =        "CPHCBZ",
  DOI =          "https://doi.org/10.1016/S0010-4655(02)00697-5",
  ISSN =         "0010-4655 (print), 1879-2944 (electronic)",
  ISSN-L =       "0010-4655",
  bibdate =      "Mon Feb 13 23:41:27 MST 2012",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/compphyscomm2000.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/mathematica.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0010465502006975",
  abstract =     "A useful method of computing the integral order Bessel
                 functions of the second kind $ Y_n(x + i y) $ when
                 either, the absolute value of the real part, or the
                 imaginary part of the argument $ z = x + i y $ is
                 small, is described. This method is based on computing
                 the Bessel functions for extreme parameter regimes when
                 $ x \sim 0 $ (or $ y \sim 0 $ ) and is useful because a
                 number existing algorithms and methods fail to give
                 correct results for small $x$ or small $y$. The
                 approximating equations are derived by expanding the
                 Bessel function in Taylor series, are tested and
                 discussed. The present work is a continuation of the
                 previous one conducted in regard to the Bessel function
                 of the first kind. The results of our formalism are
                 compared to the available existing numerical methods
                 used in Mathematica, IMSL, MATLAB, and the Amos
                 library. Our numerical method is easy to implement,
                 efficient, and produces reliable results. In addition,
                 this method reduces the computation of the Bessel
                 functions of the second complex argument to that of
                 real argument which simplify the computation
                 considerably.",
  acknowledgement = ack-nhfb,
  fjournal =     "Computer Physics Communications",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00104655",
}


bibtype: book
filename: elefunt.bib
label: Zwillinger:2003:CSM
editor: Daniel Zwillinger
title: CRC Standard Mathematical Tables and Formulae
publisher: pub-CHAPMAN-HALL-CRC
address: pub-CHAPMAN-HALL-CRC:adr
edition: 31st
pages: xiv + 910
year: 2003
ISBN: 1-58488-291-3 (hardcover), 1-4200-3534-7 (e-book) OR 1584882913 (hardcover), 1420035347 (ebook)
LCCN: QA47 .M315 2003
bibdate: Thu Nov 25 11:07:20 MST 2010
bibtimestamp: 2010.11.25 11:07:20 MST
subject: mathematics; tables
acknowledgement: ack-nhfb
raweditor: Daniel Zwillinger
rawtitle: {CRC} Standard Mathematical Tables and Formulae
ISBN13:
978-1-58488-291-6 (hardcover), 978-1-4200-3534-6 (e-book) OR 9781584882916 (hardcover), 9781420035346 (ebook)

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

TOC:
Preface Contributors Table of Contents 1 Analysis 2 Algebra 3 Discrete Mathematics 4 Geometry 5 Continuous Mathematics 6 Special Functions 7 Probability and Statistics 8 Scientific Computing 9 Financial Analysis 10 Miscellaneous List of References List of Figures List of Notation Index

rawTOC:
Preface \\ Contributors \\ Table of Contents \\ 1 Analysis \\ 2 Algebra \\ 3 Discrete Mathematics \\ 4 Geometry \\ 5 Continuous Mathematics \\ 6 Special Functions \\ 7 Probability and Statistics \\ 8 Scientific Computing \\ 9 Financial Analysis \\ 10 Miscellaneous \\ List of References \\ List of Figures \\ List of Notation \\ Index

entry:
@Book{Zwillinger:2003:CSM,
  editor =       "Daniel Zwillinger",
  title =        "{CRC} Standard Mathematical Tables and Formulae",
  publisher =    pub-CHAPMAN-HALL-CRC,
  address =      pub-CHAPMAN-HALL-CRC:adr,
  edition =      "31st",
  pages =        "xiv + 910",
  year =         "2003",
  ISBN =         "1-58488-291-3 (hardcover), 1-4200-3534-7 (e-book)",
  ISBN-13 =      "978-1-58488-291-6 (hardcover), 978-1-4200-3534-6
                 (e-book)",
  LCCN =         "QA47 .M315 2003",
  bibdate =      "Thu Nov 25 11:07:20 MST 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",
  acknowledgement = ack-nhfb,
  subject =      "mathematics; tables",
  tableofcontents = "Preface \\
                 Contributors \\
                 Table of Contents \\
                 1 Analysis \\
                 2 Algebra \\
                 3 Discrete Mathematics \\
                 4 Geometry \\
                 5 Continuous Mathematics \\
                 6 Special Functions \\
                 7 Probability and Statistics \\
                 8 Scientific Computing \\
                 9 Financial Analysis \\
                 10 Miscellaneous \\
                 List of References \\
                 List of Figures \\
                 List of Notation \\
                 Index",
}


bibtype: book
filename: elefunt.bib
label: Bell:2004:SFS
author: W. W. (William Wallace) Bell
title: Special Functions for Scientists and Engineers
publisher: pub-DOVER
address: pub-DOVER:adr
pages: xiv + 247
year: 2004
ISBN: 0-486-43521-0 OR 0486435210
ISBN13: 978-0-486-43521-3 OR 9780486435213
LCCN: QA351 .B4 2004
bibdate: Sat Oct 30 16:30:44 MDT 2010
bibsource: fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2010.10.30 16:30:44 MDT
series: Dover books on mathematics
remark: Reprinting of cite Bell:1968:SFS.
subject: Functions, Special
acknowledgement: ack-nhfb
rawauthor: W. W. (William Wallace) Bell
rawtitle: Special Functions for Scientists and Engineers
TOC:
Preface / v List of Symbols / ix Series Solution of Differential Equations Method of Frobenius / 1 Examples / 8 Problems / 21 Gamma and Beta Functions Definitions / 23 Properties of the beta and gamma functions / 24 Definition of the gamma function for negative values of the argument / 30 Examples / 37 Problems / 40 Legendre Polynomials and Functions Legendre's equation and its solution / 42 Generating function for the Legendre polynomials / 46 Further expressions for the Legendre polynomials / 48 Explicit expressions for and special values of the Legendre polynomials / 50 Orthogonality properties of the Legendre polynomials / 52 Legendre series / 55 Relations between the Legendre polynomials and their derivatives; / 58 recurrence relations Associated Legendre functions / 62 Properties of the associated Legendre functions / 65 Legendre functions of the second kind / 70 Spherical harmonics / 78 Graphs of the Legendre functions / 82 Examples / 85 Problems / 90 Bessel Functions Bessel's equation and its solutions; Bessel functions of the first and second kind / 92 Generating function for the Bessel functions / 99 Integral representations for Bessel functions / 101 Recurrence relations / 104 Hankel functions / 107 Equations reducible to Bessel's equation / 108 Modified Bessel functions / 110 Recurrence relations for the modified Bessel functions / 113 Integral representations for the modified Bessel functions / 116 Kelvin's functions / 120 Spherical Bessel functions / 121 Behaviour of the Bessel functions for large and small values of the argument / 127 Graphs of the Bessel functions / 133 Orthonormality of the Bessel functions; Bessel series / 137 Integrals involving Bessel functions / 141 Examples / 148 Problems / 154 Hermite Polynomials Hermite's equation and its solution / 156 Generating function / 157 Other expressions for the Hermite polynomials / 158 Explicit expressions for, and special values of, the Hermite polynomials / 160 Orthogonality properties of the Hermite polynomials / 161 Relations between Hermite polynomials and their derivatives; recurrence relations / 162 Weber--Hermite functions / 163 Examples / 164 Problems / 166 Laguerre Polynomials Laguerre's equation and its solution / 168 Generating function / 169 Alternative expression for the Laguerre polynomials / 170 Explicit expressions for, and special values of, the Laguerre polynomials / 171 Orthogonality properties of the Laguerre polynomials / 172 Relations between Laguerre polynomials and their derivatives; recurrence relations / 173 Associated Laguerre polynomials / 176 Properties of the associated Laguerre polynomials / 177 Notation / 182 Examples / 182 Problems / 185 Chebyshev Polynomials Definition of Chebyshev polynomials; Chebyshev's equation / 187 Generating function / 190 Orthogonality properties / 192 Recurrence relations / 193 Examples / 194 Problems / 196 Gegenbauer and Jacobi Polynomials Gegenbauer polynomials / 197 Jacobi polynomials / 198 Examples / 200 Problems / 201 Hypergeometric Functions Definition of hypergeometric functions / 203 Properties of the hypergeometric function / 207 Properties of the confluent hypergeometric function / 210 Examples / 212 Problem / 216 Other Special Functions Incomplete gamma functions / 218 Exponential integral and related functions / 218 The error function and related functions / 221 Riemann's zeta function / 223 Debye functions / 224 Elliptic integrals / 224 Examples / 225 Problems / 228 Appendices Convergence of Legendre series / 230 Euler's constant / 231 Differential equations / 233 Orthogonality relations / 234 Generating functions / 236 Hints and Solutions to Problems / 237 Bibliography / 243 Index / 245

rawTOC:
Preface / v List of Symbols / ix Series Solution of Differential Equations Method of Frobenius / 1 \\ Examples / 8 \\ Problems / 21 \\ Gamma and Beta Functions Definitions / 23 \\ Properties of the beta and gamma functions / 24 \\ Definition of the gamma function for negative values of the argument / 30 \\ Examples / 37 \\ Problems / 40 \\ Legendre Polynomials and Functions Legendre's equation and its solution / 42 \\ Generating function for the Legendre polynomials / 46 \\ Further expressions for the Legendre polynomials / 48 \\ Explicit expressions for and special values of the Legendre polynomials / 50 \\ Orthogonality properties of the Legendre polynomials / 52 \\ Legendre series / 55 \\ Relations between the Legendre polynomials and their derivatives; / 58 \\ recurrence relations Associated Legendre functions / 62 \\ Properties of the associated Legendre functions / 65 \\ Legendre functions of the second kind / 70 \\ Spherical harmonics / 78 \\ Graphs of the Legendre functions / 82 \\ Examples / 85 \\ Problems / 90 \\ Bessel Functions Bessel's equation and its solutions; Bessel functions of the first and second kind / 92 \\ Generating function for the Bessel functions / 99 \\ Integral representations for Bessel functions / 101 \\ Recurrence relations / 104 \\ Hankel functions / 107 \\ Equations reducible to Bessel's equation / 108 \\ Modified Bessel functions / 110 \\ Recurrence relations for the modified Bessel functions / 113 \\ Integral representations for the modified Bessel functions / 116 \\ Kelvin's functions / 120 \\ Spherical Bessel functions / 121 \\ Behaviour of the Bessel functions for large and small values of the argument / 127 \\ Graphs of the Bessel functions / 133 \\ Orthonormality of the Bessel functions; Bessel series / 137 \\ Integrals involving Bessel functions / 141 \\ Examples / 148 \\ Problems / 154 \\ Hermite Polynomials Hermite's equation and its solution / 156 \\ Generating function / 157 \\ Other expressions for the Hermite polynomials / 158 \\ Explicit expressions for, and special values of, the Hermite polynomials / 160 \\ Orthogonality properties of the Hermite polynomials / 161 \\ Relations between Hermite polynomials and their derivatives; recurrence relations / 162 \\ Weber--Hermite functions / 163 \\ Examples / 164 \\ Problems / 166 \\ Laguerre Polynomials Laguerre's equation and its solution / 168 \\ Generating function / 169 \\ Alternative expression for the Laguerre polynomials / 170 \\ Explicit expressions for, and special values of, the Laguerre polynomials / 171 \\ Orthogonality properties of the Laguerre polynomials / 172 \\ Relations between Laguerre polynomials and their derivatives; recurrence relations / 173 \\ Associated Laguerre polynomials / 176 \\ Properties of the associated Laguerre polynomials / 177 \\ Notation / 182 \\ Examples / 182 \\ Problems / 185 \\ Chebyshev Polynomials Definition of Chebyshev polynomials; Chebyshev's equation / 187 \\ Generating function / 190 \\ Orthogonality properties / 192 \\ Recurrence relations / 193 \\ Examples / 194 \\ Problems / 196 \\ Gegenbauer and Jacobi Polynomials Gegenbauer polynomials / 197 \\ Jacobi polynomials / 198 \\ Examples / 200 \\ Problems / 201 \\ Hypergeometric Functions Definition of hypergeometric functions / 203 \\ Properties of the hypergeometric function / 207 \\ Properties of the confluent hypergeometric function / 210 \\ Examples / 212 \\ Problem / 216 \\ Other Special Functions Incomplete gamma functions / 218 \\ Exponential integral and related functions / 218 \\ The error function and related functions / 221 \\ Riemann's zeta function / 223 \\ Debye functions / 224 \\ Elliptic integrals / 224 \\ Examples / 225 \\ Problems / 228 \\ Appendices Convergence of Legendre series / 230 \\ Euler's constant / 231 \\ Differential equations / 233 \\ Orthogonality relations / 234 \\ Generating functions / 236 \\ Hints and Solutions to Problems / 237 \\ Bibliography / 243 \\ Index / 245

entry:
@Book{Bell:2004:SFS,
  author =       "W. W. (William Wallace) Bell",
  title =        "Special Functions for Scientists and Engineers",
  publisher =    pub-DOVER,
  address =      pub-DOVER:adr,
  pages =        "xiv + 247",
  year =         "2004",
  ISBN =         "0-486-43521-0",
  ISBN-13 =      "978-0-486-43521-3",
  LCCN =         "QA351 .B4 2004",
  bibdate =      "Sat Oct 30 16:30:44 MDT 2010",
  bibsource =    "fsz3950.oclc.org:210/WorldCat;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  series =       "Dover books on mathematics",
  acknowledgement = ack-nhfb,
  remark =       "Reprinting of \cite{Bell:1968:SFS}.",
  subject =      "Functions, Special",
  tableofcontents = "Preface / v List of Symbols / ix Series Solution of
                 Differential Equations Method of Frobenius / 1 \\
                 Examples / 8 \\
                 Problems / 21 \\
                 Gamma and Beta Functions Definitions / 23 \\
                 Properties of the beta and gamma functions / 24 \\
                 Definition of the gamma function for negative values of
                 the argument / 30 \\
                 Examples / 37 \\
                 Problems / 40 \\
                 Legendre Polynomials and Functions Legendre's equation
                 and its solution / 42 \\
                 Generating function for the Legendre polynomials / 46
                 \\
                 Further expressions for the Legendre polynomials / 48
                 \\
                 Explicit expressions for and special values of the
                 Legendre polynomials / 50 \\
                 Orthogonality properties of the Legendre polynomials /
                 52 \\
                 Legendre series / 55 \\
                 Relations between the Legendre polynomials and their
                 derivatives; / 58 \\
                 recurrence relations Associated Legendre functions / 62
                 \\
                 Properties of the associated Legendre functions / 65
                 \\
                 Legendre functions of the second kind / 70 \\
                 Spherical harmonics / 78 \\
                 Graphs of the Legendre functions / 82 \\
                 Examples / 85 \\
                 Problems / 90 \\
                 Bessel Functions Bessel's equation and its solutions;
                 Bessel functions of the first and second kind / 92 \\
                 Generating function for the Bessel functions / 99 \\
                 Integral representations for Bessel functions / 101 \\
                 Recurrence relations / 104 \\
                 Hankel functions / 107 \\
                 Equations reducible to Bessel's equation / 108 \\
                 Modified Bessel functions / 110 \\
                 Recurrence relations for the modified Bessel functions
                 / 113 \\
                 Integral representations for the modified Bessel
                 functions / 116 \\
                 Kelvin's functions / 120 \\
                 Spherical Bessel functions / 121 \\
                 Behaviour of the Bessel functions for large and small
                 values of the argument / 127 \\
                 Graphs of the Bessel functions / 133 \\
                 Orthonormality of the Bessel functions; Bessel series /
                 137 \\
                 Integrals involving Bessel functions / 141 \\
                 Examples / 148 \\
                 Problems / 154 \\
                 Hermite Polynomials Hermite's equation and its solution
                 / 156 \\
                 Generating function / 157 \\
                 Other expressions for the Hermite polynomials / 158 \\
                 Explicit expressions for, and special values of, the
                 Hermite polynomials / 160 \\
                 Orthogonality properties of the Hermite polynomials /
                 161 \\
                 Relations between Hermite polynomials and their
                 derivatives; recurrence relations / 162 \\
                 Weber--Hermite functions / 163 \\
                 Examples / 164 \\
                 Problems / 166 \\
                 Laguerre Polynomials Laguerre's equation and its
                 solution / 168 \\
                 Generating function / 169 \\
                 Alternative expression for the Laguerre polynomials /
                 170 \\
                 Explicit expressions for, and special values of, the
                 Laguerre polynomials / 171 \\
                 Orthogonality properties of the Laguerre polynomials /
                 172 \\
                 Relations between Laguerre polynomials and their
                 derivatives; recurrence relations / 173 \\
                 Associated Laguerre polynomials / 176 \\
                 Properties of the associated Laguerre polynomials / 177
                 \\
                 Notation / 182 \\
                 Examples / 182 \\
                 Problems / 185 \\
                 Chebyshev Polynomials Definition of Chebyshev
                 polynomials; Chebyshev's equation / 187 \\
                 Generating function / 190 \\
                 Orthogonality properties / 192 \\
                 Recurrence relations / 193 \\
                 Examples / 194 \\
                 Problems / 196 \\
                 Gegenbauer and Jacobi Polynomials Gegenbauer
                 polynomials / 197 \\
                 Jacobi polynomials / 198 \\
                 Examples / 200 \\
                 Problems / 201 \\
                 Hypergeometric Functions Definition of hypergeometric
                 functions / 203 \\
                 Properties of the hypergeometric function / 207 \\
                 Properties of the confluent hypergeometric function /
                 210 \\
                 Examples / 212 \\
                 Problem / 216 \\
                 Other Special Functions Incomplete gamma functions /
                 218 \\
                 Exponential integral and related functions / 218 \\
                 The error function and related functions / 221 \\
                 Riemann's zeta function / 223 \\
                 Debye functions / 224 \\
                 Elliptic integrals / 224 \\
                 Examples / 225 \\
                 Problems / 228 \\
                 Appendices Convergence of Legendre series / 230 \\
                 Euler's constant / 231 \\
                 Differential equations / 233 \\
                 Orthogonality relations / 234 \\
                 Generating functions / 236 \\
                 Hints and Solutions to Problems / 237 \\
                 Bibliography / 243 \\
                 Index / 245",
}


bibtype: article
filename: elefunt.bib
label: Berrut:2004:APS
author: Jean-Paul Berrut and Hans D. Mittelmann
title: Adaptive Point Shifts in Rational Approximation with Optimized Denominator
journal: j-J-COMPUT-APPL-MATH
volume: 164--165
pages: 81--92
day: 1
month: mar
monthnumber: 3
year: 2004
CODEN: JCAMDI
DOI: https://doi.org/10.1016/S0377-0427(03)00485-0
ISSN: 0377-0427 (print), 1879-1778 (electronic) OR 03770427 (print), 18791778 (electronic)
ISSNL: 0377-0427 OR 03770427
bibdate: Tue Mar 24 21:10:48 2009
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2009.03.24 21:10:48 ???
fjournal: Journal of Computational and Applied Mathematics
keywords: interpolation; optimal interpolation; point shifts; rational approximation
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/03770427
rawauthor: Jean-Paul Berrut and Hans D. Mittelmann
rawtitle: Adaptive Point Shifts in Rational Approximation with Optimized Denominator
note:
Proceedings of the 10th International Congress on Computational and Applied Mathematics University of Leuven, Belgium, 22--26 July 2002. Edited by M. J. Goovaerts, S. Vandewalle, and L. Wuytack.

abstract:
Classical rational interpolation is known to suffer from several drawbacks, such as unattainable points and randomly located poles for a small number of nodes, as well as an erratic behavior of the error as this number grows larger. In a former article, we have suggested to obtain rational interpolants by a procedure that attaches optimally placed poles to the interpolating polynomial, using the barycentric representation of the interpolants. In order to improve upon the condition of the derivatives in the solution of differential equations, we have then experimented with a conformal point shift suggested by Kosloff and Tal-Ezer. As it turned out, such shifts can achieve a spectacular improvement in the quality of the approximation itself for functions with a large gradient in the center of the interval. This leads us to the present work which combines the pole attachment method with shifts optimally adjusted to the interpolated function. Such shifts are also constructed for functions with several shocks away from the extremities of the interval.

rawabstract:
Classical rational interpolation is known to suffer from several drawbacks, such as unattainable points and randomly located poles for a small number of nodes, as well as an erratic behavior of the error as this number grows larger. In a former article, we have suggested to obtain rational interpolants by a procedure that attaches optimally placed poles to the interpolating polynomial, using the barycentric representation of the interpolants. In order to improve upon the condition of the derivatives in the solution of differential equations, we have then experimented with a conformal point shift suggested by Kosloff and Tal-Ezer. As it turned out, such shifts can achieve a spectacular improvement in the quality of the approximation itself for functions with a large gradient in the center of the interval. This leads us to the present work which combines the pole attachment method with shifts optimally adjusted to the interpolated function. Such shifts are also constructed for functions with several shocks away from the extremities of the interval.

rawnote:
Proceedings of the 10th International Congress on Computational and Applied Mathematics University of Leuven, Belgium, 22--26 July 2002. Edited by M. J. Goovaerts, S. Vandewalle, and L. Wuytack.

entry:
@Article{Berrut:2004:APS,
  author =       "Jean-Paul Berrut and Hans D. Mittelmann",
  title =        "Adaptive Point Shifts in Rational Approximation with
                 Optimized Denominator",
  journal =      j-J-COMPUT-APPL-MATH,
  volume =       "164--165",
  number =       "??",
  pages =        "81--92",
  day =          "1",
  month =        mar,
  year =         "2004",
  CODEN =        "JCAMDI",
  DOI =          "https://doi.org/10.1016/S0377-0427(03)00485-0",
  ISSN =         "0377-0427 (print), 1879-1778 (electronic)",
  ISSN-L =       "0377-0427",
  bibdate =      "Tue Mar 24 21:10:48 2009",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  note =         "Proceedings of the 10th International Congress on
                 Computational and Applied Mathematics University of
                 Leuven, Belgium, 22--26 July 2002. Edited by M. J.
                 Goovaerts, S. Vandewalle, and L. Wuytack.",
  abstract =     "Classical rational interpolation is known to suffer
                 from several drawbacks, such as unattainable points and
                 randomly located poles for a small number of nodes, as
                 well as an erratic behavior of the error as this number
                 grows larger. In a former article, we have suggested to
                 obtain rational interpolants by a procedure that
                 attaches optimally placed poles to the interpolating
                 polynomial, using the barycentric representation of the
                 interpolants. In order to improve upon the condition of
                 the derivatives in the solution of differential
                 equations, we have then experimented with a conformal
                 point shift suggested by Kosloff and Tal-Ezer. As it
                 turned out, such shifts can achieve a spectacular
                 improvement in the quality of the approximation itself
                 for functions with a large gradient in the center of
                 the interval. This leads us to the present work which
                 combines the pole attachment method with shifts
                 optimally adjusted to the interpolated function. Such
                 shifts are also constructed for functions with several
                 shocks away from the extremities of the interval.",
  acknowledgement = ack-nhfb,
  fjournal =     "Journal of Computational and Applied Mathematics",
  journal-URL =  "http://www.sciencedirect.com/science/journal/03770427",
  keywords =     "interpolation; optimal interpolation; point shifts;
                 rational approximation",
}


bibtype: incollection
filename: elefunt.bib
label: Borwein:2004:AGMa
author: J. M. Borwein and P. B. Borwein
title: The Arithmetic--Geometric Mean and Fast Computation of Elementary Functions
crossref: Berggren:2004:PSB
pages: 537--552
year: 2004
DOI: https://doi.org/10.1007/978-1-4757-4217-6_56
bibdate: Thu Aug 11 09:36:22 MDT 2016
bibtimestamp: 2016.08.11 09:36:22 MDT
note: Reprint of cite Borwein:1984:AGM.
URL: http://link.springer.com/chapter/10.1007/978-1-4757-4217-6_56
acknowledgement: ack-nhfb
authordates: Jonathan Michael Borwein (20 May 1951--2 August 2016)
rawauthor: J. M. Borwein and P. B. Borwein
rawnote: Reprint of \cite{Borwein:1984:AGM}.
rawtitle: The Arithmetic--Geometric Mean and Fast Computation of Elementary Functions
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/agm.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

entry:
@InCollection{Borwein:2004:AGMa,
  author =       "J. M. Borwein and P. B. Borwein",
  title =        "The Arithmetic--Geometric Mean and Fast Computation of
                 Elementary Functions",
  crossref =     "Berggren:2004:PSB",
  pages =        "537--552",
  year =         "2004",
  DOI =          "https://doi.org/10.1007/978-1-4757-4217-6_56",
  bibdate =      "Thu Aug 11 09:36:22 MDT 2016",
  bibsource =    "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib;
                 https://www.math.utah.edu/pub/tex/bib/agm.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  note =         "Reprint of \cite{Borwein:1984:AGM}.",
  URL =          "http://link.springer.com/chapter/10.1007/978-1-4757-4217-6_56",
  acknowledgement = ack-nhfb,
  author-dates = "Jonathan Michael Borwein (20 May 1951--2 August
                 2016)",
  ORCID-numbers = "Borwein, Jonathan/0000-0002-1263-0646",
}


bibtype: article
filename: elefunt.bib
label: Brisebarre:2004:ACR
author: N. Brisebarre and J.-M. Muller and Saurabh Kumar Raina
title: Accelerating correctly rounded floating-point division when the divisor is known in advance
journal: j-IEEE-TRANS-COMPUT
volume: 53
number: 8
pages: 1069--1072
month: aug
monthnumber: 8
year: 2004
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2004.37
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Sat Jul 16 08:40:52 MDT 2005
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2005.07.16 08:40:52 MDT
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: N. Brisebarre and J.-M. Muller and Saurabh Kumar Raina
rawtitle: Accelerating correctly rounded floating-point division when the divisor is known in advance
abstract:
We present techniques for accelerating the floating-point computation of $ x / y $ when $y$ is known before $x$. The proposed algorithms are oriented toward architectures with available fused-mac operations. The goal is to get exactly the same result as with ldots 

rawabstract:
We present techniques for accelerating the floating-point computation of $ x / y $ when $y$ is known before $x$. The proposed algorithms are oriented toward architectures with available fused-mac operations. The goal is to get exactly the same result as with \ldots{}

entry:
@Article{Brisebarre:2004:ACR,
  author =       "N. Brisebarre and J.-M. Muller and Saurabh Kumar
                 Raina",
  title =        "Accelerating correctly rounded floating-point division
                 when the divisor is known in advance",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "53",
  number =       "8",
  pages =        "1069--1072",
  month =        aug,
  year =         "2004",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2004.37",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Sat Jul 16 08:40:52 MDT 2005",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  abstract =     "We present techniques for accelerating the
                 floating-point computation of $ x / y $ when $y$ is
                 known before $x$. The proposed algorithms are oriented
                 toward architectures with available fused-mac
                 operations. The goal is to get exactly the same result
                 as with \ldots{}",
  acknowledgement = ack-nhfb,
  ajournal =     "IEEE Trans. Comput.",
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
  summary =      "We present techniques for accelerating the
                 floating-point computation of x/y when y is known
                 before x. The proposed algorithms are oriented toward
                 architectures with available fused-mac operations. The
                 goal is to get exactly the same result as with
                 \ldots{}",
}


bibtype: article
filename: elefunt.bib
label: Chaudhry:2004:EHC
author: M. Aslam Chaudhry and Asghar Qadir and H. M. Srivastava and R. B. Paris
title: Extended hypergeometric and confluent hypergeometric functions
journal: j-APPL-MATH-COMP
volume: 159
number: 2
pages: 589--602
day: 6
month: dec
monthnumber: 12
year: 2004
CODEN: AMHCBQ
ISSN: 0096-3003 (print), 1873-5649 (electronic) OR 00963003 (print), 18735649 (electronic)
ISSNL: 0096-3003 OR 00963003
bibdate: Mon Jul 4 09:15:38 MDT 2005
bibtimestamp: 2005.07.04 09:15:38 MDT
fjournal: Applied Mathematics and Computation
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00963003
rawauthor: M. Aslam Chaudhry and Asghar Qadir and H. M. Srivastava and R. B. Paris
rawtitle: Extended hypergeometric and confluent hypergeometric functions
bibsource:
http://www.sciencedirect.com/science/journal/00963003; https://www.math.utah.edu/pub/tex/bib/applmathcomput2000.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

entry:
@Article{Chaudhry:2004:EHC,
  author =       "M. Aslam Chaudhry and Asghar Qadir and H. M.
                 Srivastava and R. B. Paris",
  title =        "Extended hypergeometric and confluent hypergeometric
                 functions",
  journal =      j-APPL-MATH-COMP,
  volume =       "159",
  number =       "2",
  pages =        "589--602",
  day =          "6",
  month =        dec,
  year =         "2004",
  CODEN =        "AMHCBQ",
  ISSN =         "0096-3003 (print), 1873-5649 (electronic)",
  ISSN-L =       "0096-3003",
  bibdate =      "Mon Jul 4 09:15:38 MDT 2005",
  bibsource =    "http://www.sciencedirect.com/science/journal/00963003;
                 https://www.math.utah.edu/pub/tex/bib/applmathcomput2000.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "Applied Mathematics and Computation",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00963003",
}


bibtype: article
filename: elefunt.bib
label: Croot:2004:ACC
author: Ernie Croot and Ren-Cang Li and H. J. Hui June Zhu
title: The em abc conjecture and correctly rounded reciprocal square roots
journal: j-THEOR-COMP-SCI
volume: 315
number: 2--3
pages: 405--417
day: 6
month: may
monthnumber: 5
year: 2004
CODEN: TCSCDI
ISSN: 0304-3975 (print), 1879-2294 (electronic) OR 03043975 (print), 18792294 (electronic)
ISSNL: 0304-3975 OR 03043975
bibdate: Thu Nov 4 10:19:15 MST 2004
bibtimestamp: 2004.11.04 10:19:15 MST
fjournal: Theoretical Computer Science
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/03043975
rawauthor: Ernie Croot and Ren-Cang Li and H. J. Hui June Zhu
rawtitle: The {\em abc\/} conjecture and correctly rounded reciprocal square roots
bibsource:
http://www.sciencedirect.com/science/journal/03043975; 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/tcs2000.bib

abstract:
The reciprocal square root calculation $ alpha = 1 / sqrt x $ is very common in scientific computations. Having a correctly rounded implementation of it is of great importance in producing numerically predictable code among today's heterogeneous computing environment. Existing results suggest that to get the correctly rounded $ alpha $ in a floating point number system with $p$ significant bits, we may have to compute up to $ 3 p + 1 $ leading bits of $ alpha $. However, numerical evidence indicates the actual number may be as small as $ 2 p $ plus a few more bits. This paper attempts to bridge the gap by showing that this is indeed true, assuming the em abc conjecture which is widely purported to hold. (But our results do not tell exactly how many more bits beyond the $ 2 p $ bits, due to the fact that the constants involved in the conjecture are ineffective.) Along the way, rough bounds which are comparable to the existing ones are also proven. The technique used here is a combination of the classical Liouville's estimation and contemporary number theory.

rawabstract:
The reciprocal square root calculation $ \alpha = 1 / \sqrt {x} $ is very common in scientific computations. Having a correctly rounded implementation of it is of great importance in producing numerically predictable code among today's heterogeneous computing environment. Existing results suggest that to get the correctly rounded $ \alpha $ in a floating point number system with $p$ significant bits, we may have to compute up to $ 3 p + 1 $ leading bits of $ \alpha $. However, numerical evidence indicates the actual number may be as small as $ 2 p $ plus a few more bits. This paper attempts to bridge the gap by showing that this is indeed true, assuming the {\em abc\/} conjecture which is widely purported to hold. (But our results do not tell exactly how many more bits beyond the $ 2 p $ bits, due to the fact that the constants involved in the conjecture are ineffective.) Along the way, rough bounds which are comparable to the existing ones are also proven. The technique used here is a combination of the classical Liouville's estimation and contemporary number theory.

entry:
@Article{Croot:2004:ACC,
  author =       "Ernie Croot and Ren-Cang Li and H. J. Hui June Zhu",
  title =        "The {\em abc\/} conjecture and correctly rounded
                 reciprocal square roots",
  journal =      j-THEOR-COMP-SCI,
  volume =       "315",
  number =       "2--3",
  pages =        "405--417",
  day =          "6",
  month =        may,
  year =         "2004",
  CODEN =        "TCSCDI",
  ISSN =         "0304-3975 (print), 1879-2294 (electronic)",
  ISSN-L =       "0304-3975",
  bibdate =      "Thu Nov 4 10:19:15 MST 2004",
  bibsource =    "http://www.sciencedirect.com/science/journal/03043975;
                 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/tcs2000.bib",
  abstract =     "The reciprocal square root calculation $ \alpha = 1 /
                 \sqrt {x} $ is very common in scientific computations.
                 Having a correctly rounded implementation of it is of
                 great importance in producing numerically predictable
                 code among today's heterogeneous computing environment.
                 Existing results suggest that to get the correctly
                 rounded $ \alpha $ in a floating point number system
                 with $p$ significant bits, we may have to compute up to
                 $ 3 p + 1 $ leading bits of $ \alpha $. However,
                 numerical evidence indicates the actual number may be
                 as small as $ 2 p $ plus a few more bits. This paper
                 attempts to bridge the gap by showing that this is
                 indeed true, assuming the {\em abc\/} conjecture which
                 is widely purported to hold. (But our results do not
                 tell exactly how many more bits beyond the $ 2 p $
                 bits, due to the fact that the constants involved in
                 the conjecture are ineffective.) Along the way, rough
                 bounds which are comparable to the existing ones are
                 also proven. The technique used here is a combination
                 of the classical Liouville's estimation and
                 contemporary number theory.",
  acknowledgement = ack-nhfb,
  fjournal =     "Theoretical Computer Science",
  journal-URL =  "http://www.sciencedirect.com/science/journal/03043975",
}


bibtype: inproceedings
filename: elefunt.bib
label: deDinechin:2004:PCR
author: Florent de Dinechin and Catherine Loirat and Jean-Michel Muller
title: A proven correctly rounded logarithm in double-precision
crossref: Frougny:2004:RCR
pages: 71--85
year: 2004
bibdate: Fri Nov 17 07:00:31 2023
bibtimestamp: 2023.11.17 07:00:31 ???
URL: http://www.informatik.uni-trier.de/Reports/TR-08-2004/rnc6_07_dinechin.pdf
keywords: arithmetic; correct rounding; elementary functions; floating-point; libm; logarithm
acknowledgement: ack-nhfb
rawauthor: Florent de Dinechin and Catherine Loirat and Jean-Michel Muller
rawtitle: A proven correctly rounded logarithm in double-precision
abstract:
This article is a case study in the implementation of a proven, portable, and efficient correctly rounded elementary function in double-precision. We describe the methodology used in the implementation of the natural logarithm in the crlibm library. The discipline required to prove a tight bound on the overall evaluation error allows to design a very efficient implementation with moderate effort.

rawabstract:
This article is a case study in the implementation of a proven, portable, and efficient correctly rounded elementary function in double-precision. We describe the methodology used in the implementation of the natural logarithm in the crlibm library. The discipline required to prove a tight bound on the overall evaluation error allows to design a very efficient implementation with moderate effort.

entry:
@InProceedings{deDinechin:2004:PCR,
  author =       "Florent de Dinechin and Catherine Loirat and
                 Jean-Michel Muller",
  title =        "A proven correctly rounded logarithm in
                 double-precision",
  crossref =     "Frougny:2004:RCR",
  pages =        "71--85",
  year =         "2004",
  bibdate =      "Fri Nov 17 07:00:31 2023",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://www.informatik.uni-trier.de/Reports/TR-08-2004/rnc6_07_dinechin.pdf",
  abstract =     "This article is a case study in the implementation of
                 a proven, portable, and efficient correctly rounded
                 elementary function in double-precision. We describe
                 the methodology used in the implementation of the
                 natural logarithm in the crlibm library. The discipline
                 required to prove a tight bound on the overall
                 evaluation error allows to design a very efficient
                 implementation with moderate effort.",
  acknowledgement = ack-nhfb,
  keywords =     "arithmetic; correct rounding; elementary functions;
                 floating-point; libm; logarithm",
}


bibtype: article
filename: elefunt.bib
label: Defour:2004:PSM
title: Proposal for a Standardization of Mathematical Function Implementation in Floating-Point Arithmetic
journal: j-NUMER-ALGORITHMS
volume: 37
number: 1--4
pages: 367--375
month: dec
monthnumber: 12
year: 2004
CODEN: NUALEG
ISSN: 1017-1398 (print), 1572-9265 (electronic) OR 10171398 (print), 15729265 (electronic)
ISSNL: 1017-1398 OR 10171398
bibdate: Mon Dec 6 07:00:28 MST 2004
bibsource: http://www.kluweronline.com/issn/1017-1398; https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2004.12.06 07:00:28 MST
fjournal: Numerical Algorithms
remark: Special Issue: SCAN'2002 International Conference (Guest Editors: Ren e Alt and Jean-Luc Lamotte)
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/11075
rawtitle: Proposal for a Standardization of Mathematical Function Implementation in Floating-Point Arithmetic
author:
David Defour and Guillaume Hanrot and Vincent Lefevre and Jean-Michel Muller and Nathalie Revol and Paul Zimmermann

URL:
http://ipsapp009.kluweronline.com/IPS/content/ext/x/J/5058/I/58/A/30/abstract.htm; http://perso.ens-lyon.fr/jean-michel.muller/NumAlg04.pdf; http://www.loria.fr/~zimmerma/papers/PropStandFunctions.pdf

rawauthor:
David Defour and Guillaume Hanrot and Vincent Lef{\`e}vre and Jean-Michel Muller and Nathalie Revol and Paul Zimmermann

entry:
@Article{Defour:2004:PSM,
  author =       "David Defour and Guillaume Hanrot and Vincent
                 Lef{\`e}vre and Jean-Michel Muller and Nathalie Revol
                 and Paul Zimmermann",
  title =        "Proposal for a Standardization of Mathematical
                 Function Implementation in Floating-Point Arithmetic",
  journal =      j-NUMER-ALGORITHMS,
  volume =       "37",
  number =       "1--4",
  pages =        "367--375",
  month =        dec,
  year =         "2004",
  CODEN =        "NUALEG",
  ISSN =         "1017-1398 (print), 1572-9265 (electronic)",
  ISSN-L =       "1017-1398",
  bibdate =      "Mon Dec 6 07:00:28 MST 2004",
  bibsource =    "http://www.kluweronline.com/issn/1017-1398;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://ipsapp009.kluweronline.com/IPS/content/ext/x/J/5058/I/58/A/30/abstract.htm;
                 http://perso.ens-lyon.fr/jean-michel.muller/NumAlg04.pdf;
                 http://www.loria.fr/~zimmerma/papers/PropStandFunctions.pdf",
  acknowledgement = ack-nhfb,
  fjournal =     "Numerical Algorithms",
  journal-URL =  "http://link.springer.com/journal/11075",
  remark =       "Special Issue: SCAN'2002 International Conference
                 (Guest Editors: Ren {\'e} Alt and Jean-Luc Lamotte)",
}


bibtype: inproceedings
filename: elefunt.bib
label: Doss:2004:FBI
author: C. C. Doss and R. L. Riley, Jr.
title: FPGA-based implementation of a robust IEEE-754 exponential unit
publisher: pub-IEEE
address: pub-IEEE:adr
pages: 229--238
year: 2004
bibdate: Sat Jul 16 17:14:11 MDT 2005
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2005.07.16 17:14:11 MDT
acknowledgement: ack-nhfb
rawauthor: C. C. Doss and R. L. {Riley, Jr.}
rawtitle: {FPGA}-based implementation of a robust {IEEE-754} exponential unit
booktitle:
FCCM 2004. 12th Annual IEEE Symposium on Field-Programmable Custom Computing Machines, 20--23 April 2004

rawbooktitle:
{FCCM 2004}. 12th Annual {IEEE} Symposium on Field-Programmable Custom Computing Machines, 20--23 April 2004

entry:
@InProceedings{Doss:2004:FBI,
  author =       "C. C. Doss and R. L. {Riley, Jr.}",
  booktitle =    "{FCCM 2004}. 12th Annual {IEEE} Symposium on
                 Field-Programmable Custom Computing Machines, 20--23
                 April 2004",
  title =        "{FPGA}-based implementation of a robust {IEEE-754}
                 exponential unit",
  publisher =    pub-IEEE,
  address =      pub-IEEE:adr,
  pages =        "229--238",
  year =         "2004",
  CODEN =        "????",
  ISSN =         "????",
  bibdate =      "Sat Jul 16 17:14:11 MDT 2005",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
  summary =      "This work explores the feasibility of implementing a
                 floating-point exponentiation unit on reconfigurable
                 computing systems. A table-driven exponentiation unit
                 was implemented using synthesizable VHDL. The project
                 included creating pipelined \ldots{}",
}


bibtype: techreport
filename: elefunt.bib
label: Ercegovac:2004:CSRa
author: Milos Ercegovac and Jean-Michel Muller
title: Complex Square Root with Operand Prescaling
type: Research Report
number: RR2004-42
institution: Ecole Normale Superieure de Lyon
address: 69364 Lyon Cedex 07, France
pages: 2 + 12
month: sep
monthnumber: 9
year: 2004
bibdate: Mon Dec 06 11:07:40 2004
bibtimestamp: 2004.12.06 11:07:40 ???
URL: http://www.ens-lyon.fr/LIP/Pub/Rapports/RR/RR2004/RR2004-42.pdf
keywords: complex square-root; Computer arithmetic; digit-recurrence algorithm; operand prescaling.
acknowledgement: ack-nhfb
rawauthor: Milo{\v{s}} Ercegovac and Jean-Michel Muller
rawtitle: Complex Square Root with Operand Prescaling
bibsource:
http://www.ens-lyon.fr/LIP/Pub/rr2004.php; 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. Moreover, the use of digit recurrence approach allows correct rounding of the result. The algorithm, compatible with the complex division, and its design are described at a high-level. We also give rough comparisons 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. Moreover, the use of digit recurrence approach allows correct rounding of the result. The algorithm, compatible with the complex division, and its design are described at a high-level. We also give rough comparisons 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:
@TechReport{Ercegovac:2004:CSRa,
  author =       "Milo{\v{s}} Ercegovac and Jean-Michel Muller",
  title =        "Complex Square Root with Operand Prescaling",
  type =         "Research Report",
  number =       "RR2004-42",
  institution =  "{\'E}cole Normale Sup{\'e}rieure de Lyon",
  address =      "69364 Lyon Cedex 07, France",
  pages =        "2 + 12",
  month =        sep,
  year =         "2004",
  bibdate =      "Mon Dec 06 11:07:40 2004",
  bibsource =    "http://www.ens-lyon.fr/LIP/Pub/rr2004.php;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://www.ens-lyon.fr/LIP/Pub/Rapports/RR/RR2004/RR2004-42.pdf",
  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. Moreover, the use of digit recurrence
                 approach allows correct rounding of the result. The
                 algorithm, compatible with the complex division, and
                 its design are described at a high-level. We also give
                 rough comparisons 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,
  keywords =     "complex square-root; Computer arithmetic;
                 digit-recurrence algorithm; operand prescaling.",
}


bibtype: inproceedings
filename: elefunt.bib
label: Ercegovac:2004:CSRb
author: Milos Ercegovac and Jean-Michel Muller
title: Complex square root with operand prescaling
publisher: pub-IEEE
address: pub-IEEE:adr
pages: 52--62
year: 2004
DOI: https://doi.org/10.1109/ASAP.2004.1342458
ISBN: 0-7695-2226-2 OR 0769522262
ISBN13: 978-0-7695-2226-5 OR 9780769522265
ISSN: 1063-6862 OR 10636862
bibdate: Sat Jul 16 15:28:13 MDT 2005
bibtimestamp: 2005.07.16 15:28:13 MDT
acknowledgement: ack-nhfb
rawauthor: Milo{\v{s}} Ercegovac and Jean-Michel Muller
rawtitle: Complex square root with operand prescaling
booktitle:
Proceedings of the 15th IEEE International Conference on Application-Specific Systems, Architectures and Processors, 2004

rawbooktitle:
{Proceedings of the 15th IEEE International Conference on Application-Specific Systems, Architectures and Processors, 2004}

entry:
@InProceedings{Ercegovac:2004:CSRb,
  author =       "Milo{\v{s}} Ercegovac and Jean-Michel Muller",
  booktitle =    "{Proceedings of the 15th IEEE International Conference
                 on Application-Specific Systems, Architectures and
                 Processors, 2004}",
  title =        "Complex square root with operand prescaling",
  publisher =    pub-IEEE,
  address =      pub-IEEE:adr,
  pages =        "52--62",
  year =         "2004",
  CODEN =        "????",
  DOI =          "https://doi.org/10.1109/ASAP.2004.1342458",
  ISBN =         "0-7695-2226-2",
  ISBN-13 =      "978-0-7695-2226-5",
  ISSN =         "1063-6862",
  bibdate =      "Sat Jul 16 15:28:13 MDT 2005",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  summary =      "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. Moreover, the use of digit \ldots{}",
}


bibtype: article
filename: elefunt.bib
label: Fabijonas:2004:AAF
author: B. R. Fabijonas
title: Algorithm 838: Airy Functions
journal: j-TOMS
volume: 30
number: 4
pages: 491--501
month: dec
monthnumber: 12
year: 2004
CODEN: ACMSCU
DOI: https://doi.org/10.1145/1039813.1039819
ISSN: 0098-3500 (print), 1557-7295 (electronic) OR 00983500 (print), 15577295 (electronic)
ISSNL: 0098-3500 OR 00983500
bibdate: Tue Apr 12 06:34:31 MDT 2005
bibtimestamp: 2005.04.12 06:34:31 MDT
fjournal: ACM Transactions on Mathematical Software (TOMS)
acknowledgement: ack-nhfb
journalURL: https://dl.acm.org/loi/toms
rawauthor: B. R. Fabijonas
rawtitle: {Algorithm 838}: {Airy} Functions
abstract:
We present a Fortran 90 module, which computes the solutions and their derivatives of Airy's differential equation, both on the real line and in the complex plane. The module also computes the zeros and associated values of the solutions and their derivatives, and the modulus and phase functions on the negative real axis. The computational methods are numerical integration of the differential equation and summation of asymptotic expansions for large argument. These methods were chosen because they are simple, adaptable to any precision, and amenable to rigorous error analysis. The module can be used to validate other codes or as a component in programs that require Airy functions.

rawabstract:
We present a Fortran 90 module, which computes the solutions and their derivatives of Airy's differential equation, both on the real line and in the complex plane. The module also computes the zeros and associated values of the solutions and their derivatives, and the modulus and phase functions on the negative real axis. The computational methods are numerical integration of the differential equation and summation of asymptotic expansions for large argument. These methods were chosen because they are simple, adaptable to any precision, and amenable to rigorous error analysis. The module can be used to validate other codes or as a component in programs that require Airy functions.

entry:
@Article{Fabijonas:2004:AAF,
  author =       "B. R. Fabijonas",
  title =        "{Algorithm 838}: {Airy} Functions",
  journal =      j-TOMS,
  volume =       "30",
  number =       "4",
  pages =        "491--501",
  month =        dec,
  year =         "2004",
  CODEN =        "ACMSCU",
  DOI =          "https://doi.org/10.1145/1039813.1039819",
  ISSN =         "0098-3500 (print), 1557-7295 (electronic)",
  ISSN-L =       "0098-3500",
  bibdate =      "Tue Apr 12 06:34:31 MDT 2005",
  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 a Fortran 90 module, which computes the
                 solutions and their derivatives of Airy's differential
                 equation, both on the real line and in the complex
                 plane. The module also computes the zeros and
                 associated values of the solutions and their
                 derivatives, and the modulus and phase functions on the
                 negative real axis. The computational methods are
                 numerical integration of the differential equation and
                 summation of asymptotic expansions for large argument.
                 These methods were chosen because they are simple,
                 adaptable to any precision, and amenable to rigorous
                 error analysis. The module can be used to validate
                 other codes or as a component in programs that require
                 Airy functions.",
  acknowledgement = ack-nhfb,
  fjournal =     "ACM Transactions on Mathematical Software (TOMS)",
  journal-URL =  "https://dl.acm.org/loi/toms",
}


bibtype: article
filename: elefunt.bib
label: Fabijonas:2004:CCA
author: B. R. Fabijonas and D. W. Lozier and F. W. J. Olver
journal: j-TOMS
volume: 30
number: 4
pages: 471--490
month: dec
monthnumber: 12
year: 2004
CODEN: ACMSCU
DOI: https://doi.org/10.1145/1039813.1039818
ISSN: 0098-3500 (print), 1557-7295 (electronic) OR 00983500 (print), 15577295 (electronic)
ISSNL: 0098-3500 OR 00983500
bibdate: Tue Apr 12 06:34:31 MDT 2005
bibtimestamp: 2005.04.12 06:34:31 MDT
fjournal: ACM Transactions on Mathematical Software (TOMS)
acknowledgement: ack-nhfb
journalURL: https://dl.acm.org/loi/toms
rawauthor: B. R. Fabijonas and D. W. Lozier and F. W. J. Olver
title:
Computation of complex Airy functions and their zeros using asymptotics and the differential equation

abstract:
We describe a method by which one can compute the solutions of Airy's differential equation, and their derivatives, both on the real line and in the complex plane. The computational methods are numerical integration of the differential equation and summation of asymptotic expansions for large argument. We give details involved in obtaining all of the parameter values, and we control the truncation errors rigorously. Using the same computational methods, we describe an algorithm that computes the zeros and associated values of the Airy functions and their derivatives, and the modulus and phase functions on the negative real axis.

rawabstract:
We describe a method by which one can compute the solutions of Airy's differential equation, and their derivatives, both on the real line and in the complex plane. The computational methods are numerical integration of the differential equation and summation of asymptotic expansions for large argument. We give details involved in obtaining all of the parameter values, and we control the truncation errors rigorously. Using the same computational methods, we describe an algorithm that computes the zeros and associated values of the Airy functions and their derivatives, and the modulus and phase functions on the negative real axis.

rawtitle:
Computation of complex {Airy} functions and their zeros using asymptotics and the differential equation

entry:
@Article{Fabijonas:2004:CCA,
  author =       "B. R. Fabijonas and D. W. Lozier and F. W. J. Olver",
  title =        "Computation of complex {Airy} functions and their
                 zeros using asymptotics and the differential equation",
  journal =      j-TOMS,
  volume =       "30",
  number =       "4",
  pages =        "471--490",
  month =        dec,
  year =         "2004",
  CODEN =        "ACMSCU",
  DOI =          "https://doi.org/10.1145/1039813.1039818",
  ISSN =         "0098-3500 (print), 1557-7295 (electronic)",
  ISSN-L =       "0098-3500",
  bibdate =      "Tue Apr 12 06:34:31 MDT 2005",
  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 describe a method by which one can compute the
                 solutions of Airy's differential equation, and their
                 derivatives, both on the real line and in the complex
                 plane. The computational methods are numerical
                 integration of the differential equation and summation
                 of asymptotic expansions for large argument. We give
                 details involved in obtaining all of the parameter
                 values, and we control the truncation errors
                 rigorously. Using the same computational methods, we
                 describe an algorithm that computes the zeros and
                 associated values of the Airy functions and their
                 derivatives, and the modulus and phase functions on the
                 negative real axis.",
  acknowledgement = ack-nhfb,
  fjournal =     "ACM Transactions on Mathematical Software (TOMS)",
  journal-URL =  "https://dl.acm.org/loi/toms",
}


bibtype: inproceedings
filename: elefunt.bib
label: Gebali:2004:EAF
author: Fayez Gebali and Mohamed Watheq El-Kharashi
title: ERL: an algorithm for fast evaluation of exponential, reciprocal, and logarithmic functions
crossref: Wahdan:2004:IHE
pages: 269--272
year: 2004
bibdate: Sat Jul 16 18:04:58 MDT 2005
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2005.07.16 18:04:58 MDT
acknowledgement: ack-nhfb
rawauthor: Fayez Gebali and Mohamed Watheq El-Kharashi
rawtitle: {ERL}: an algorithm for fast evaluation of exponential, reciprocal, and logarithmic functions
abstract:
A fast algorithm (ERL) is proposed for evaluating Exponential, Reciprocal, and Logarithmic functions. The algorithm requires two to three iterations to complete using simple operations such as multiply, accumulate, and table lookup. The algorithm is independent of the number format used by the machine. Thus it can be implemented using the IEEE 754 floating-point standard or any other special format used by special-purpose processors. The dynamic range of the algorithm is limited only by the dynamic range of the machine on which it is implemented Numerical simulations are performed which verifies the speed and accuracy of the algorithm.

rawabstract:
A fast algorithm (ERL) is proposed for evaluating Exponential, Reciprocal, and Logarithmic functions. The algorithm requires two to three iterations to complete using simple operations such as multiply, accumulate, and table lookup. The algorithm is independent of the number format used by the machine. Thus it can be implemented using the IEEE 754 floating-point standard or any other special format used by special-purpose processors. The dynamic range of the algorithm is limited only by the dynamic range of the machine on which it is implemented Numerical simulations are performed which verifies the speed and accuracy of the algorithm.

entry:
@InProceedings{Gebali:2004:EAF,
  author =       "Fayez Gebali and Mohamed Watheq El-Kharashi",
  title =        "{ERL}: an algorithm for fast evaluation of
                 exponential, reciprocal, and logarithmic functions",
  crossref =     "Wahdan:2004:IHE",
  pages =        "269--272",
  year =         "2004",
  bibdate =      "Sat Jul 16 18:04:58 MDT 2005",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  abstract =     "A fast algorithm (ERL) is proposed for evaluating
                 Exponential, Reciprocal, and Logarithmic functions. The
                 algorithm requires two to three iterations to complete
                 using simple operations such as multiply, accumulate,
                 and table lookup. The algorithm is independent of the
                 number format used by the machine. Thus it can be
                 implemented using the IEEE 754 floating-point standard
                 or any other special format used by special-purpose
                 processors. The dynamic range of the algorithm is
                 limited only by the dynamic range of the machine on
                 which it is implemented Numerical simulations are
                 performed which verifies the speed and accuracy of the
                 algorithm.",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: elefunt.bib
label: Gil:2004:AMB
author: Amparo Gil and Javier Segura and Nico M. Temme
title: Algorithm 831: Modified Bessel functions of imaginary order and positive argument
journal: j-TOMS
volume: 30
number: 2
pages: 159--164
month: jun
monthnumber: 6
year: 2004
CODEN: ACMSCU
DOI: https://doi.org/10.1145/992200.992204
ISSN: 0098-3500 (print), 1557-7295 (electronic) OR 00983500 (print), 15577295 (electronic)
ISSNL: 0098-3500 OR 00983500
bibdate: Thu Jun 10 07:24:58 MDT 2004
bibtimestamp: 2004.06.10 07:24:58 MDT
fjournal: ACM Transactions on Mathematical Software (TOMS)
acknowledgement: ack-nhfb
journalURL: https://dl.acm.org/loi/toms
rawauthor: Amparo Gil and Javier Segura and Nico M. Temme
rawtitle: {Algorithm 831}: {Modified} {Bessel} functions of imaginary order and positive argument
abstract:
Fortran 77 programs for the computation of modified Bessel functions of purely imaginary order are presented. The codes compute the functions $ K_ia(x) $, $ L_ia(x) $ and their derivatives for real $a$ and positive $x$; these functions are independent solutions of the differential equation $ x^2 w'' + x w' + (a^2 - x^2)w = 0 $. The code also computes exponentially scaled functions. The range of computation is $ (x, a) in (0, 1500] times [ - 1500, 1500] $ when scaled functions are considered and it is larger than $ (0, 500] times [ - 400, 400] $ for standard IEEE double precision arithmetic. The relative accuracy is better than $ 10^-13 $ in the range $ (0, 200] times [ - 200, 200] $ and close to $ 10^-12 $ in $ (0, 1500] times [ - 1500, 1500] $.

rawabstract:
Fortran 77 programs for the computation of modified Bessel functions of purely imaginary order are presented. The codes compute the functions $ K_{ia}(x) $, $ L_{ia}(x) $ and their derivatives for real $a$ and positive $x$; these functions are independent solutions of the differential equation $ x^2 w'' + x w' + (a^2 - x^2)w = 0 $. The code also computes exponentially scaled functions. The range of computation is $ (x, a) \in (0, 1500] \times [ - 1500, 1500] $ when scaled functions are considered and it is larger than $ (0, 500] \times [ - 400, 400] $ for standard IEEE double precision arithmetic. The relative accuracy is better than $ 10^{-13} $ in the range $ (0, 200] \times [ - 200, 200] $ and close to $ 10^{-12} $ in $ (0, 1500] \times [ - 1500, 1500] $.

entry:
@Article{Gil:2004:AMB,
  author =       "Amparo Gil and Javier Segura and Nico M. Temme",
  title =        "{Algorithm 831}: {Modified} {Bessel} functions of
                 imaginary order and positive argument",
  journal =      j-TOMS,
  volume =       "30",
  number =       "2",
  pages =        "159--164",
  month =        jun,
  year =         "2004",
  CODEN =        "ACMSCU",
  DOI =          "https://doi.org/10.1145/992200.992204",
  ISSN =         "0098-3500 (print), 1557-7295 (electronic)",
  ISSN-L =       "0098-3500",
  bibdate =      "Thu Jun 10 07:24:58 MDT 2004",
  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 =     "Fortran 77 programs for the computation of modified
                 Bessel functions of purely imaginary order are
                 presented. The codes compute the functions $ K_{ia}(x)
                 $, $ L_{ia}(x) $ and their derivatives for real $a$ and
                 positive $x$; these functions are independent solutions
                 of the differential equation $ x^2 w'' + x w' + (a^2 -
                 x^2)w = 0 $. The code also computes exponentially
                 scaled functions. The range of computation is $ (x, a)
                 \in (0, 1500] \times [ - 1500, 1500] $ when scaled
                 functions are considered and it is larger than $ (0,
                 500] \times [ - 400, 400] $ for standard IEEE double
                 precision arithmetic. The relative accuracy is better
                 than $ 10^{-13} $ in the range $ (0, 200] \times [ -
                 200, 200] $ and close to $ 10^{-12} $ in $ (0, 1500]
                 \times [ - 1500, 1500] $.",
  acknowledgement = ack-nhfb,
  fjournal =     "ACM Transactions on Mathematical Software (TOMS)",
  journal-URL =  "https://dl.acm.org/loi/toms",
}


bibtype: article
filename: elefunt.bib
label: Gil:2004:CRZ
author: Amparo Gil and Wolfram Koepf and Javier Segura
title: Computing the Real Zeros of Hypergeometric Functions
journal: j-NUMER-ALGORITHMS
volume: 36
number: 2
pages: 113--134
month: jun
monthnumber: 6
year: 2004
CODEN: NUALEG
ISSN: 1017-1398 (print), 1572-9265 (electronic) OR 10171398 (print), 15729265 (electronic)
ISSNL: 1017-1398 OR 10171398
bibdate: Mon Dec 6 07:00:32 MST 2004
bibtimestamp: 2004.12.06 07:00:32 MST
URL: http://ipsapp009.kluweronline.com/IPS/content/ext/x/J/5058/I/54/A/2/abstract.htm
fjournal: Numerical Algorithms
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/11075
rawauthor: Amparo Gil and Wolfram Koepf and Javier Segura
rawtitle: Computing the Real Zeros of Hypergeometric Functions
bibsource:
http://www.kluweronline.com/issn/1017-1398; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib

entry:
@Article{Gil:2004:CRZ,
  author =       "Amparo Gil and Wolfram Koepf and Javier Segura",
  title =        "Computing the Real Zeros of Hypergeometric Functions",
  journal =      j-NUMER-ALGORITHMS,
  volume =       "36",
  number =       "2",
  pages =        "113--134",
  month =        jun,
  year =         "2004",
  CODEN =        "NUALEG",
  ISSN =         "1017-1398 (print), 1572-9265 (electronic)",
  ISSN-L =       "1017-1398",
  bibdate =      "Mon Dec 6 07:00:32 MST 2004",
  bibsource =    "http://www.kluweronline.com/issn/1017-1398;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib",
  URL =          "http://ipsapp009.kluweronline.com/IPS/content/ext/x/J/5058/I/54/A/2/abstract.htm",
  acknowledgement = ack-nhfb,
  fjournal =     "Numerical Algorithms",
  journal-URL =  "http://link.springer.com/journal/11075",
}


bibtype: article
filename: elefunt.bib
label: Gil:2004:CSM
author: Amparo Gil and Javier Segura and Nico M. Temme
journal: j-TOMS
volume: 30
number: 2
pages: 145--158
month: jun
monthnumber: 6
year: 2004
CODEN: ACMSCU
DOI: https://doi.org/10.1145/992200.992203
ISSN: 0098-3500 (print), 1557-7295 (electronic) OR 00983500 (print), 15577295 (electronic)
ISSNL: 0098-3500 OR 00983500
bibdate: Thu Jun 10 07:24:58 MDT 2004
bibtimestamp: 2004.06.10 07:24:58 MDT
fjournal: ACM Transactions on Mathematical Software (TOMS)
acknowledgement: ack-nhfb
journalURL: https://dl.acm.org/loi/toms
rawauthor: Amparo Gil and Javier Segura and Nico M. Temme
title:
Computing solutions of the modified Bessel differential equation for imaginary orders and positive arguments

abstract:
We describe a variety of methods to compute the functions $ K_ia(x) $, $ L_ia(x) $ and their derivatives for real $a$ and positive $x$. These functions are numerically satisfactory independent solutions of the differential equation $ x^2 w'' + x w' + (a^2 - x^2)w = 0 $. In the accompanying paper [Gil et al. 2004], we describe the implementation of these methods in Fortran 77 codes.

rawabstract:
We describe a variety of methods to compute the functions $ K_{ia}(x) $, $ L_{ia}(x) $ and their derivatives for real $a$ and positive $x$. These functions are numerically satisfactory independent solutions of the differential equation $ x^2 w'' + x w' + (a^2 - x^2)w = 0 $. In the accompanying paper [Gil et al. 2004], we describe the implementation of these methods in Fortran 77 codes.

rawtitle:
Computing solutions of the modified {Bessel} differential equation for imaginary orders and positive arguments

entry:
@Article{Gil:2004:CSM,
  author =       "Amparo Gil and Javier Segura and Nico M. Temme",
  title =        "Computing solutions of the modified {Bessel}
                 differential equation for imaginary orders and positive
                 arguments",
  journal =      j-TOMS,
  volume =       "30",
  number =       "2",
  pages =        "145--158",
  month =        jun,
  year =         "2004",
  CODEN =        "ACMSCU",
  DOI =          "https://doi.org/10.1145/992200.992203",
  ISSN =         "0098-3500 (print), 1557-7295 (electronic)",
  ISSN-L =       "0098-3500",
  bibdate =      "Thu Jun 10 07:24:58 MDT 2004",
  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 describe a variety of methods to compute the
                 functions $ K_{ia}(x) $, $ L_{ia}(x) $ and their
                 derivatives for real $a$ and positive $x$. These
                 functions are numerically satisfactory independent
                 solutions of the differential equation $ x^2 w'' + x w'
                 + (a^2 - x^2)w = 0 $. In the accompanying paper [Gil et
                 al. 2004], we describe the implementation of these
                 methods in Fortran 77 codes.",
  acknowledgement = ack-nhfb,
  fjournal =     "ACM Transactions on Mathematical Software (TOMS)",
  journal-URL =  "https://dl.acm.org/loi/toms",
}


bibtype: article
filename: elefunt.bib
label: Guseinov:2004:EIG
author: I. I. Guseinov and B. A. Mamedov
title: Evaluation of Incomplete Gamma Functions Using Downward Recursion and Analytical Relations
journal: j-J-MATH-CHEM
volume: 36
number: 4
pages: 341--346
month: aug
monthnumber: 8
year: 2004
CODEN: JMCHEG
DOI: https://doi.org/10.1023/B:JOMC.0000044521.18885.d3
ISSN: 0259-9791 (print), 1572-8897 (electronic) OR 02599791 (print), 15728897 (electronic)
ISSNL: 0259-9791 OR 02599791
bibdate: Thu Apr 9 18:14:03 MDT 2015
bibtimestamp: 2015.04.09 18:14:03 MDT
URL: http://link.springer.com/article/10.1023/B:JOMC.0000044521.18885.d3
fjournal: Journal of Mathematical Chemistry
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/10910
journalabr: J. Math. Chem.
rawauthor: I. I. Guseinov and B. A. Mamedov
rawtitle: Evaluation of Incomplete Gamma Functions Using Downward Recursion and Analytical Relations
bibsource:
http://link.springer.com/journal/10910/36/4; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jmathchem.bib

entry:
@Article{Guseinov:2004:EIG,
  author =       "I. I. Guseinov and B. A. Mamedov",
  title =        "Evaluation of Incomplete Gamma Functions Using
                 Downward Recursion and Analytical Relations",
  journal =      j-J-MATH-CHEM,
  volume =       "36",
  number =       "4",
  pages =        "341--346",
  month =        aug,
  year =         "2004",
  CODEN =        "JMCHEG",
  DOI =          "https://doi.org/10.1023/B:JOMC.0000044521.18885.d3",
  ISSN =         "0259-9791 (print), 1572-8897 (electronic)",
  ISSN-L =       "0259-9791",
  bibdate =      "Thu Apr 9 18:14:03 MDT 2015",
  bibsource =    "http://link.springer.com/journal/10910/36/4;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jmathchem.bib",
  URL =          "http://link.springer.com/article/10.1023/B:JOMC.0000044521.18885.d3",
  acknowledgement = ack-nhfb,
  fjournal =     "Journal of Mathematical Chemistry",
  journal-URL =  "http://link.springer.com/journal/10910",
  journalabr =   "J. Math. Chem.",
}


bibtype: book
filename: elefunt.bib
label: Jeffrey:2004:HMF
author: Alan Jeffrey
title: Handbook of Mathematical Formulas and Integrals
publisher: pub-ELSEVIER-ACADEMIC
address: pub-ELSEVIER-ACADEMIC:adr
edition: Third
pages: xxvi + 453
year: 2004
ISBN: 0-12-382256-4 OR 0123822564
ISBN13: 978-0-12-382256-7 OR 9780123822567
LCCN: QA47 .J38 2004
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
rawtitle: Handbook of Mathematical Formulas and Integrals
bibsource:
http://www.math.utah.edu/pub/tex/bib/mathcw.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager

TOC:
0. Quick Reference List of Frequently Used Data 1 1. Numerical, Algebraic, and Analytical Results for Series and Calculus 25 2. Functions and Identities 101 3. Derivatives of Elementary Functions 139 4. Indefinite Integrals of Algebraic Functions 145 5. Indefinite Integrals of Exponential Functions 167 6. Indefinite Integrals of logarithmic Functions 173 7. Indefinite Integrals of Hyperbolic Functions 179 8. Indefinite Integrals Involving Inverse Hyperbolic Functions 191 9. Indefinite Integrals of Trigonometric Functions 197 10. Indefinite Integrals of Inverse Trigonometric Functions 215 11. The Gamma, Beta, Pi, and Psi Functions 221 12. Elliptic Integrals and Functions 229 13. Probability Integrals and the Error Function 239 14. Fresnel Integrals, Sine and Cosine Integrals 245 15. Definite Integrals 249 16. Different Forms of Fourier Series 257 17. Bessel Functions 269 18. Orthogonal Polynomials 285 19. Laplace Transformation 299 20. Fourier Transforms 307 21. Numerical Integration 315 22. Solutions of Standard Ordinary Differential Equations 321 23. Vector Analysis 353 24. Systems of Orthogonal Coordinates 369 25. Partial Differential Equations and Special Functions 381 26. the z-Transform 403 27. Numerical Approximation 409 28. Solutions of Elliptic, Parabolic, and Hyperbolic Equations 419 29. Qualitative Properties of the Heat and Laplace Equation 437

rawTOC:
0. Quick Reference List of Frequently Used Data 1 \\ 1. Numerical, Algebraic, and Analytical Results for Series and Calculus 25 \\ 2. Functions and Identities 101 \\ 3. Derivatives of Elementary Functions 139 \\ 4. Indefinite Integrals of Algebraic Functions 145 \\ 5. Indefinite Integrals of Exponential Functions 167 \\ 6. Indefinite Integrals of logarithmic Functions 173 \\ 7. Indefinite Integrals of Hyperbolic Functions 179 \\ 8. Indefinite Integrals Involving Inverse Hyperbolic Functions 191 \\ 9. Indefinite Integrals of Trigonometric Functions 197 \\ 10. Indefinite Integrals of Inverse Trigonometric Functions 215 \\ 11. The Gamma, Beta, Pi, and Psi Functions 221 \\ 12. Elliptic Integrals and Functions 229 \\ 13. Probability Integrals and the Error Function 239 \\ 14. Fresnel Integrals, Sine and Cosine Integrals 245 \\ 15. Definite Integrals 249 \\ 16. Different Forms of Fourier Series 257 \\ 17. Bessel Functions 269 \\ 18. Orthogonal Polynomials 285 \\ 19. Laplace Transformation 299 \\ 20. Fourier Transforms 307 \\ 21. Numerical Integration 315 \\ 22. Solutions of Standard Ordinary Differential Equations 321 \\ 23. Vector Analysis 353 \\ 24. Systems of Orthogonal Coordinates 369 \\ 25. Partial Differential Equations and Special Functions 381 \\ 26. the z-Transform 403 \\ 27. Numerical Approximation 409 \\ 28. Solutions of Elliptic, Parabolic, and Hyperbolic Equations 419 \\ 29. Qualitative Properties of the Heat and Laplace Equation 437

entry:
@Book{Jeffrey:2004:HMF,
  author =       "Alan Jeffrey",
  title =        "Handbook of Mathematical Formulas and Integrals",
  publisher =    pub-ELSEVIER-ACADEMIC,
  address =      pub-ELSEVIER-ACADEMIC:adr,
  edition =      "Third",
  pages =        "xxvi + 453",
  year =         "2004",
  ISBN =         "0-12-382256-4",
  ISBN-13 =      "978-0-12-382256-7",
  LCCN =         "QA47 .J38 2004",
  bibdate =      "Thu May 8 16:02:52 MDT 2008",
  bibsource =    "http://www.math.utah.edu/pub/tex/bib/mathcw.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 z3950.loc.gov:7090/Voyager",
  acknowledgement = ack-nhfb,
  subject =      "mathematics; tables; formulae",
  tableofcontents = "0. Quick Reference List of Frequently Used Data 1
                 \\
                 1. Numerical, Algebraic, and Analytical Results for
                 Series and Calculus 25 \\
                 2. Functions and Identities 101 \\
                 3. Derivatives of Elementary Functions 139 \\
                 4. Indefinite Integrals of Algebraic Functions 145 \\
                 5. Indefinite Integrals of Exponential Functions 167
                 \\
                 6. Indefinite Integrals of logarithmic Functions 173
                 \\
                 7. Indefinite Integrals of Hyperbolic Functions 179 \\
                 8. Indefinite Integrals Involving Inverse Hyperbolic
                 Functions 191 \\
                 9. Indefinite Integrals of Trigonometric Functions 197
                 \\
                 10. Indefinite Integrals of Inverse Trigonometric
                 Functions 215 \\
                 11. The Gamma, Beta, Pi, and Psi Functions 221 \\
                 12. Elliptic Integrals and Functions 229 \\
                 13. Probability Integrals and the Error Function 239
                 \\
                 14. Fresnel Integrals, Sine and Cosine Integrals 245
                 \\
                 15. Definite Integrals 249 \\
                 16. Different Forms of Fourier Series 257 \\
                 17. Bessel Functions 269 \\
                 18. Orthogonal Polynomials 285 \\
                 19. Laplace Transformation 299 \\
                 20. Fourier Transforms 307 \\
                 21. Numerical Integration 315 \\
                 22. Solutions of Standard Ordinary Differential
                 Equations 321 \\
                 23. Vector Analysis 353 \\
                 24. Systems of Orthogonal Coordinates 369 \\
                 25. Partial Differential Equations and Special
                 Functions 381 \\
                 26. the z-Transform 403 \\
                 27. Numerical Approximation 409 \\
                 28. Solutions of Elliptic, Parabolic, and Hyperbolic
                 Equations 419 \\
                 29. Qualitative Properties of the Heat and Laplace
                 Equation 437",
  xxURL =        "http://www.loc.gov/catdir/description/els041/2003049507.html;
                 http://www.loc.gov/catdir/toc/els041/2003049507.html;
                 http://www.e-streams.com/es0710/es0710_3628.html",
}


bibtype: misc
filename: elefunt.bib
label: Kahan:2004:LTC
author: W. Kahan
title: A Logarithm Too Clever by Half
pages: 9
day: 9
month: aug
monthnumber: 8
year: 2004
bibdate: Mon Apr 25 17:39:08 2005
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2005.04.25 17:39:08 ???
URL: http://www.cs.berkeley.edu/~wkahan/LOG10HAF.TXT
remark: Careful analysis of the problem of computing tt log10(x) accurately from tt log(x).
acknowledgement: ack-nhfb
howpublished: World-Wide Web document
rawauthor: W. Kahan
rawtitle: A Logarithm Too Clever by Half
entry:
@Misc{Kahan:2004:LTC,
  author =       "W. Kahan",
  title =        "A Logarithm Too Clever by Half",
  howpublished = "World-Wide Web document",
  pages =        "9",
  day =          "9",
  month =        aug,
  year =         "2004",
  bibdate =      "Mon Apr 25 17:39:08 2005",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.cs.berkeley.edu/~wkahan/LOG10HAF.TXT",
  acknowledgement = ack-nhfb,
  remark =       "Careful analysis of the problem of computing {\tt
                 log10(x)} accurately from {\tt log(x)}.",
}


bibtype: article
filename: elefunt.bib
label: Kalmykov:2004:SEH
author: M. Y. Kalmykov
title: Series and $ epsilon $-expansion of the hypergeometric functions
journal: j-NUCL-PHYS-B-PROC-SUPPL
volume: 135
pages: 280--284
year: 2004
CODEN: NPBSE7
ISSN: 0920-5632 (print), 1873-3832 (electronic) OR 09205632 (print), 18733832 (electronic)
ISSNL: 0920-5632 OR 09205632
bibdate: Thu Dec 01 09:14:29 2011
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2011.12.01 09:14:29 ???
fjournal: Nuclear Physics B, Proceedings Supplements
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/09205632
rawauthor: M. Y. Kalmykov
rawtitle: Series and $ \epsilon $-expansion of the hypergeometric functions
entry:
@Article{Kalmykov:2004:SEH,
  author =       "M. Y. Kalmykov",
  title =        "Series and $ \epsilon $-expansion of the
                 hypergeometric functions",
  journal =      j-NUCL-PHYS-B-PROC-SUPPL,
  volume =       "135",
  number =       "??",
  pages =        "280--284",
  month =        "????",
  year =         "2004",
  CODEN =        "NPBSE7",
  ISSN =         "0920-5632 (print), 1873-3832 (electronic)",
  ISSN-L =       "0920-5632",
  bibdate =      "Thu Dec 01 09:14:29 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "Nuclear Physics B, Proceedings Supplements",
  journal-URL =  "http://www.sciencedirect.com/science/journal/09205632",
}


bibtype: book
filename: elefunt.bib
label: Kilbas:2004:TTA
author: A. A. (Anatolii Aleksandrovich) Kilbas and Megumi Saigo
title: $H$-transforms: theory and applications
volume: 9
publisher: pub-CHAPMAN-HALL-CRC
address: pub-CHAPMAN-HALL-CRC:adr
pages: xii + 389
year: 2004
ISBN: 0-203-48737-0, 0-415-29916-0, 1-58488-116-X OR 0203487370, 0415299160, 158488116X
bibdate: Sat Oct 30 17:20:21 MDT 2010
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.bibsys.no:2100/BIBSYS
bibtimestamp: 2010.10.30 17:20:21 MDT
series: Analytical methods and special functions
subject: $H$-functions; integral transforms
acknowledgement: ack-nhfb
rawauthor: A. A. (Anatolii Aleksandrovich) Kilbas and Megumi Saigo
rawtitle: {$H$}-transforms: theory and applications
ISBN13:
978-0-203-48737-2, 978-1-58488-116-2, 978-0-415-29916-9 OR 9780203487372, 9781584881162, 9780415299169

TOC:
1. Definition, representations and expansions of the H-function 2. Properties of the H-function 3. H-transform on the space [symbol] 4. H-transform on the space [symbol] 5. Modified H-transforms on the space [symbol] 6. G-transform and modified G-transforms on the space [symbol] 7. Hypergeometric type integral transforms on the space [symbol] 8. Bessel type integral transforms on the space [symbol] Bibliography Author Index Subject Index Symbol Index

rawTOC:
1. Definition, representations and expansions of the H-function \\ 2. Properties of the H-function \\ 3. H-transform on the space [symbol] \\ 4. H-transform on the space [symbol] \\ 5. Modified H-transforms on the space [symbol] \\ 6. G-transform and modified G-transforms on the space [symbol] \\ 7. Hypergeometric type integral transforms on the space [symbol] \\ 8. Bessel type integral transforms on the space [symbol] \\ Bibliography \\ Author Index \\ Subject Index \\ Symbol Index

entry:
@Book{Kilbas:2004:TTA,
  author =       "A. A. (Anatolii Aleksandrovich) Kilbas and Megumi
                 Saigo",
  title =        "{$H$}-transforms: theory and applications",
  volume =       "9",
  publisher =    pub-CHAPMAN-HALL-CRC,
  address =      pub-CHAPMAN-HALL-CRC:adr,
  pages =        "xii + 389",
  year =         "2004",
  ISBN =         "0-203-48737-0, 0-415-29916-0, 1-58488-116-X",
  ISBN-13 =      "978-0-203-48737-2, 978-1-58488-116-2,
                 978-0-415-29916-9",
  LCCN =         "????",
  bibdate =      "Sat Oct 30 17:20:21 MDT 2010",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 z3950.bibsys.no:2100/BIBSYS",
  series =       "Analytical methods and special functions",
  acknowledgement = ack-nhfb,
  subject =      "$H$-functions; integral transforms",
  tableofcontents = "1. Definition, representations and expansions of
                 the H-function \\
                 2. Properties of the H-function \\
                 3. H-transform on the space [symbol] \\
                 4. H-transform on the space [symbol] \\
                 5. Modified H-transforms on the space [symbol] \\
                 6. G-transform and modified G-transforms on the space
                 [symbol] \\
                 7. Hypergeometric type integral transforms on the space
                 [symbol] \\
                 8. Bessel type integral transforms on the space
                 [symbol] \\
                 Bibliography \\
                 Author Index \\
                 Subject Index \\
                 Symbol Index",
}


bibtype: article
filename: elefunt.bib
label: Kyurkchiev:2004:FCN
author: N. Kyurkchiev and A. Iliev
journal: j-COMPUT-MATH-APPL
volume: 47
number: 2--3
pages: 441--446
month: jan
monthnumber: 1
year: 2004
CODEN: CMAPDK
ISSN: 0898-1221 (print), 1873-7668 (electronic) OR 08981221 (print), 18737668 (electronic)
ISSNL: 0898-1221 OR 08981221
bibdate: Wed Mar 1 21:49:35 MST 2017
bibtimestamp: 2017.03.01 21:49:35 MST
URL: http://www.sciencedirect.com/science/article/pii/S0898122104900363
fjournal: Computers and Mathematics with Applications
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/08981221
rawauthor: N. Kyurkchiev and A. Iliev
title:
Failure of convergence of the Newton--Weierstrass iterative method for simultaneous root finding of generalized polynomials

rawtitle:
Failure of convergence of the {Newton--Weierstrass} iterative method for simultaneous root finding of generalized polynomials

entry:
@Article{Kyurkchiev:2004:FCN,
  author =       "N. Kyurkchiev and A. Iliev",
  title =        "Failure of convergence of the {Newton--Weierstrass}
                 iterative method for simultaneous root finding of
                 generalized polynomials",
  journal =      j-COMPUT-MATH-APPL,
  volume =       "47",
  number =       "2--3",
  pages =        "441--446",
  month =        jan # "\slash " # feb,
  year =         "2004",
  CODEN =        "CMAPDK",
  ISSN =         "0898-1221 (print), 1873-7668 (electronic)",
  ISSN-L =       "0898-1221",
  bibdate =      "Wed Mar 1 21:49:35 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/S0898122104900363",
  acknowledgement = ack-nhfb,
  fjournal =     "Computers and Mathematics with Applications",
  journal-URL =  "http://www.sciencedirect.com/science/journal/08981221",
}


bibtype: book
filename: elefunt.bib
label: Lau:2004:NLJ
author: H. T. (Hang Tong) Lau
title: A Numerical Library in Java for Scientists and Engineers
publisher: pub-CHAPMAN-HALL-CRC
address: pub-CHAPMAN-HALL-CRC:adr
pages: xxiii + 1063
year: 2004
DOI: https://doi.org/10.1201/9780203507643
ISBN: 1-58488-430-4 OR 1584884304
ISBN13: 978-1-58488-430-9 OR 9781584884309
LCCN: QA76.73.J38 L363 2004
bibdate: Fri Sep 26 14:28:47 MDT 2008
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib; z3950.loc.gov:7090/Voyager
bibtimestamp: 2008.09.26 14:28:47 MDT
URL: http://www.loc.gov/catdir/enhancements/fy0646/2003055149-d.html
subject: Java (Computer program language)
acknowledgement: ack-nhfb
rawauthor: H. T. (Hang Tong) Lau
rawtitle: A Numerical Library in {Java} for Scientists and Engineers
TOC:
1: Elementary Procedures 1.1: Real vector and matrix Initialization 1.2: Real vector and matrix Duplication 1.3: Real vector and matrix Multiplication 1.4: Real vector vector products 1.5: Real matrix vector products 1.6: Real matrix matrix products 1.7: Real vector and matrix Elimination 1.8: Real vector and matrix Interchanging 1.9: Real vector and matrix Rotation 1.10: Real vector and matrix Norms 1.11: Real vector and matrix Scaling 1.12: Complex vector and matrix Multiplication 1.13: Complex vector and matrix Scalar products 1.14: Complex vector and matrix Elimination 1.15: Complex vector and matrix Rotation 1.16: Complex vector and matrix Norms 1.17: Complex vector and matrix Scaling 1.18: Complex monadic operations 1.19: Complex dyadic operations 1.20: Long integer arithmetic 2: Algebraic Evaluations 2.1: Evaluation of polynomials in Grunert form 2.2: Evaluation of general orthogonal polynomials 2.3: Evaluation of Chebyshev polynomials 2.4: Evaluation of Fourier series 2.5: Evaluation of continued fractions 2.6: Transformation of polynomial representation 2.7: Operations on orthogonal polynomials 3: Linear Algebra 3.1: Full real general matrices 3.2: Real symmetric positive definite matrices 3.3: General real symmetric matrices 3.4: Real full rank overdetermined systems 3.5: Other real matrix problems 3.6: Real sparse nonsymmetric band matrices 3.7: Real sparse nonsymmetric tridiagonal matrices 3.8: Sparse symmetric positive definite band matrices 3.9: Symmetric positive definite tridiagonal matrices 3.10: Sparse real matrices Iterative methods 3.11: Similarity transformation 3.12: Other transformations 3.13: The (ordinary) eigenvalue problem 3.14: The generalized eigenvalue problem 3.15: Singular values 3.16: Zeros of polynomials 4: Analytic Evaluations 4.1: Evaluation of an infinite series 4.2: Quadrature 4.3: Numerical differentiation 5: Analytic Problems 5.1: Nonlinear equations 5.2: Unconstrained optimization 5.3: Overdetermined nonlinear systems 5.4: Differential equations Initial value problems 5.5: Two point boundary value problems 5.6: Two-dimensional boundary value problems 5.7: Parameter estimation in differential equations 6: Special Functions 6.1: Elementary functions 6.2: Exponential Integral 6.3: Gamma function 6.4: Error function 6.5: Bessel functions of integer order 6.6: Bessel functions of real order 7: Interpolation and Approximation 7.1: Real data in one dimension I: Fast Fourier transforms II: Time series analysis Worked Examples Examples for chapter 1 procedures Examples for chapter 2 procedures Examples for chapter 3 procedures Examples for chapter 4 procedures Examples for chapter 5 procedures Examples for chapter 6 procedures Examples for chapter 7 procedures App. B: Procedures Description

rawTOC:
1: Elementary Procedures \\ 1.1: Real vector and matrix \\ Initialization \\ 1.2: Real vector and matrix \\ Duplication \\ 1.3: Real vector and matrix \\ Multiplication \\ 1.4: Real vector vector products \\ 1.5: Real matrix vector products \\ 1.6: Real matrix matrix products \\ 1.7: Real vector and matrix \\ Elimination \\ 1.8: Real vector and matrix \\ Interchanging \\ 1.9: Real vector and matrix \\ Rotation \\ 1.10: Real vector and matrix \\ Norms \\ 1.11: Real vector and matrix \\ Scaling \\ 1.12: Complex vector and matrix \\ Multiplication \\ 1.13: Complex vector and matrix \\ Scalar products \\ 1.14: Complex vector and matrix \\ Elimination \\ 1.15: Complex vector and matrix \\ Rotation \\ 1.16: Complex vector and matrix \\ Norms \\ 1.17: Complex vector and matrix \\ Scaling \\ 1.18: Complex monadic operations \\ 1.19: Complex dyadic operations \\ 1.20: Long integer arithmetic \\ 2: Algebraic Evaluations \\ 2.1: Evaluation of polynomials in Grunert form \\ 2.2: Evaluation of general orthogonal polynomials \\ 2.3: Evaluation of Chebyshev polynomials \\ 2.4: Evaluation of Fourier series \\ 2.5: Evaluation of continued fractions \\ 2.6: Transformation of polynomial representation \\ 2.7: Operations on orthogonal polynomials \\ 3: Linear Algebra \\ 3.1: Full real general matrices \\ 3.2: Real symmetric positive definite matrices \\ 3.3: General real symmetric matrices \\ 3.4: Real full rank overdetermined systems \\ 3.5: Other real matrix problems \\ 3.6: Real sparse nonsymmetric band matrices \\ 3.7: Real sparse nonsymmetric tridiagonal matrices \\ 3.8: Sparse symmetric positive definite band matrices \\ 3.9: Symmetric positive definite tridiagonal matrices \\ 3.10: Sparse real matrices \\ Iterative methods \\ 3.11: Similarity transformation \\ 3.12: Other transformations \\ 3.13: The (ordinary) eigenvalue problem \\ 3.14: The generalized eigenvalue problem \\ 3.15: Singular values \\ 3.16: Zeros of polynomials \\ 4: Analytic Evaluations \\ 4.1: Evaluation of an infinite series \\ 4.2: Quadrature \\ 4.3: Numerical differentiation \\ 5: Analytic Problems \\ 5.1: Nonlinear equations \\ 5.2: Unconstrained optimization \\ 5.3: Overdetermined nonlinear systems \\ 5.4: Differential equations \\ Initial value problems \\ 5.5: Two point boundary value problems \\ 5.6: Two-dimensional boundary value problems \\ 5.7: Parameter estimation in differential equations \\ 6: Special Functions \\ 6.1: Elementary functions \\ 6.2: Exponential Integral \\ 6.3: Gamma function \\ 6.4: Error function \\ 6.5: Bessel functions of integer order \\ 6.6: Bessel functions of real order \\ 7: Interpolation and Approximation \\ 7.1: Real data in one dimension \\ I: Fast Fourier transforms \\ II: Time series analysis \\ Worked Examples \\ Examples for chapter 1 procedures \\ Examples for chapter 2 procedures \\ Examples for chapter 3 procedures \\ Examples for chapter 4 procedures \\ Examples for chapter 5 procedures \\ Examples for chapter 6 procedures \\ Examples for chapter 7 procedures \\ App. B: Procedures Description

entry:
@Book{Lau:2004:NLJ,
  author =       "H. T. (Hang Tong) Lau",
  title =        "A Numerical Library in {Java} for Scientists and
                 Engineers",
  publisher =    pub-CHAPMAN-HALL-CRC,
  address =      pub-CHAPMAN-HALL-CRC:adr,
  pages =        "xxiii + 1063",
  year =         "2004",
  DOI =          "https://doi.org/10.1201/9780203507643",
  ISBN =         "1-58488-430-4",
  ISBN-13 =      "978-1-58488-430-9",
  LCCN =         "QA76.73.J38 L363 2004",
  bibdate =      "Fri Sep 26 14:28:47 MDT 2008",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 z3950.loc.gov:7090/Voyager",
  URL =          "http://www.loc.gov/catdir/enhancements/fy0646/2003055149-d.html",
  acknowledgement = ack-nhfb,
  subject =      "Java (Computer program language)",
  tableofcontents = "1: Elementary Procedures \\
                 1.1: Real vector and matrix \\
                 Initialization \\
                 1.2: Real vector and matrix \\
                 Duplication \\
                 1.3: Real vector and matrix \\
                 Multiplication \\
                 1.4: Real vector vector products \\
                 1.5: Real matrix vector products \\
                 1.6: Real matrix matrix products \\
                 1.7: Real vector and matrix \\
                 Elimination \\
                 1.8: Real vector and matrix \\
                 Interchanging \\
                 1.9: Real vector and matrix \\
                 Rotation \\
                 1.10: Real vector and matrix \\
                 Norms \\
                 1.11: Real vector and matrix \\
                 Scaling \\
                 1.12: Complex vector and matrix \\
                 Multiplication \\
                 1.13: Complex vector and matrix \\
                 Scalar products \\
                 1.14: Complex vector and matrix \\
                 Elimination \\
                 1.15: Complex vector and matrix \\
                 Rotation \\
                 1.16: Complex vector and matrix \\
                 Norms \\
                 1.17: Complex vector and matrix \\
                 Scaling \\
                 1.18: Complex monadic operations \\
                 1.19: Complex dyadic operations \\
                 1.20: Long integer arithmetic \\
                 2: Algebraic Evaluations \\
                 2.1: Evaluation of polynomials in Grunert form \\
                 2.2: Evaluation of general orthogonal polynomials \\
                 2.3: Evaluation of Chebyshev polynomials \\
                 2.4: Evaluation of Fourier series \\
                 2.5: Evaluation of continued fractions \\
                 2.6: Transformation of polynomial representation \\
                 2.7: Operations on orthogonal polynomials \\
                 3: Linear Algebra \\
                 3.1: Full real general matrices \\
                 3.2: Real symmetric positive definite matrices \\
                 3.3: General real symmetric matrices \\
                 3.4: Real full rank overdetermined systems \\
                 3.5: Other real matrix problems \\
                 3.6: Real sparse nonsymmetric band matrices \\
                 3.7: Real sparse nonsymmetric tridiagonal matrices \\
                 3.8: Sparse symmetric positive definite band matrices
                 \\
                 3.9: Symmetric positive definite tridiagonal matrices
                 \\
                 3.10: Sparse real matrices \\
                 Iterative methods \\
                 3.11: Similarity transformation \\
                 3.12: Other transformations \\
                 3.13: The (ordinary) eigenvalue problem \\
                 3.14: The generalized eigenvalue problem \\
                 3.15: Singular values \\
                 3.16: Zeros of polynomials \\
                 4: Analytic Evaluations \\
                 4.1: Evaluation of an infinite series \\
                 4.2: Quadrature \\
                 4.3: Numerical differentiation \\
                 5: Analytic Problems \\
                 5.1: Nonlinear equations \\
                 5.2: Unconstrained optimization \\
                 5.3: Overdetermined nonlinear systems \\
                 5.4: Differential equations \\
                 Initial value problems \\
                 5.5: Two point boundary value problems \\
                 5.6: Two-dimensional boundary value problems \\
                 5.7: Parameter estimation in differential equations \\
                 6: Special Functions \\
                 6.1: Elementary functions \\
                 6.2: Exponential Integral \\
                 6.3: Gamma function \\
                 6.4: Error function \\
                 6.5: Bessel functions of integer order \\
                 6.6: Bessel functions of real order \\
                 7: Interpolation and Approximation \\
                 7.1: Real data in one dimension \\
                 I: Fast Fourier transforms \\
                 II: Time series analysis \\
                 Worked Examples \\
                 Examples for chapter 1 procedures \\
                 Examples for chapter 2 procedures \\
                 Examples for chapter 3 procedures \\
                 Examples for chapter 4 procedures \\
                 Examples for chapter 5 procedures \\
                 Examples for chapter 6 procedures \\
                 Examples for chapter 7 procedures \\
                 App. B: Procedures Description",
}


bibtype: inproceedings
filename: elefunt.bib
label: Markstein:2004:SDS
author: Peter Markstein
title: Software Division and Square Root Using Goldschmidt's Algorithms
crossref: Frougny:2004:RCR
pages: 146--157
year: 2004
bibdate: Fri Nov 17 07:00:31 2023
bibtimestamp: 2023.11.17 07:00:31 ???
URL: http://www.informatik.uni-trier.de/Reports/TR-08-2004/rnc6_12_markstein.pdf
keywords: division; floating-point; Goldschmidt; square root
acknowledgement: ack-nhfb
rawauthor: Peter Markstein
rawtitle: Software Division and Square Root Using {Goldschmidt}'s Algorithms
abstract:
Goldschmidt's Algorithms for division and square root are often characterized as being useful for hardware implementation, and lacking self-correction. A reexamination of these algorithms show that there are good software counterparts that retain the speed advantage of Goldschmidt's Algorithm over the Newton--Raphson iteration. A final step is needed, however, to get the last bit rounded correctly.

rawabstract:
Goldschmidt's Algorithms for division and square root are often characterized as being useful for hardware implementation, and lacking self-correction. A reexamination of these algorithms show that there are good software counterparts that retain the speed advantage of Goldschmidt's Algorithm over the Newton--Raphson iteration. A final step is needed, however, to get the last bit rounded correctly.

entry:
@InProceedings{Markstein:2004:SDS,
  author =       "Peter Markstein",
  title =        "Software Division and Square Root Using
                 {Goldschmidt}'s Algorithms",
  crossref =     "Frougny:2004:RCR",
  pages =        "146--157",
  year =         "2004",
  bibdate =      "Fri Nov 17 07:00:31 2023",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://www.informatik.uni-trier.de/Reports/TR-08-2004/rnc6_12_markstein.pdf",
  abstract =     "Goldschmidt's Algorithms for division and square root
                 are often characterized as being useful for hardware
                 implementation, and lacking self-correction. A
                 reexamination of these algorithms show that there are
                 good software counterparts that retain the speed
                 advantage of Goldschmidt's Algorithm over the
                 Newton--Raphson iteration. A final step is needed,
                 however, to get the last bit rounded correctly.",
  acknowledgement = ack-nhfb,
  keywords =     "division; floating-point; Goldschmidt; square root",
}


bibtype: article
filename: elefunt.bib
label: Marsaglia:2004:END
author: George Marsaglia
title: Evaluating the Normal Distribution
journal: j-J-STAT-SOFT
volume: 11
number: 4
pages: 1--7
year: 2004
CODEN: JSSOBK
ISSN: 1548-7660 OR 15487660
bibdate: Sat Dec 04 09:18:40 2004
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2004.12.04 09:18:40 ???
URL: http://www.jstatsoft.org/counter.php?id=100&url=v11/i04/cphi.pdf&ct=1
fjournal: Journal of Statistical Software
acknowledgement: ack-nhfb
journalURL: http://www.jstatsoft.org/
rawauthor: George Marsaglia
rawtitle: Evaluating the Normal Distribution
remark:
This article exhibits accurate, compact, and fast algorithms for computation of the normal distribution function and the complementary normal distribution, which have a simple relation to the error function and the complementary error function. They appear to be improvements on almost all previously-published algorithms for these functions. However, closer study shows that the complementary normal distribution function has an unchecked out-of-bounds array access for $ |x| geq 17 $, and its Taylor series sum has poor convergence because the tabulated intervals are twice too wide. The Taylor series sum for the normal distribution function is expanded around $ x = 0 $, and thus has poor convergence for large $ |x| $. Neither function takes into account the accuracy loss when the computed result is the larger of the two (their sum is one, and their range is $ [ - infty , + infty ] $ ), although the text discusses the problem. The article also discusses the historical origin of the term ``error function'', tracing it to J. W. Glaisher in 1871.

entry:
@Article{Marsaglia:2004:END,
  author =       "George Marsaglia",
  title =        "Evaluating the Normal Distribution",
  journal =      j-J-STAT-SOFT,
  volume =       "11",
  number =       "4",
  pages =        "1--7",
  month =        "????",
  year =         "2004",
  CODEN =        "JSSOBK",
  ISSN =         "1548-7660",
  bibdate =      "Sat Dec 04 09:18:40 2004",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.jstatsoft.org/counter.php?id=100&url=v11/i04/cphi.pdf&ct=1",
  accepted =     "2004-07-18",
  acknowledgement = ack-nhfb,
  fjournal =     "Journal of Statistical Software",
  journal-URL =  "http://www.jstatsoft.org/",
  remark =       "This article exhibits accurate, compact, and fast
                 algorithms for computation of the normal distribution
                 function and the complementary normal distribution,
                 which have a simple relation to the error function and
                 the complementary error function. They appear to be
                 improvements on almost all previously-published
                 algorithms for these functions. However, closer study
                 shows that the complementary normal distribution
                 function has an unchecked out-of-bounds array access
                 for $ |x| \geq 17 $, and its Taylor series sum has poor
                 convergence because the tabulated intervals are twice
                 too wide. The Taylor series sum for the normal
                 distribution function is expanded around $ x = 0 $, and
                 thus has poor convergence for large $ |x| $. Neither
                 function takes into account the accuracy loss when the
                 computed result is the larger of the two (their sum is
                 one, and their range is $ [ - \infty, + \infty] $ ),
                 although the text discusses the problem. The article
                 also discusses the historical origin of the term
                 ``error function'', tracing it to J. W. Glaisher in
                 1871.",
  submitted =    "2004-06-05",
}


bibtype: article
filename: elefunt.bib
label: Mathar:2004:NRI
author: Richard J. Mathar
title: Numerical Representations of the Incomplete Gamma Function of Complex-Valued Argument
journal: j-NUMER-ALGORITHMS
volume: 36
number: 3
pages: 247--264
month: jul
monthnumber: 7
year: 2004
CODEN: NUALEG
ISSN: 1017-1398 (print), 1572-9265 (electronic) OR 10171398 (print), 15729265 (electronic)
ISSNL: 1017-1398 OR 10171398
bibdate: Mon Dec 6 07:00:32 MST 2004
bibsource: http://www.kluweronline.com/issn/1017-1398; https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2004.12.06 07:00:32 MST
URL: http://ipsapp009.kluweronline.com/IPS/content/ext/x/J/5058/I/57/A/4/abstract.htm
fjournal: Numerical Algorithms
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/11075
rawauthor: Richard J. Mathar
rawtitle: Numerical Representations of the Incomplete Gamma Function of Complex-Valued Argument
entry:
@Article{Mathar:2004:NRI,
  author =       "Richard J. Mathar",
  title =        "Numerical Representations of the Incomplete Gamma
                 Function of Complex-Valued Argument",
  journal =      j-NUMER-ALGORITHMS,
  volume =       "36",
  number =       "3",
  pages =        "247--264",
  month =        jul,
  year =         "2004",
  CODEN =        "NUALEG",
  ISSN =         "1017-1398 (print), 1572-9265 (electronic)",
  ISSN-L =       "1017-1398",
  bibdate =      "Mon Dec 6 07:00:32 MST 2004",
  bibsource =    "http://www.kluweronline.com/issn/1017-1398;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://ipsapp009.kluweronline.com/IPS/content/ext/x/J/5058/I/57/A/4/abstract.htm",
  acknowledgement = ack-nhfb,
  fjournal =     "Numerical Algorithms",
  journal-URL =  "http://link.springer.com/journal/11075",
}


bibtype: misc
filename: elefunt.bib
label: Miller:2004:AMF
author: Alan Miller
title: Alan Miller's Fortran Software
pages: 247--264
day: 4
month: feb
monthnumber: 2
year: 2004
bibdate: Tue Jun 13 12:03:37 2023
bibtimestamp: 2023.06.13 12:03:37 ???
note: From the Web site: All code written by Alan Miller is released into the public domain.
URL: https://jblevins.org/mirror/amiller/
acknowledgement: ack-nhfb
howpublished: Web site
rawauthor: Alan Miller
rawnote: From the Web site: All code written by Alan Miller is released into the public domain.
rawtitle: {Alan Miller}'s {Fortran} Software
bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fortran3.bib

remark:
The Web site contains a section ``Code converted from the Naval Surface Warfare Center Math. Library'' with links to individual Fortran 90 source files.

entry:
@Misc{Miller:2004:AMF,
  author =       "Alan Miller",
  title =        "{Alan Miller}'s {Fortran} Software",
  howpublished = "Web site",
  day =          "4",
  month =        feb,
  year =         "2004",
  bibdate =      "Tue Jun 13 12:03:37 2023",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fortran3.bib",
  note =         "From the Web site: All code written by Alan Miller is
                 released into the public domain.",
  URL =          "https://jblevins.org/mirror/amiller/",
  acknowledgement = ack-nhfb,
  remark =       "The Web site contains a section ``Code converted from
                 the Naval Surface Warfare Center Math. Library'' with
                 links to individual Fortran 90 source files.",
}


bibtype: article
filename: elefunt.bib
label: Moore:2004:PSW
author: Ian C. Moore and Michael Cada
title: Prolate spheroidal wave functions, an introduction to the Slepian series and its properties
journal: j-APPL-COMPUT-HARMON-ANAL
volume: 16
number: 3
pages: 208--230
month: may
monthnumber: 5
year: 2004
DOI: https://doi.org/10.1016/j.acha.2004.03.004
ISSN: 1063-5203 (print), 1096-603x (electronic) OR 10635203 (print), 1096603x (electronic)
ISSNL: 1063-5203 OR 10635203
bibdate: Sun Oct 31 09:58:00 2010
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2010.10.31 09:58:00 ???
keywords: Interpolation; Orthogonal expansion; Prolate spheroidal wave function
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/10635203
rawauthor: Ian C. Moore and Michael Cada
rawtitle: Prolate spheroidal wave functions, an introduction to the {Slepian} series and its properties
abstract:
For decades mathematicians, physicists, and engineers have relied on various orthogonal expansions such as Fourier, Legendre, and Chebyschev to solve a variety of problems. In this paper we exploit the orthogonal properties of prolate spheroidal wave functions (PSWF) in the form of a new orthogonal expansion which we have named the Slepian series. We empirically show that the Slepian series is potentially optimal over more conventional orthogonal expansions for discontinuous functions such as the square wave among others. With regards to interpolation, we explore the connections the Slepian series has to the Shannon sampling theorem. By utilizing Euler's equation, a relationship between the even and odd ordered PSWFs is investigated. We also establish several other key advantages the Slepian series has such as the presence of a free tunable bandwidth parameter.

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

rawabstract:
For decades mathematicians, physicists, and engineers have relied on various orthogonal expansions such as Fourier, Legendre, and Chebyschev to solve a variety of problems. In this paper we exploit the orthogonal properties of prolate spheroidal wave functions (PSWF) in the form of a new orthogonal expansion which we have named the Slepian series. We empirically show that the Slepian series is potentially optimal over more conventional orthogonal expansions for discontinuous functions such as the square wave among others. With regards to interpolation, we explore the connections the Slepian series has to the Shannon sampling theorem. By utilizing Euler's equation, a relationship between the even and odd ordered PSWFs is investigated. We also establish several other key advantages the Slepian series has such as the presence of a free tunable bandwidth parameter.

entry:
@Article{Moore:2004:PSW,
  author =       "Ian C. Moore and Michael Cada",
  title =        "Prolate spheroidal wave functions, an introduction to
                 the {Slepian} series and its properties",
  journal =      j-APPL-COMPUT-HARMON-ANAL,
  volume =       "16",
  number =       "3",
  pages =        "208--230",
  month =        may,
  year =         "2004",
  DOI =          "https://doi.org/10.1016/j.acha.2004.03.004",
  ISSN =         "1063-5203 (print), 1096-603x (electronic)",
  ISSN-L =       "1063-5203",
  bibdate =      "Sun Oct 31 09:58:00 2010",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  abstract =     "For decades mathematicians, physicists, and engineers
                 have relied on various orthogonal expansions such as
                 Fourier, Legendre, and Chebyschev to solve a variety of
                 problems. In this paper we exploit the orthogonal
                 properties of prolate spheroidal wave functions (PSWF)
                 in the form of a new orthogonal expansion which we have
                 named the Slepian series. We empirically show that the
                 Slepian series is potentially optimal over more
                 conventional orthogonal expansions for discontinuous
                 functions such as the square wave among others. With
                 regards to interpolation, we explore the connections
                 the Slepian series has to the Shannon sampling theorem.
                 By utilizing Euler's equation, a relationship between
                 the even and odd ordered PSWFs is investigated. We also
                 establish several other key advantages the Slepian
                 series has such as the presence of a free tunable
                 bandwidth parameter.",
  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 =     "Interpolation; Orthogonal expansion; Prolate
                 spheroidal wave function",
}


bibtype: article
filename: elefunt.bib
label: Muller:2004:CSR
author: Siguna Muller
title: On the Computation of Square Roots in Finite Fields
journal: j-DESIGNS-CODES-CRYPTOGR
volume: 31
number: 3
pages: 301--312
month: mar
monthnumber: 3
year: 2004
CODEN: DCCREC
ISSN: 0925-1022 (print), 1573-7586 (electronic) OR 09251022 (print), 15737586 (electronic)
ISSNL: 0925-1022 OR 09251022
bibdate: Tue Aug 3 16:38:18 MDT 2004
bibtimestamp: 2004.08.03 16:38:18 MDT
URL: http://ipsapp008.kluweronline.com/IPS/content/ext/x/J/4630/I/61/A/8/abstract.htm
fjournal: Designs, codes, and cryptography
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/10623
rawauthor: Siguna M{\"u}ller
rawtitle: On the Computation of Square Roots in Finite Fields
bibsource:
http://www.wkap.nl/jrnltoc.htm/0925-1022; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib

entry:
@Article{Muller:2004:CSR,
  author =       "Siguna M{\"u}ller",
  title =        "On the Computation of Square Roots in Finite Fields",
  journal =      j-DESIGNS-CODES-CRYPTOGR,
  volume =       "31",
  number =       "3",
  pages =        "301--312",
  month =        mar,
  year =         "2004",
  CODEN =        "DCCREC",
  ISSN =         "0925-1022 (print), 1573-7586 (electronic)",
  ISSN-L =       "0925-1022",
  bibdate =      "Tue Aug 3 16:38:18 MDT 2004",
  bibsource =    "http://www.wkap.nl/jrnltoc.htm/0925-1022;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://ipsapp008.kluweronline.com/IPS/content/ext/x/J/4630/I/61/A/8/abstract.htm",
  acknowledgement = ack-nhfb,
  fjournal =     "Designs, codes, and cryptography",
  journal-URL =  "http://link.springer.com/journal/10623",
}


bibtype: article
filename: elefunt.bib
label: Nagel:2004:CEG
author: Bengt Nagel
title: Confluence expansions of the generalized hypergeometric function
journal: j-J-MATH-PHYS
volume: 45
number: 1
pages: 495--508
month: jan
monthnumber: 1
year: 2004
CODEN: JMAPAQ
DOI: https://doi.org/10.1063/1.1629777
ISSNL: 0022-2488 OR 00222488
bibdate: Tue Oct 25 18:16:52 MDT 2011
bibtimestamp: 2011.10.25 18:16:52 MDT
URL: http://jmp.aip.org/resource/1/jmapaq/v45/i1/p495_s1
fjournal: Journal of Mathematical Physics
acknowledgement: ack-nhfb
journalURL: http://jmp.aip.org/
onlinedate: 19 December 2003
rawauthor: Bengt Nagel
rawtitle: Confluence expansions of the generalized hypergeometric function
entry:
@Article{Nagel:2004:CEG,
  author =       "Bengt Nagel",
  title =        "Confluence expansions of the generalized
                 hypergeometric function",
  journal =      j-J-MATH-PHYS,
  volume =       "45",
  number =       "1",
  pages =        "495--508",
  month =        jan,
  year =         "2004",
  CODEN =        "JMAPAQ",
  DOI =          "https://doi.org/10.1063/1.1629777",
  ISSN =         "0022-2488 (print), 1089-7658 (electronic), 1527-2427",
  ISSN-L =       "0022-2488",
  bibdate =      "Tue Oct 25 18:16:52 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/jmathphys2000.bib",
  URL =          "http://jmp.aip.org/resource/1/jmapaq/v45/i1/p495_s1",
  acknowledgement = ack-nhfb,
  fjournal =     "Journal of Mathematical Physics",
  journal-URL =  "http://jmp.aip.org/",
  onlinedate =   "19 December 2003",
  pagecount =    "14",
}


bibtype: inproceedings
filename: elefunt.bib
label: Ortiz:2004:SPI
author: I. Ortiz and M. Jimenez
booktitle: MWSCAS '04. The 2004 47th Midwest Symposium on Circuits and Systems. 25--28 July 2004
title: Scalable pipeline insertion in floating-point division and square root units
volume: 2
publisher: pub-IEEE
address: pub-IEEE:adr
pages: II-225--II-228
year: 2004
bibdate: Sat Jul 16 15:28:14 MDT 2005
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2005.07.16 15:28:14 MDT
acknowledgement: ack-nhfb
rawauthor: I. Ortiz and M. Jimenez
rawbooktitle: {MWSCAS '04. The 2004 47th Midwest Symposium on Circuits and Systems. 25--28 July 2004}
rawtitle: Scalable pipeline insertion in floating-point division and square root units
abstract:
Division and square root are important operations in a number of data processing algorithms. They are inherently time consuming operations and can require a significant amount of resources when implemented in hardware. This work reports the development of scalable, floating-point (FP) division and square root operators with adjustable precision, range, and pipeline granularity. An algorithm for pipeline insertion was used for both operators, enabling speeds up to 204MFLOPS when implemented on a Xilinx Virtex II FPGA.

rawabstract:
Division and square root are important operations in a number of data processing algorithms. They are inherently time consuming operations and can require a significant amount of resources when implemented in hardware. This work reports the development of scalable, floating-point (FP) division and square root operators with adjustable precision, range, and pipeline granularity. An algorithm for pipeline insertion was used for both operators, enabling speeds up to 204MFLOPS when implemented on a Xilinx Virtex II FPGA.

entry:
@InProceedings{Ortiz:2004:SPI,
  author =       "I. Ortiz and M. Jimenez",
  booktitle =    "{MWSCAS '04. The 2004 47th Midwest Symposium on
                 Circuits and Systems. 25--28 July 2004}",
  title =        "Scalable pipeline insertion in floating-point division
                 and square root units",
  volume =       "2",
  publisher =    pub-IEEE,
  address =      pub-IEEE:adr,
  pages =        "II-225--II-228",
  year =         "2004",
  CODEN =        "????",
  ISSN =         "????",
  bibdate =      "Sat Jul 16 15:28:14 MDT 2005",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  abstract =     "Division and square root are important operations in a
                 number of data processing algorithms. They are
                 inherently time consuming operations and can require a
                 significant amount of resources when implemented in
                 hardware. This work reports the development of
                 scalable, floating-point (FP) division and square root
                 operators with adjustable precision, range, and
                 pipeline granularity. An algorithm for pipeline
                 insertion was used for both operators, enabling speeds
                 up to 204MFLOPS when implemented on a Xilinx Virtex II
                 FPGA.",
  acknowledgement = ack-nhfb,
  summary =      "Division and square root are important operations in a
                 number of data processing algorithms. They are
                 inherently time consuming operations and can require a
                 significant amount of resources when implemented in
                 hardware. This work reports the \ldots{}",
}


bibtype: article
filename: elefunt.bib
label: Petkovic:2004:GCS
author: M. S. Petkovic and L. Rancic
title: On the guaranteed convergence of the square-root iteration method
journal: j-J-COMPUT-APPL-MATH
volume: 170
number: 1
pages: 169--179
day: 1
month: sep
monthnumber: 9
year: 2004
CODEN: JCAMDI
ISSN: 0377-0427 (print), 1879-1778 (electronic) OR 03770427 (print), 18791778 (electronic)
ISSNL: 0377-0427 OR 03770427
bibdate: Sat Feb 25 13:00:00 MST 2017
bibtimestamp: 2017.02.25 13:00:00 MST
URL: http://www.sciencedirect.com/science/article/pii/S0377042704000184
fjournal: Journal of Computational and Applied Mathematics
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/03770427
rawauthor: M. S. Petkovi{\'c} and L. Ranci{\'c}
rawtitle: On the guaranteed convergence of the square-root iteration method
entry:
@Article{Petkovic:2004:GCS,
  author =       "M. S. Petkovi{\'c} and L. Ranci{\'c}",
  title =        "On the guaranteed convergence of the square-root
                 iteration method",
  journal =      j-J-COMPUT-APPL-MATH,
  volume =       "170",
  number =       "1",
  pages =        "169--179",
  day =          "1",
  month =        sep,
  year =         "2004",
  CODEN =        "JCAMDI",
  ISSN =         "0377-0427 (print), 1879-1778 (electronic)",
  ISSN-L =       "0377-0427",
  bibdate =      "Sat Feb 25 13:00:00 MST 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2000.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0377042704000184",
  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: Pineiro:2004:AAL
author: J. A. Pineiro and M. D. Ercegovac and J. D. Bruguera
title: Algorithm and Architecture for Logarithm, Exponential and Powering Computation
journal: j-IEEE-TRANS-COMPUT
volume: 53
number: 9
pages: 1085--1096
year: 2004
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2004.53
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Fri Jun 24 10:05:48 2005
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2005.06.24 10:05:48 ???
URL: http://www.ac.usc.es/arquivos/articulos/2004/gac2004-j05.ps
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: J. A. Pi{\~n}eiro and M. D. Ercegovac and J. D. Bruguera
rawtitle: Algorithm and Architecture for Logarithm, Exponential and Powering Computation
entry:
@Article{Pineiro:2004:AAL,
  author =       "J. A. Pi{\~n}eiro and M. D. Ercegovac and J. D.
                 Bruguera",
  title =        "Algorithm and Architecture for Logarithm, Exponential
                 and Powering Computation",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "53",
  number =       "9",
  pages =        "1085--1096",
  year =         "2004",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2004.53",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Fri Jun 24 10:05:48 2005",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.ac.usc.es/arquivos/articulos/2004/gac2004-j05.ps",
  acknowledgement = ack-nhfb,
  ajournal =     "IEEE Trans. Comput.",
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
}


bibtype: phdthesis
filename: elefunt.bib
label: Pugh:2004:ALG
author: Glendon Ralpha Pugh
title: An Analysis of the Lanczos Gamma Approximation
type: Ph.D. thesis
school: Department of Mathematics, University of British Columbia
address: Vancouver, BC, Canada
pages: viii + 154
year: 2004
ISBN: 0-612-99536-4 OR 0612995364
ISBN13: 978-0-612-99536-9 OR 9780612995369
LCCN: AW5 .B7 2005-995364
bibdate: Mon Nov 24 20:55:30 2008
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2008.11.24 20:55:30 ???
acknowledgement: ack-nhfb
rawauthor: Glendon Ralpha Pugh
rawtitle: An Analysis of the {Lanczos} Gamma Approximation
entry:
@PhdThesis{Pugh:2004:ALG,
  author =       "Glendon Ralpha Pugh",
  title =        "An Analysis of the {Lanczos} Gamma Approximation",
  type =         "Ph.D. thesis",
  school =       "Department of Mathematics, University of British
                 Columbia",
  address =      "Vancouver, BC, Canada",
  pages =        "viii + 154",
  month =        "????",
  year =         "2004",
  ISBN =         "0-612-99536-4",
  ISBN-13 =      "978-0-612-99536-9",
  LCCN =         "AW5 .B7 2005-995364",
  bibdate =      "Mon Nov 24 20:55:30 2008",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
}


bibtype: book
filename: elefunt.bib
label: Raade:2004:MHS
author: Lennart Raa de and Bertil Westergren
title: Mathematics Handbook for Science and Engineering
publisher: pub-SV
address: pub-SV:adr
edition: Fifth
pages: 562
year: 2004
ISBN: 3-540-21141-1 (hardcover) OR 3540211411 (hardcover)
ISBN13: 978-3-540-21141-9 (hardcover) OR 9783540211419 (hardcover)
LCCN: QA41 .R34 2004
bibdate: Sat May 15 09:15:39 MDT 2010
bibtimestamp: 2010.05.15 09:15:39 MDT
subject: mathematics; formulae; tables; handbooks, manuals, etc.
acknowledgement: ack-nhfb
rawauthor: Lennart R{\aa}de and Bertil Westergren
rawtitle: Mathematics Handbook for Science and Engineering
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

URL:
http://www.loc.gov/catdir/enhancements/fy0818/2006286513-d.html; http://www.loc.gov/catdir/toc/fy0704/2006286513.html

TOC:
1. Fundamentals Discrete Mathematics / 9 1.1 Logic / 9 1.2 Set Theory / 14 1.3 Binary Relations and Functions / 17 1.4 Algebraic Structures / 21 1.5 Graph Theory / 33 1.6 Codes / 37 2: Algebra / 43 2.1 Basic Algebra of Real Numbers / 43 2.2 Number Theory / 49 2.3 Complex Numbers / 61 2.4 Algebraic Equations / 63 3: Geometry and Trigonometry / 66 3.1 Plane Figures / 66 3.2 Solids / 71 3.3 Spherical Trigonometry / 75 3.4 Geometrical Vectors / 77 3.5 Plane Analytic Geometry / 79 3.6 Analytic Geometry in Space / 83 3.7 Fractals / 87 4: Linear Algebra / 90 4.1 Matrices / 90 4.2 Determinants / 93 4.3 Systems of Linear Equations / 95 4.4 Linear Coordinate Transformations / 97 4.5 Eigenvalues. Diagonalization / 98 4.6 Quadratic Forms / 103 4.7 Linear Spaces / 106 4.8 Linear Mappings / 108 4.9 Tensors / 114 4.10 Complex matrices / 114 5: The Elementary Functions / 118 5.1 A Survey of the Elementary Functions / 118 5.2 Polynomials and Rational Functions / 119 5.3 Logarithmic, Exponential, Power and Hyperbolic Functions / 121 5.4 Trigonometric and Inverse Trigonometric Functions / 125 6: Differential Calculus (one variable) / 132 6.1 Some Basic Concepts / 132 6.2 Limits and Continuity / 133 6.3 Derivatives / 136 6.4 Monotonicity. Extremes of Functions / 139 7: Integral Calculus / 141 7.1 Indefinite Integrals / 141 7.2 Definite Integrals / 146 7.3 Applications of Differential and Integral Calculus / 148 7.4 Table of Indefinite Integral / 153 7.5 Tables of Definite Integrals / 178 8: Sequences and Series / 183 8.1 Sequences of Numbers / 183 8.2 Sequences of Functions / 184 8.3 Series of Constant Terms / 185 8.4 Series of Functions / 187 8.5 Taylor Series / 189 8.6 Special Sums and Series / 192 9: Ordinary Differential Equations (ODE) / 200 9.1 Differential Equations of the First Order / 200 9.2 Differential Equations of the Second Order / 202 9.3 Linear Differential Equations / 205 9.4 Autonomous systems / 2313 9.5 General Concepts and Results / 216 9.6 Linear Difference Equations / 218 10: Multidimensional Calculus / 221 10.1 The Space Rn / 221 10.2 Surfaces. Tangent Planes / 222 10.3 Limits and Continuity / 223 10.4 Partial Derivatives / 224 10.5 Extremes of Functions / 227 10.6 Functions $f: R^n to R^m (R^n to R^n)$ / 229 10.7 Double Integrals / 231 10.8 Triple Integrals / 234 10.9 Partial Differential Equations / 239 11: Vector Analysis / 246 11.1 Curves / 246 11.2 Vector Fields / 248 11.3 Line Integrals / 253 11.4 Surface Integrals / 256 12: Orthogonal Series and Special Functions / 259 12.1 Orthogonal Systems / 259 12.2 Orthogonal Polynomials / 263 12.3 Bernoulli and Euler Polynomials / 269 12.4 Bessel Functions / 270 12.5 Functions Defined by Transcendental Integrals / 287 12.6 Step and Impulse Functions / 297 12.7 Functional Analysis / 298 12.8 Lebesgue Integrals / 303 12.9 Generalized functions (Distributions) / 308 13: Transforms / 310 13.1 Trigonometric Fourier Series / 310 13.2 Fourier Transforms / 315 13.3 Discrete Fourier Transforms / 325 13.4 The $z$-transform / 327 13.5 Laplace Transforms / 330 13.6 Dynamical Systems (Filters) / 338 13.7 Hankel and Hilbert transforms / 341 13.8 Wavelets / 344 14: Complex Analysis / 349 14.1 Functions of a Complex Variable / 349 14.2 Complex Integration / 352 14.3 Power Series Expansions / 354 14.4 Zeros and Singularities / 355 14.5 Conformal Mappings / 356 15: Optimization / 365 15.1 Calculus of Variations / 365 15.2 Linear Optimization / 371 15.3 Integer and Combinatorial Optimization / 379 15.4 Nonlinear Optimization / 383 15.5 Dynamic Optimization / 389 16: Numerical Analysis / 391 16.1 Approximations and Errors / 391 16.2 Numerical Solution of Equations / 392 16.3 Perturbation analysis / 397 16.4 Interpolation / 398 16.5 Numerical Integration and Differentiation / 404 16.6 Numerical Solutions of Differential Equations / 412 16.7 Numerical summation / 421 17: Probability Theory / 424 17.1 Basic Probability Theory / 424 17.2 Probability Distributions / 434 17.3 Stochastic Processes / 439 17.4 Algorithms for Calculation of Probability Distributions / 443 17.5 Simulation / 445 17.6 Queueing Systems / 449 17.7 Reliability / 452 17.8 Tables / 459 18: Statistics / 479 18.1 Descriptive Statistics / 479 18.2 Point Estimation / 488 18.3 Confidence Intervals / 491 18.4 Tables for Confidence Intervals / 495 18.5 Tests of Significance / 501 18.6 Linear Models / 507 18.7 Distribution-free Methods / 512 18.8 Statistical Quality Control / 518 18.9 Factorial Experiments / 522 18.10 Analysis of life time (failure time) data / 525 18.11 Statistical glossary / 526 19: Miscellaneous / 530

rawTOC:
1. Fundamentals \\ Discrete Mathematics / 9 \\ 1.1 Logic / 9 \\ 1.2 Set Theory / 14 \\ 1.3 Binary Relations and Functions / 17 \\ 1.4 Algebraic Structures / 21 \\ 1.5 Graph Theory / 33 \\ 1.6 Codes / 37 \\ 2: Algebra / 43 \\ 2.1 Basic Algebra of Real Numbers / 43 \\ 2.2 Number Theory / 49 \\ 2.3 Complex Numbers / 61 \\ 2.4 Algebraic Equations / 63 \\ 3: Geometry and Trigonometry / 66 \\ 3.1 Plane Figures / 66 \\ 3.2 Solids / 71 \\ 3.3 Spherical Trigonometry / 75 \\ 3.4 Geometrical Vectors / 77 \\ 3.5 Plane Analytic Geometry / 79 \\ 3.6 Analytic Geometry in Space / 83 \\ 3.7 Fractals / 87 \\ 4: Linear Algebra / 90 \\ 4.1 Matrices / 90 \\ 4.2 Determinants / 93 \\ 4.3 Systems of Linear Equations / 95 \\ 4.4 Linear Coordinate Transformations / 97 \\ 4.5 Eigenvalues. Diagonalization / 98 \\ 4.6 Quadratic Forms / 103 \\ 4.7 Linear Spaces / 106 \\ 4.8 Linear Mappings / 108 \\ 4.9 Tensors / 114 \\ 4.10 Complex matrices / 114 \\ 5: The Elementary Functions / 118 \\ 5.1 A Survey of the Elementary Functions / 118 \\ 5.2 Polynomials and Rational Functions / 119 \\ 5.3 Logarithmic, Exponential, Power and Hyperbolic Functions / 121 \\ 5.4 Trigonometric and Inverse Trigonometric Functions / 125 \\ 6: Differential Calculus (one variable) / 132 \\ 6.1 Some Basic Concepts / 132 \\ 6.2 Limits and Continuity / 133 \\ 6.3 Derivatives / 136 \\ 6.4 Monotonicity. Extremes of Functions / 139 \\ 7: Integral Calculus / 141 \\ 7.1 Indefinite Integrals / 141 \\ 7.2 Definite Integrals / 146 \\ 7.3 Applications of Differential and Integral Calculus / 148 \\ 7.4 Table of Indefinite Integral / 153 \\ 7.5 Tables of Definite Integrals / 178 \\ 8: Sequences and Series / 183 \\ 8.1 Sequences of Numbers / 183 \\ 8.2 Sequences of Functions / 184 \\ 8.3 Series of Constant Terms / 185 \\ 8.4 Series of Functions / 187 \\ 8.5 Taylor Series / 189 \\ 8.6 Special Sums and Series / 192 \\ 9: Ordinary Differential Equations (ODE) / 200 \\ 9.1 Differential Equations of the First Order / 200 \\ 9.2 Differential Equations of the Second Order / 202 \\ 9.3 Linear Differential Equations / 205 \\ 9.4 Autonomous systems / 2313 \\ 9.5 General Concepts and Results / 216 \\ 9.6 Linear Difference Equations / 218 \\ 10: Multidimensional Calculus / 221 \\ 10.1 The Space Rn / 221 \\ 10.2 Surfaces. Tangent Planes / 222 \\ 10.3 Limits and Continuity / 223 \\ 10.4 Partial Derivatives / 224 \\ 10.5 Extremes of Functions / 227 \\ 10.6 Functions $f: R^n \to R^m (R^n \to R^n)$ / 229 \\ 10.7 Double Integrals / 231 \\ 10.8 Triple Integrals / 234 \\ 10.9 Partial Differential Equations / 239 \\ 11: Vector Analysis / 246 \\ 11.1 Curves / 246 \\ 11.2 Vector Fields / 248 \\ 11.3 Line Integrals / 253 \\ 11.4 Surface Integrals / 256 \\ 12: Orthogonal Series and Special Functions / 259 \\ 12.1 Orthogonal Systems / 259 \\ 12.2 Orthogonal Polynomials / 263 \\ 12.3 Bernoulli and Euler Polynomials / 269 \\ 12.4 Bessel Functions / 270 \\ 12.5 Functions Defined by Transcendental Integrals / 287 \\ 12.6 Step and Impulse Functions / 297 \\ 12.7 Functional Analysis / 298 \\ 12.8 Lebesgue Integrals / 303 \\ 12.9 Generalized functions (Distributions) / 308 \\ 13: Transforms / 310 \\ 13.1 Trigonometric Fourier Series / 310 \\ 13.2 Fourier Transforms / 315 \\ 13.3 Discrete Fourier Transforms / 325 \\ 13.4 The $z$-transform / 327 \\ 13.5 Laplace Transforms / 330 \\ 13.6 Dynamical Systems (Filters) / 338 \\ 13.7 Hankel and Hilbert transforms / 341 \\ 13.8 Wavelets / 344 \\ 14: Complex Analysis / 349 \\ 14.1 Functions of a Complex Variable / 349 \\ 14.2 Complex Integration / 352 \\ 14.3 Power Series Expansions / 354 \\ 14.4 Zeros and Singularities / 355 \\ 14.5 Conformal Mappings / 356 \\ 15: Optimization / 365 \\ 15.1 Calculus of Variations / 365 \\ 15.2 Linear Optimization / 371 \\ 15.3 Integer and Combinatorial Optimization / 379 \\ 15.4 Nonlinear Optimization / 383 \\ 15.5 Dynamic Optimization / 389 \\ 16: Numerical Analysis / 391 \\ 16.1 Approximations and Errors / 391 \\ 16.2 Numerical Solution of Equations / 392 \\ 16.3 Perturbation analysis / 397 \\ 16.4 Interpolation / 398 \\ 16.5 Numerical Integration and Differentiation / 404 \\ 16.6 Numerical Solutions of Differential Equations / 412 \\ 16.7 Numerical summation / 421 \\ 17: Probability Theory / 424 \\ 17.1 Basic Probability Theory / 424 \\ 17.2 Probability Distributions / 434 \\ 17.3 Stochastic Processes / 439 \\ 17.4 Algorithms for Calculation of Probability Distributions / 443 \\ 17.5 Simulation / 445 \\ 17.6 Queueing Systems / 449 \\ 17.7 Reliability / 452 \\ 17.8 Tables / 459 \\ 18: Statistics / 479 \\ 18.1 Descriptive Statistics / 479 \\ 18.2 Point Estimation / 488 \\ 18.3 Confidence Intervals / 491 \\ 18.4 Tables for Confidence Intervals / 495 \\ 18.5 Tests of Significance / 501 \\ 18.6 Linear Models / 507 \\ 18.7 Distribution-free Methods / 512 \\ 18.8 Statistical Quality Control / 518 \\ 18.9 Factorial Experiments / 522 \\ 18.10 Analysis of life time (failure time) data / 525 \\ 18.11 Statistical glossary / 526 \\ 19: Miscellaneous / 530

entry:
@Book{Raade:2004:MHS,
  author =       "Lennart R{\aa}de and Bertil Westergren",
  title =        "Mathematics Handbook for Science and Engineering",
  publisher =    pub-SV,
  address =      pub-SV:adr,
  edition =      "Fifth",
  pages =        "562",
  year =         "2004",
  ISBN =         "3-540-21141-1 (hardcover)",
  ISBN-13 =      "978-3-540-21141-9 (hardcover)",
  LCCN =         "QA41 .R34 2004",
  bibdate =      "Sat May 15 09:15:39 MDT 2010",
  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",
  URL =          "http://www.loc.gov/catdir/enhancements/fy0818/2006286513-d.html;
                 http://www.loc.gov/catdir/toc/fy0704/2006286513.html",
  acknowledgement = ack-nhfb,
  libnote =      "Not in my library.",
  subject =      "mathematics; formulae; tables; handbooks, manuals,
                 etc.",
  tableofcontents = "1. Fundamentals \\
                 Discrete Mathematics / 9 \\
                 1.1 Logic / 9 \\
                 1.2 Set Theory / 14 \\
                 1.3 Binary Relations and Functions / 17 \\
                 1.4 Algebraic Structures / 21 \\
                 1.5 Graph Theory / 33 \\
                 1.6 Codes / 37 \\
                 2: Algebra / 43 \\
                 2.1 Basic Algebra of Real Numbers / 43 \\
                 2.2 Number Theory / 49 \\
                 2.3 Complex Numbers / 61 \\
                 2.4 Algebraic Equations / 63 \\
                 3: Geometry and Trigonometry / 66 \\
                 3.1 Plane Figures / 66 \\
                 3.2 Solids / 71 \\
                 3.3 Spherical Trigonometry / 75 \\
                 3.4 Geometrical Vectors / 77 \\
                 3.5 Plane Analytic Geometry / 79 \\
                 3.6 Analytic Geometry in Space / 83 \\
                 3.7 Fractals / 87 \\
                 4: Linear Algebra / 90 \\
                 4.1 Matrices / 90 \\
                 4.2 Determinants / 93 \\
                 4.3 Systems of Linear Equations / 95 \\
                 4.4 Linear Coordinate Transformations / 97 \\
                 4.5 Eigenvalues. Diagonalization / 98 \\
                 4.6 Quadratic Forms / 103 \\
                 4.7 Linear Spaces / 106 \\
                 4.8 Linear Mappings / 108 \\
                 4.9 Tensors / 114 \\
                 4.10 Complex matrices / 114 \\
                 5: The Elementary Functions / 118 \\
                 5.1 A Survey of the Elementary Functions / 118 \\
                 5.2 Polynomials and Rational Functions / 119 \\
                 5.3 Logarithmic, Exponential, Power and Hyperbolic
                 Functions / 121 \\
                 5.4 Trigonometric and Inverse Trigonometric Functions /
                 125 \\
                 6: Differential Calculus (one variable) / 132 \\
                 6.1 Some Basic Concepts / 132 \\
                 6.2 Limits and Continuity / 133 \\
                 6.3 Derivatives / 136 \\
                 6.4 Monotonicity. Extremes of Functions / 139 \\
                 7: Integral Calculus / 141 \\
                 7.1 Indefinite Integrals / 141 \\
                 7.2 Definite Integrals / 146 \\
                 7.3 Applications of Differential and Integral Calculus
                 / 148 \\
                 7.4 Table of Indefinite Integral / 153 \\
                 7.5 Tables of Definite Integrals / 178 \\
                 8: Sequences and Series / 183 \\
                 8.1 Sequences of Numbers / 183 \\
                 8.2 Sequences of Functions / 184 \\
                 8.3 Series of Constant Terms / 185 \\
                 8.4 Series of Functions / 187 \\
                 8.5 Taylor Series / 189 \\
                 8.6 Special Sums and Series / 192 \\
                 9: Ordinary Differential Equations (ODE) / 200 \\
                 9.1 Differential Equations of the First Order / 200 \\
                 9.2 Differential Equations of the Second Order / 202
                 \\
                 9.3 Linear Differential Equations / 205 \\
                 9.4 Autonomous systems / 2313 \\
                 9.5 General Concepts and Results / 216 \\
                 9.6 Linear Difference Equations / 218 \\
                 10: Multidimensional Calculus / 221 \\
                 10.1 The Space Rn / 221 \\
                 10.2 Surfaces. Tangent Planes / 222 \\
                 10.3 Limits and Continuity / 223 \\
                 10.4 Partial Derivatives / 224 \\
                 10.5 Extremes of Functions / 227 \\
                 10.6 Functions $f: R^n \to R^m (R^n \to R^n)$ / 229 \\
                 10.7 Double Integrals / 231 \\
                 10.8 Triple Integrals / 234 \\
                 10.9 Partial Differential Equations / 239 \\
                 11: Vector Analysis / 246 \\
                 11.1 Curves / 246 \\
                 11.2 Vector Fields / 248 \\
                 11.3 Line Integrals / 253 \\
                 11.4 Surface Integrals / 256 \\
                 12: Orthogonal Series and Special Functions / 259 \\
                 12.1 Orthogonal Systems / 259 \\
                 12.2 Orthogonal Polynomials / 263 \\
                 12.3 Bernoulli and Euler Polynomials / 269 \\
                 12.4 Bessel Functions / 270 \\
                 12.5 Functions Defined by Transcendental Integrals /
                 287 \\
                 12.6 Step and Impulse Functions / 297 \\
                 12.7 Functional Analysis / 298 \\
                 12.8 Lebesgue Integrals / 303 \\
                 12.9 Generalized functions (Distributions) / 308 \\
                 13: Transforms / 310 \\
                 13.1 Trigonometric Fourier Series / 310 \\
                 13.2 Fourier Transforms / 315 \\
                 13.3 Discrete Fourier Transforms / 325 \\
                 13.4 The $z$-transform / 327 \\
                 13.5 Laplace Transforms / 330 \\
                 13.6 Dynamical Systems (Filters) / 338 \\
                 13.7 Hankel and Hilbert transforms / 341 \\
                 13.8 Wavelets / 344 \\
                 14: Complex Analysis / 349 \\
                 14.1 Functions of a Complex Variable / 349 \\
                 14.2 Complex Integration / 352 \\
                 14.3 Power Series Expansions / 354 \\
                 14.4 Zeros and Singularities / 355 \\
                 14.5 Conformal Mappings / 356 \\
                 15: Optimization / 365 \\
                 15.1 Calculus of Variations / 365 \\
                 15.2 Linear Optimization / 371 \\
                 15.3 Integer and Combinatorial Optimization / 379 \\
                 15.4 Nonlinear Optimization / 383 \\
                 15.5 Dynamic Optimization / 389 \\
                 16: Numerical Analysis / 391 \\
                 16.1 Approximations and Errors / 391 \\
                 16.2 Numerical Solution of Equations / 392 \\
                 16.3 Perturbation analysis / 397 \\
                 16.4 Interpolation / 398 \\
                 16.5 Numerical Integration and Differentiation / 404
                 \\
                 16.6 Numerical Solutions of Differential Equations /
                 412 \\
                 16.7 Numerical summation / 421 \\
                 17: Probability Theory / 424 \\
                 17.1 Basic Probability Theory / 424 \\
                 17.2 Probability Distributions / 434 \\
                 17.3 Stochastic Processes / 439 \\
                 17.4 Algorithms for Calculation of Probability
                 Distributions / 443 \\
                 17.5 Simulation / 445 \\
                 17.6 Queueing Systems / 449 \\
                 17.7 Reliability / 452 \\
                 17.8 Tables / 459 \\
                 18: Statistics / 479 \\
                 18.1 Descriptive Statistics / 479 \\
                 18.2 Point Estimation / 488 \\
                 18.3 Confidence Intervals / 491 \\
                 18.4 Tables for Confidence Intervals / 495 \\
                 18.5 Tests of Significance / 501 \\
                 18.6 Linear Models / 507 \\
                 18.7 Distribution-free Methods / 512 \\
                 18.8 Statistical Quality Control / 518 \\
                 18.9 Factorial Experiments / 522 \\
                 18.10 Analysis of life time (failure time) data / 525
                 \\
                 18.11 Statistical glossary / 526 \\
                 19: Miscellaneous / 530",
}


bibtype: article
filename: elefunt.bib
label: Skorokhodov:2004:STP
author: S. L. Skorokhodov
journal: j-PROG-COMP-SOFT
volume: 30
number: 3
pages: 150--156
year: 2004
CODEN: PCSODA
ISSN: 0361-7688 (print), 1608-3261 (electronic) OR 03617688 (print), 16083261 (electronic)
ISSNL: 0361-7688 OR 03617688
MRclass: 3C20 (33B15 33C05 33F05 33F10)
MRnumber: MR2082811 (2005f:33013)
bibdate: Thu Dec 01 09:18:16 2011
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2011.12.01 09:18:16 ???
fjournal: Programming and Computer Software; translation of Programmirovaniye (Moscow, USSR) Plenum
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/11086
rawauthor: S. L. Skorokhodov
title:
Symbolic transformations in the problem of analytic continuation of the hypergeometric function $_p F_p - 1(z)$ in the neighborhood of the point $ z = 1 $ in the logarithmic case

rawtitle:
Symbolic transformations in the problem of analytic continuation of the hypergeometric function {$_p F_{p - 1}(z)$} in the neighborhood of the point $ z = 1 $ in the logarithmic case

entry:
@Article{Skorokhodov:2004:STP,
  author =       "S. L. Skorokhodov",
  title =        "Symbolic transformations in the problem of analytic
                 continuation of the hypergeometric function {$_p F_{p -
                 1}(z)$} in the neighborhood of the point $ z = 1 $ in
                 the logarithmic case",
  journal =      j-PROG-COMP-SOFT,
  volume =       "30",
  number =       "3",
  pages =        "150--156",
  month =        "????",
  year =         "2004",
  CODEN =        "PCSODA",
  ISSN =         "0361-7688 (print), 1608-3261 (electronic)",
  ISSN-L =       "0361-7688",
  MRclass =      "3C20 (33B15 33C05 33F05 33F10)",
  MRnumber =     "MR2082811 (2005f:33013)",
  bibdate =      "Thu Dec 01 09:18:16 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "Programming and Computer Software; translation of
                 Programmirovaniye (Moscow, USSR) Plenum",
  journal-URL =  "http://link.springer.com/journal/11086",
}


bibtype: article
filename: elefunt.bib
label: Thompson:2004:EBB
author: I. J. Thompson
journal: j-COMP-PHYS-COMM
volume: 159
number: 3
pages: 243--244
day: 1
month: jun
monthnumber: 6
year: 2004
CODEN: CPHCBZ
DOI: https://doi.org/10.1016/j.cpc.2004.02.007
ISSN: 0010-4655 (print), 1879-2944 (electronic) OR 00104655 (print), 18792944 (electronic)
ISSNL: 0010-4655 OR 00104655
bibdate: Thu Apr 24 10:35:27 2008
bibtimestamp: 2008.04.24 10:35:27 ???
note: See cite Thompson:1987:MBF.
URL: http://www.sciencedirect.com/science/article/pii/S0010465504001067
fjournal: Computer Physics Communications
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00104655
rawauthor: I. J. Thompson
rawnote: See \cite{Thompson:1987:MBF}.
title:
Erratum to booktitle Modified Bessel functions $ I _ n u(z) $ and $ K_nu (z) $ of real order and complex argument [Comput. Phys. Commun. bf 47 (1987) 245--257]

bibsource:
https://www.math.utah.edu/pub/tex/bib/compphyscomm1980.bib; https://www.math.utah.edu/pub/tex/bib/compphyscomm2000.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

rawtitle:
Erratum to {{\booktitle{Modified Bessel functions $ I \_ n u(z) $ and $ K_\nu (z) $ of real order and complex argument}} [Comput. Phys. Commun. {\bf 47} (1987) 245--257]}

entry:
@Article{Thompson:2004:EBB,
  author =       "I. J. Thompson",
  title =        "Erratum to {{\booktitle{Modified Bessel functions $ I
                 \_ n u(z) $ and $ K_\nu (z) $ of real order and complex
                 argument}} [Comput. Phys. Commun. {\bf 47} (1987)
                 245--257]}",
  journal =      j-COMP-PHYS-COMM,
  volume =       "159",
  number =       "3",
  pages =        "243--244",
  day =          "1",
  month =        jun,
  year =         "2004",
  CODEN =        "CPHCBZ",
  DOI =          "https://doi.org/10.1016/j.cpc.2004.02.007",
  ISSN =         "0010-4655 (print), 1879-2944 (electronic)",
  ISSN-L =       "0010-4655",
  bibdate =      "Thu Apr 24 10:35:27 2008",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/compphyscomm1980.bib;
                 https://www.math.utah.edu/pub/tex/bib/compphyscomm2000.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  note =         "See \cite{Thompson:1987:MBF}.",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0010465504001067",
  acknowledgement = ack-nhfb,
  fjournal =     "Computer Physics Communications",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00104655",
}


bibtype: book
filename: elefunt.bib
label: Vallee:2004:AFA
author: Olivier Vallee and Manuel Soares
title: Airy Functions and Applications to Physics
publisher: pub-WORLD-SCI
address: pub-WORLD-SCI:adr
pages: x + 194
year: 2004
ISBN: 1-86094-478-7 (hardcover) OR 1860944787 (hardcover)
ISBN13: 978-1-86094-478-9 (hardcover) OR 9781860944789 (hardcover)
LCCN: QA351 .V35 2004
bibdate: Tue Dec 5 10:16:05 2023
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2023.12.05 10:16:05 ???
remark: See also second edition cite Vallee:2010:AFA.
acknowledgement: ack-nhfb
rawauthor: Olivier Vall{\'e}e and Manuel Soares
rawtitle: {Airy} Functions and Applications to Physics
abstract:
The use of special functions, and in particular Airy functions, is rather common in physics. The reason may be found in the need, and even in the necessity, to express a physical phenomenon in terms of an effective and comprehensive analytical form for the whole scientific community. However, for the past twenty years, many physical problems have been resolved by computers. This trend is now becoming the norm as the importance of computers continues to grow. As a last resort, the special functions employed in physics will have to be calculated numerically, even if the analytic formulation of physics is of primary importance.par Airy functions have periodically been the subject of many review articles, but no noteworthy compilation on this subject has been published since the 1950s. In this work, we provide an exhaustive compilation of the current knowledge on the analytical properties of Airy functions, developing with care the calculus implying the Airy functions.

TOC:
Preface / v 1. A Historical Introduction: Sir George Biddell Airy / 1 2. Definitions and Properties / 5 2.1 The Homogeneous Airy Functions / 5 2.1.1 The Airy's equation / 5 2.1.2 Elementary properties / 8 2.1.2.1 Wronskians of homogeneous Airy functions / 8 2.1.2.2 Particular values of Airy functions / 8 2.1.2.3 Relations between Airy functions / 9 2.1.3 Integral representations / 9 2.1.4 Ascending and asymptotic series / 11 2.1.4.1 Expansion of $Ai $ near the origin / 11 2.1.4.2 Ascending series of $Ai $ and $Bi $ / 12 2.1.4.3 Asymptotic series of $Ai $ and $Bi $ / 13 2.2 Properties of Airy Functions / 15 2.2.1 Zeros of Airy functions / 15 2.2.2 The spectral zeta function / 18 2.2.3 Inequalities / 20 2.2.4 Connection with Bessel functions / 20 2.2.5 Modulus and phase of Airy functions / 21 2.2.5.1 Definitions / 21 2.2.5.2 Differential equations / 22 2.2.5.3 Asymptotic expansions / 23 2.2.5.4 Functions of positive arguments / 24 2.3 The Inhomogeneous Airy Functions / 25 2.3.1 Definitions / 25 2.3.2 Properties of inhomogeneous Airy functions / 27 2.3.2.1 Values at the origin / 27 2.3.2.2 Other integral representations / 27 2.3.3 Ascending and asymptotic series / 28 2.3.3.1 Ascending series / 28 2.3.3.2 Asymptotic series / 29 2.3.4 Zeros of the Scorer functions / 29 2.4 Squares and Products of Airy Functions / 30 2.4.1 Differential equation and integral representation / 30 2.4.2 A remarkable identity / 32 2.4.3 The product $Ai (x) Ai (-x)$: Airy wavelets / 32 3. Primitives and Integrals of Airy Functions / 37 3.1 Primitives Containing One Airy Function / 37 3.1.1 In terms of Airy functions / 37 3.1.2 Ascending series / 38 3.1.3 Asymptotic series / 38 3.1.4 Primitive of Scorer functions / 39 3.1.5 Repeated primitives / 40 3.2 Product of Airy Functions / 40 3.2.1 The method of Albright / 41 3.2.2 Some primitives / 43 3.3 Other Primitives / 48 3.4 Miscellaneous / 49 3.5 Elementary Integrals / 50 3.5.1 Particular integrals / 50 3.5.2 Integrals containing a single Airy function / 51 3.5.2.1 Integrals involving algebraic functions / 51 3.5.2.2 Integrals involving transcendental functions / 54 3.5.3 Integrals of products of two Airy functions / 56 3.6 Other Integrals / 60 3.6.1 Integrals involving the Volterra $mu $-function / 60 3.6.2 Canonisation of cubic form / 64 3.6.3 Integrals with three Airy functions / 65 3.6.4 Integrals with four Airy functions / 67 3.6.5 Double integrals / 68 4. Transformations of Airy Functions / 71 4.1 Causal Properties of Airy Functions / 71 4.1.1 Causal relations / 71 4.1.2 Green function of the Airy equation / 73 4.2 The Airy Transform / 74 4.2.1 Definitions and elementary properties / 74 4.2.2 Some examples / 77 4.2.3 Airy polynomials / 82 4.2.4 Summary of Airy transform / 84 4.2.5 Airy averaging / 85 4.3 Other Kinds of Transformations / 85 4.3.1 Laplace transform of Airy functions / 85 4.3.2 Mellin transform of Airy function / 86 4.3.3 Fourier transform of Airy functions / 87 4.4 Expansion into Fourier-Airy Series / 88 5. The Uniform Approximation / 91 5.1 Oscillating Integrals / 91 5.1.1 The method of stationary phase / 91 5.1.2 The uniform approximation of oscillating integrals / 93 5.1.3 The Airy uniform approximation / 94 5.2 Differential Equation of the Second Order / 95 5.2.1 The JWKB method / 95 5.2.2 The generalisation of Langer / 97 5.3 Inhomogeneous Differential Equations / 98 6. Generalisation of Airy Functions / 101 6.1 Generalisation of the Airy Integral / 101 6.2 Third Order Differential Equations / 105 6.2.1 The linear third order differential equation / 105 6.2.2 Asymptotic solutions / 106 6.2.3 The comparison equation / 107 6.3 Differential Equation of the Fourth Order / 111 7. Applications to Classical Physics / 115 7.1 Optics and Electromagnetism / 115 7.2 Fluid Mechanics / 119 7.2.1 The Tricomi equation / 119 7.2.2 The Orr--Sommerfeld equation / 121 7.3 Elasticity / 124 7.4 The Heat Equation / 127 7.5 Nonlinear Physics / 129 7.5.1 Korteweg--de Vries equation / 129 7.5.1.1 The linearised Korteweg--de Vries equation / 129 7.5.1.2 Similarity solutions / 131 7.5.2 The second Painleve equation / 132 7.5.2.1 The Painleve equations / 132 7.5.2.2 An integral equation / 134 7.5.2.3 Rational solutions 135 8. Applications to Quantum Physics / 137 8.1 The Schrodinger Equation / 137 8.1.1 Particle in a uniform field / 137 8.1.2 The $|x|$ potential / 140 8.1.3 Uniform approximation of the Schrodinger equation / 144 8.1.3.1 The JWKB approximation / 145 8.1.3.2 The Airy uniform approximation / 146 8.1.3.3 Exact vs approximate wave functions / 148 8.2 Evaluation of the Franck--Condon Factors / 152 8.2.1 The Franck--Condon principle / 153 8.2.2 The JWKB approximation / 154 8.2.3 The uniform approximation / 157 8.3 The Semiclassical Wigner Distribution / 162 8.3.1 The Weyl--Wigner formalism / 163 8.3.2 The one-dimensional Wigner distribution / 164 8.3.3 The two-dimensional Wigner distribution / 166 8.3.4 Configuration of the Wigner distribution / 169 8.4 Airy Transform of the Schrodinger Equation / 173 Appendix A: Numerical Computation of the Airy Functions / 177 A.1 The Homogeneous Functions / 177 A.2 The Inhomogeneous Functions / 180 Bibliography / 183 Index / 193

rawabstract:
The use of special functions, and in particular Airy functions, is rather common in physics. The reason may be found in the need, and even in the necessity, to express a physical phenomenon in terms of an effective and comprehensive analytical form for the whole scientific community. However, for the past twenty years, many physical problems have been resolved by computers. This trend is now becoming the norm as the importance of computers continues to grow. As a last resort, the special functions employed in physics will have to be calculated numerically, even if the analytic formulation of physics is of primary importance.\par Airy functions have periodically been the subject of many review articles, but no noteworthy compilation on this subject has been published since the 1950s. In this work, we provide an exhaustive compilation of the current knowledge on the analytical properties of Airy functions, developing with care the calculus implying the Airy functions.

shortTOC:
1: A historical introduction: Sir George Biddell Airy / 1 2: Definitions and properties / 5 3: Primitives and integrals of Airy functions / 37 4: Transformations of Airy functions / 71 5: The uniform approximation / 91 6: Generalisation of Airy functions / 101 7: Applications to classical physics / 115 8: Applications to quantum physics / 137 Appendix A: Numerical computation of the Airy functions / 177 Bibliography / 183 Index / 193

rawshortTOC:
1: A historical introduction: Sir George Biddell Airy / 1 \\ 2: Definitions and properties / 5 \\ 3: Primitives and integrals of Airy functions / 37 \\ 4: Transformations of Airy functions / 71 \\ 5: The uniform approximation / 91 \\ 6: Generalisation of Airy functions / 101 \\ 7: Applications to classical physics / 115 \\ 8: Applications to quantum physics / 137 \\ Appendix A: Numerical computation of the Airy functions / 177 \\ Bibliography / 183 \\ Index / 193

rawTOC:
Preface / v \\ 1. A Historical Introduction: Sir George Biddell Airy / 1 \\ 2. Definitions and Properties / 5 \\ 2.1 The Homogeneous Airy Functions / 5 \\ 2.1.1 The Airy's equation / 5 \\ 2.1.2 Elementary properties / 8 \\ 2.1.2.1 Wronskians of homogeneous Airy functions / 8 \\ 2.1.2.2 Particular values of Airy functions / 8 \\ 2.1.2.3 Relations between Airy functions / 9 \\ 2.1.3 Integral representations / 9 \\ 2.1.4 Ascending and asymptotic series / 11 \\ 2.1.4.1 Expansion of $\Ai$ near the origin / 11 \\ 2.1.4.2 Ascending series of $\Ai$ and $\Bi$ / 12 \\ 2.1.4.3 Asymptotic series of $\Ai$ and $\Bi$ / 13 \\ 2.2 Properties of Airy Functions / 15 \\ 2.2.1 Zeros of Airy functions / 15 \\ 2.2.2 The spectral zeta function / 18 \\ 2.2.3 Inequalities / 20 \\ 2.2.4 Connection with Bessel functions / 20 \\ 2.2.5 Modulus and phase of Airy functions / 21 \\ 2.2.5.1 Definitions / 21 \\ 2.2.5.2 Differential equations / 22 \\ 2.2.5.3 Asymptotic expansions / 23 \\ 2.2.5.4 Functions of positive arguments / 24 \\ 2.3 The Inhomogeneous Airy Functions / 25 \\ 2.3.1 Definitions / 25 \\ 2.3.2 Properties of inhomogeneous Airy functions / 27 \\ 2.3.2.1 Values at the origin / 27 \\ 2.3.2.2 Other integral representations / 27 \\ 2.3.3 Ascending and asymptotic series / 28 \\ 2.3.3.1 Ascending series / 28 \\ 2.3.3.2 Asymptotic series / 29 \\ 2.3.4 Zeros of the Scorer functions / 29 \\ 2.4 Squares and Products of Airy Functions / 30 \\ 2.4.1 Differential equation and integral representation / 30 \\ 2.4.2 A remarkable identity / 32 \\ 2.4.3 The product $\Ai(x) \Ai(-x)$: Airy wavelets / 32 \\ 3. Primitives and Integrals of Airy Functions / 37 \\ 3.1 Primitives Containing One Airy Function / 37 \\ 3.1.1 In terms of Airy functions / 37 \\ 3.1.2 Ascending series / 38 \\ 3.1.3 Asymptotic series / 38 \\ 3.1.4 Primitive of Scorer functions / 39 \\ 3.1.5 Repeated primitives / 40 \\ 3.2 Product of Airy Functions / 40 \\ 3.2.1 The method of Albright / 41 \\ 3.2.2 Some primitives / 43 \\ 3.3 Other Primitives / 48 \\ 3.4 Miscellaneous / 49 \\ 3.5 Elementary Integrals / 50 \\ 3.5.1 Particular integrals / 50 \\ 3.5.2 Integrals containing a single Airy function / 51 \\ 3.5.2.1 Integrals involving algebraic functions / 51 \\ 3.5.2.2 Integrals involving transcendental functions / 54 \\ 3.5.3 Integrals of products of two Airy functions / 56 \\ 3.6 Other Integrals / 60 \\ 3.6.1 Integrals involving the Volterra $\mu$-function / 60 \\ 3.6.2 Canonisation of cubic form / 64 \\ 3.6.3 Integrals with three Airy functions / 65 \\ 3.6.4 Integrals with four Airy functions / 67 \\ 3.6.5 Double integrals / 68 \\ 4. Transformations of Airy Functions / 71 \\ 4.1 Causal Properties of Airy Functions / 71 \\ 4.1.1 Causal relations / 71 \\ 4.1.2 Green function of the Airy equation / 73 \\ 4.2 The Airy Transform / 74 \\ 4.2.1 Definitions and elementary properties / 74 \\ 4.2.2 Some examples / 77 \\ 4.2.3 Airy polynomials / 82 \\ 4.2.4 Summary of Airy transform / 84 \\ 4.2.5 Airy averaging / 85 \\ 4.3 Other Kinds of Transformations / 85 \\ 4.3.1 Laplace transform of Airy functions / 85 \\ 4.3.2 Mellin transform of Airy function / 86 \\ 4.3.3 Fourier transform of Airy functions / 87 \\ 4.4 Expansion into Fourier-Airy Series / 88 \\ 5. The Uniform Approximation / 91 \\ 5.1 Oscillating Integrals / 91 \\ 5.1.1 The method of stationary phase / 91 \\ 5.1.2 The uniform approximation of oscillating integrals / 93 \\ 5.1.3 The Airy uniform approximation / 94 \\ 5.2 Differential Equation of the Second Order / 95 \\ 5.2.1 The JWKB method / 95 \\ 5.2.2 The generalisation of Langer / 97 \\ 5.3 Inhomogeneous Differential Equations / 98 \\ 6. Generalisation of Airy Functions / 101 \\ 6.1 Generalisation of the Airy Integral / 101 \\ 6.2 Third Order Differential Equations / 105 \\ 6.2.1 The linear third order differential equation / 105 \\ 6.2.2 Asymptotic solutions / 106 \\ 6.2.3 The comparison equation / 107 \\ 6.3 Differential Equation of the Fourth Order / 111 \\ 7. Applications to Classical Physics / 115 \\ 7.1 Optics and Electromagnetism / 115 \\ 7.2 Fluid Mechanics / 119 \\ 7.2.1 The Tricomi equation / 119 \\ 7.2.2 The Orr--Sommerfeld equation / 121 \\ 7.3 Elasticity / 124 \\ 7.4 The Heat Equation / 127 \\ 7.5 Nonlinear Physics / 129 \\ 7.5.1 Korteweg--de Vries equation / 129 \\ 7.5.1.1 The linearised Korteweg--de Vries equation / 129 \\ 7.5.1.2 Similarity solutions / 131 \\ 7.5.2 The second Painlev{\'e} equation / 132 \\ 7.5.2.1 The Painlev{\'e} equations / 132 \\ 7.5.2.2 An integral equation / 134 \\ 7.5.2.3 Rational solutions 135 \\ 8. Applications to Quantum Physics / 137 \\ 8.1 The Schr{\"o}dinger Equation / 137 \\ 8.1.1 Particle in a uniform field / 137 \\ 8.1.2 The $|x|$ potential / 140 \\ 8.1.3 Uniform approximation of the Schr{\"o}dinger equation / 144 \\ 8.1.3.1 The JWKB approximation / 145 \\ 8.1.3.2 The Airy uniform approximation / 146 \\ 8.1.3.3 Exact vs approximate wave functions / 148 \\ 8.2 Evaluation of the Franck--Condon Factors / 152 \\ 8.2.1 The Franck--Condon principle / 153 \\ 8.2.2 The JWKB approximation / 154 \\ 8.2.3 The uniform approximation / 157 \\ 8.3 The Semiclassical Wigner Distribution / 162 \\ 8.3.1 The Weyl--Wigner formalism / 163 \\ 8.3.2 The one-dimensional Wigner distribution / 164 \\ 8.3.3 The two-dimensional Wigner distribution / 166 \\ 8.3.4 Configuration of the Wigner distribution / 169 \\ 8.4 Airy Transform of the Schr{\"o}dinger Equation / 173 \\ Appendix A: Numerical Computation of the Airy Functions / 177 \\ A.1 The Homogeneous Functions / 177 \\ A.2 The Inhomogeneous Functions / 180 \\ Bibliography / 183 \\ Index / 193

entry:
@Book{Vallee:2004:AFA,
  author =       "Olivier Vall{\'e}e and Manuel Soares",
  title =        "{Airy} Functions and Applications to Physics",
  publisher =    pub-WORLD-SCI,
  address =      pub-WORLD-SCI:adr,
  pages =        "x + 194",
  year =         "2004",
  ISBN =         "1-86094-478-7 (hardcover)",
  ISBN-13 =      "978-1-86094-478-9 (hardcover)",
  LCCN =         "QA351 .V35 2004",
  bibdate =      "Tue Dec 5 10:16:05 2023",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  abstract =     "The use of special functions, and in particular Airy
                 functions, is rather common in physics. The reason may
                 be found in the need, and even in the necessity, to
                 express a physical phenomenon in terms of an effective
                 and comprehensive analytical form for the whole
                 scientific community. However, for the past twenty
                 years, many physical problems have been resolved by
                 computers. This trend is now becoming the norm as the
                 importance of computers continues to grow. As a last
                 resort, the special functions employed in physics will
                 have to be calculated numerically, even if the analytic
                 formulation of physics is of primary
                 importance.\par

                 Airy functions have periodically been the subject of
                 many review articles, but no noteworthy compilation on
                 this subject has been published since the 1950s. In
                 this work, we provide an exhaustive compilation of the
                 current knowledge on the analytical properties of Airy
                 functions, developing with care the calculus implying
                 the Airy functions.",
  acknowledgement = ack-nhfb,
  remark =       "See also second edition \cite{Vallee:2010:AFA}.",
  shorttableofcontents = "1: A historical introduction: Sir George
                 Biddell Airy / 1 \\
                 2: Definitions and properties / 5 \\
                 3: Primitives and integrals of Airy functions / 37 \\
                 4: Transformations of Airy functions / 71 \\
                 5: The uniform approximation / 91 \\
                 6: Generalisation of Airy functions / 101 \\
                 7: Applications to classical physics / 115 \\
                 8: Applications to quantum physics / 137 \\
                 Appendix A: Numerical computation of the Airy functions
                 / 177 \\
                 Bibliography / 183 \\
                 Index / 193",
  tableofcontents = "Preface / v \\
                 1. A Historical Introduction: Sir George Biddell Airy /
                 1 \\
                 2. Definitions and Properties / 5 \\
                 2.1 The Homogeneous Airy Functions / 5 \\
                 2.1.1 The Airy's equation / 5 \\
                 2.1.2 Elementary properties / 8 \\
                 2.1.2.1 Wronskians of homogeneous Airy functions / 8
                 \\
                 2.1.2.2 Particular values of Airy functions / 8 \\
                 2.1.2.3 Relations between Airy functions / 9 \\
                 2.1.3 Integral representations / 9 \\
                 2.1.4 Ascending and asymptotic series / 11 \\
                 2.1.4.1 Expansion of $\Ai$ near the origin / 11 \\
                 2.1.4.2 Ascending series of $\Ai$ and $\Bi$ / 12 \\
                 2.1.4.3 Asymptotic series of $\Ai$ and $\Bi$ / 13 \\
                 2.2 Properties of Airy Functions / 15 \\
                 2.2.1 Zeros of Airy functions / 15 \\
                 2.2.2 The spectral zeta function / 18 \\
                 2.2.3 Inequalities / 20 \\
                 2.2.4 Connection with Bessel functions / 20 \\
                 2.2.5 Modulus and phase of Airy functions / 21 \\
                 2.2.5.1 Definitions / 21 \\
                 2.2.5.2 Differential equations / 22 \\
                 2.2.5.3 Asymptotic expansions / 23 \\
                 2.2.5.4 Functions of positive arguments / 24 \\
                 2.3 The Inhomogeneous Airy Functions / 25 \\
                 2.3.1 Definitions / 25 \\
                 2.3.2 Properties of inhomogeneous Airy functions / 27
                 \\
                 2.3.2.1 Values at the origin / 27 \\
                 2.3.2.2 Other integral representations / 27 \\
                 2.3.3 Ascending and asymptotic series / 28 \\
                 2.3.3.1 Ascending series / 28 \\
                 2.3.3.2 Asymptotic series / 29 \\
                 2.3.4 Zeros of the Scorer functions / 29 \\
                 2.4 Squares and Products of Airy Functions / 30 \\
                 2.4.1 Differential equation and integral representation
                 / 30 \\
                 2.4.2 A remarkable identity / 32 \\
                 2.4.3 The product $\Ai(x) \Ai(-x)$: Airy wavelets / 32
                 \\
                 3. Primitives and Integrals of Airy Functions / 37 \\
                 3.1 Primitives Containing One Airy Function / 37 \\
                 3.1.1 In terms of Airy functions / 37 \\
                 3.1.2 Ascending series / 38 \\
                 3.1.3 Asymptotic series / 38 \\
                 3.1.4 Primitive of Scorer functions / 39 \\
                 3.1.5 Repeated primitives / 40 \\
                 3.2 Product of Airy Functions / 40 \\
                 3.2.1 The method of Albright / 41 \\
                 3.2.2 Some primitives / 43 \\
                 3.3 Other Primitives / 48 \\
                 3.4 Miscellaneous / 49 \\
                 3.5 Elementary Integrals / 50 \\
                 3.5.1 Particular integrals / 50 \\
                 3.5.2 Integrals containing a single Airy function / 51
                 \\
                 3.5.2.1 Integrals involving algebraic functions / 51
                 \\
                 3.5.2.2 Integrals involving transcendental functions /
                 54 \\
                 3.5.3 Integrals of products of two Airy functions / 56
                 \\
                 3.6 Other Integrals / 60 \\
                 3.6.1 Integrals involving the Volterra $\mu$-function /
                 60 \\
                 3.6.2 Canonisation of cubic form / 64 \\
                 3.6.3 Integrals with three Airy functions / 65 \\
                 3.6.4 Integrals with four Airy functions / 67 \\
                 3.6.5 Double integrals / 68 \\
                 4. Transformations of Airy Functions / 71 \\
                 4.1 Causal Properties of Airy Functions / 71 \\
                 4.1.1 Causal relations / 71 \\
                 4.1.2 Green function of the Airy equation / 73 \\
                 4.2 The Airy Transform / 74 \\
                 4.2.1 Definitions and elementary properties / 74 \\
                 4.2.2 Some examples / 77 \\
                 4.2.3 Airy polynomials / 82 \\
                 4.2.4 Summary of Airy transform / 84 \\
                 4.2.5 Airy averaging / 85 \\
                 4.3 Other Kinds of Transformations / 85 \\
                 4.3.1 Laplace transform of Airy functions / 85 \\
                 4.3.2 Mellin transform of Airy function / 86 \\
                 4.3.3 Fourier transform of Airy functions / 87 \\
                 4.4 Expansion into Fourier-Airy Series / 88 \\
                 5. The Uniform Approximation / 91 \\
                 5.1 Oscillating Integrals / 91 \\
                 5.1.1 The method of stationary phase / 91 \\
                 5.1.2 The uniform approximation of oscillating
                 integrals / 93 \\
                 5.1.3 The Airy uniform approximation / 94 \\
                 5.2 Differential Equation of the Second Order / 95 \\
                 5.2.1 The JWKB method / 95 \\
                 5.2.2 The generalisation of Langer / 97 \\
                 5.3 Inhomogeneous Differential Equations / 98 \\
                 6. Generalisation of Airy Functions / 101 \\
                 6.1 Generalisation of the Airy Integral / 101 \\
                 6.2 Third Order Differential Equations / 105 \\
                 6.2.1 The linear third order differential equation /
                 105 \\
                 6.2.2 Asymptotic solutions / 106 \\
                 6.2.3 The comparison equation / 107 \\
                 6.3 Differential Equation of the Fourth Order / 111 \\
                 7. Applications to Classical Physics / 115 \\
                 7.1 Optics and Electromagnetism / 115 \\
                 7.2 Fluid Mechanics / 119 \\
                 7.2.1 The Tricomi equation / 119 \\
                 7.2.2 The Orr--Sommerfeld equation / 121 \\
                 7.3 Elasticity / 124 \\
                 7.4 The Heat Equation / 127 \\
                 7.5 Nonlinear Physics / 129 \\
                 7.5.1 Korteweg--de Vries equation / 129 \\
                 7.5.1.1 The linearised Korteweg--de Vries equation /
                 129 \\
                 7.5.1.2 Similarity solutions / 131 \\
                 7.5.2 The second Painlev{\'e} equation / 132 \\
                 7.5.2.1 The Painlev{\'e} equations / 132 \\
                 7.5.2.2 An integral equation / 134 \\
                 7.5.2.3 Rational solutions 135 \\
                 8. Applications to Quantum Physics / 137 \\
                 8.1 The Schr{\"o}dinger Equation / 137 \\
                 8.1.1 Particle in a uniform field / 137 \\
                 8.1.2 The $|x|$ potential / 140 \\
                 8.1.3 Uniform approximation of the Schr{\"o}dinger
                 equation / 144 \\
                 8.1.3.1 The JWKB approximation / 145 \\
                 8.1.3.2 The Airy uniform approximation / 146 \\
                 8.1.3.3 Exact vs approximate wave functions / 148 \\
                 8.2 Evaluation of the Franck--Condon Factors / 152 \\
                 8.2.1 The Franck--Condon principle / 153 \\
                 8.2.2 The JWKB approximation / 154 \\
                 8.2.3 The uniform approximation / 157 \\
                 8.3 The Semiclassical Wigner Distribution / 162 \\
                 8.3.1 The Weyl--Wigner formalism / 163 \\
                 8.3.2 The one-dimensional Wigner distribution / 164 \\
                 8.3.3 The two-dimensional Wigner distribution / 166 \\
                 8.3.4 Configuration of the Wigner distribution / 169
                 \\
                 8.4 Airy Transform of the Schr{\"o}dinger Equation /
                 173 \\
                 Appendix A: Numerical Computation of the Airy Functions
                 / 177 \\
                 A.1 The Homogeneous Functions / 177 \\
                 A.2 The Inhomogeneous Functions / 180 \\
                 Bibliography / 183 \\
                 Index / 193",
}


bibtype: article
filename: elefunt.bib
label: VanDeun:2004:IAO
author: J. Van Deun and A. Bultheel
title: An Interpolation Algorithm for Orthogonal Rational Functions
journal: j-J-COMPUT-APPL-MATH
volume: 164--165
pages: 749--762
month: mar
monthnumber: 3
year: 2004
CODEN: JCAMDI
DOI: https://doi.org/10.1016/S0377-0427(03)00493-X
ISSN: 0377-0427 (print), 1879-1778 (electronic) OR 03770427 (print), 18791778 (electronic)
ISSNL: 0377-0427 OR 03770427
bibdate: Tue Mar 24 21:14:11 2009
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2009.03.24 21:14:11 ???
fjournal: Journal of Computational and Applied Mathematics
keywords: interpolation; orthogonal polynomials; orthogonal rational functions; three-term recurrence
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/03770427
rawauthor: J. {Van Deun} and A. Bultheel
rawtitle: An Interpolation Algorithm for Orthogonal Rational Functions
note:
Proceedings of the 10th International Congress on Computational and Applied Mathematics University of Leuven, Belgium, 22--26 July 2002. Edited by M. J. Goovaerts, S. Vandewalle, and L. Wuytack.

abstract:
Let $ A = alpha _1, alpha _2, ldots $ be a sequence of numbers on the extended real line $ mathcal R = mathcal R union infty $ and $ mu $ a positive bounded Borel measure with support in (a subset of) $ mathcal R $. We introduce rational functions n with poles $ alpha _1, ldots , alpha _n $ that are orthogonal with respect to $ mu $ (if all poles are at infinity, we recover the polynomial situation). It is well known that under certain conditions on the location of the poles, the system $ phi _n $ is regular such that the orthogonal functions satisfy a three-term recurrence relation similar to the one for orthogonal polynomials. To compute the recurrence coefficients one can use explicit formulas involving inner products. We present a theoretical alternative to these explicit formulas that uses certain interpolation properties of the Riesz--Herglotz--Nevanlinna transform $ Omega _mu $ of the measure $ mu $. Error bounds are derived and some examples serve as illustration.

rawabstract:
Let $ A = \{ \alpha_1, \alpha_2, \ldots {} \} $ be a sequence of numbers on the extended real line $ \mathcal {R} = \mathcal {R} \union \{ \infty \} $ and $ \mu $ a positive bounded Borel measure with support in (a subset of) $ \mathcal {R} $. We introduce rational functions n with poles $ \{ \alpha_1, \ldots {}, \alpha_n \} $ that are orthogonal with respect to $ \mu $ (if all poles are at infinity, we recover the polynomial situation). It is well known that under certain conditions on the location of the poles, the system $ \{ \phi_n \} $ is regular such that the orthogonal functions satisfy a three-term recurrence relation similar to the one for orthogonal polynomials. To compute the recurrence coefficients one can use explicit formulas involving inner products. We present a theoretical alternative to these explicit formulas that uses certain interpolation properties of the Riesz--Herglotz--Nevanlinna transform $ \Omega_\mu $ of the measure $ \mu $. Error bounds are derived and some examples serve as illustration.

entry:
@Article{VanDeun:2004:IAO,
  author =       "J. {Van Deun} and A. Bultheel",
  title =        "An Interpolation Algorithm for Orthogonal Rational
                 Functions",
  journal =      j-J-COMPUT-APPL-MATH,
  volume =       "164--165",
  number =       "??",
  pages =        "749--762",
  month =        mar,
  year =         "2004",
  CODEN =        "JCAMDI",
  DOI =          "https://doi.org/10.1016/S0377-0427(03)00493-X",
  ISSN =         "0377-0427 (print), 1879-1778 (electronic)",
  ISSN-L =       "0377-0427",
  bibdate =      "Tue Mar 24 21:14:11 2009",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  note =         "Proceedings of the 10th International Congress on
                 Computational and Applied Mathematics University of
                 Leuven, Belgium, 22--26 July 2002. Edited by M. J.
                 Goovaerts, S. Vandewalle, and L. Wuytack.",
  abstract =     "Let $ A = \{ \alpha_1, \alpha_2, \ldots {} \} $ be a
                 sequence of numbers on the extended real line $
                 \mathcal {R} = \mathcal {R} \union \{ \infty \} $ and $
                 \mu $ a positive bounded Borel measure with support in
                 (a subset of) $ \mathcal {R} $. We introduce rational
                 functions n with poles $ \{ \alpha_1, \ldots {},
                 \alpha_n \} $ that are orthogonal with respect to $ \mu
                 $ (if all poles are at infinity, we recover the
                 polynomial situation). It is well known that under
                 certain conditions on the location of the poles, the
                 system $ \{ \phi_n \} $ is regular such that the
                 orthogonal functions satisfy a three-term recurrence
                 relation similar to the one for orthogonal polynomials.
                 To compute the recurrence coefficients one can use
                 explicit formulas involving inner products. We present
                 a theoretical alternative to these explicit formulas
                 that uses certain interpolation properties of the
                 Riesz--Herglotz--Nevanlinna transform $ \Omega_\mu $ of
                 the measure $ \mu $. Error bounds are derived and some
                 examples serve as illustration.",
  acknowledgement = ack-nhfb,
  fjournal =     "Journal of Computational and Applied Mathematics",
  journal-URL =  "http://www.sciencedirect.com/science/journal/03770427",
  keywords =     "interpolation; orthogonal polynomials; orthogonal
                 rational functions; three-term recurrence",
}


bibtype: article
filename: elefunt.bib
label: Wang:2004:CHP
author: Ren-Hong Wang and Cheng-de Zheng
title: Cubic Hermite--Pade approximation to the exponential function
journal: j-J-COMPUT-APPL-MATH
volume: 163
number: 1
pages: 259--268
day: 1
month: feb
monthnumber: 2
year: 2004
CODEN: JCAMDI
ISSN: 0377-0427 (print), 1879-1778 (electronic) OR 03770427 (print), 18791778 (electronic)
ISSNL: 0377-0427 OR 03770427
bibdate: Sat Feb 25 12:59:56 MST 2017
bibtimestamp: 2017.02.25 12:59:56 MST
URL: http://www.sciencedirect.com/science/article/pii/S0377042703008124
fjournal: Journal of Computational and Applied Mathematics
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/03770427
rawauthor: Ren-Hong Wang and Cheng-de Zheng
rawtitle: Cubic {Hermite--Pad{\'e}} approximation to the exponential function
entry:
@Article{Wang:2004:CHP,
  author =       "Ren-Hong Wang and Cheng-de Zheng",
  title =        "Cubic {Hermite--Pad{\'e}} approximation to the
                 exponential function",
  journal =      j-J-COMPUT-APPL-MATH,
  volume =       "163",
  number =       "1",
  pages =        "259--268",
  day =          "1",
  month =        feb,
  year =         "2004",
  CODEN =        "JCAMDI",
  ISSN =         "0377-0427 (print), 1879-1778 (electronic)",
  ISSN-L =       "0377-0427",
  bibdate =      "Sat Feb 25 12:59:56 MST 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2000.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0377042703008124",
  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: Zeng:2004:AMM
author: Zhonggang Zeng
title: Algorithm 835: MultRoot---a Matlab package for computing polynomial roots and multiplicities
journal: j-TOMS
volume: 30
number: 2
pages: 218--236
month: jun
monthnumber: 6
year: 2004
CODEN: ACMSCU
DOI: https://doi.org/10.1145/992200.992209
ISSN: 0098-3500 (print), 1557-7295 (electronic) OR 00983500 (print), 15577295 (electronic)
ISSNL: 0098-3500 OR 00983500
MRclass: 65H05
MRnumber: MR2075984 (2005c:65041)
bibdate: Tue Mar 30 16:16:28 2010
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2010.03.30 16:16:28 ???
fjournal: Association for Computing Machinery. Transactions on Mathematical Software
acknowledgement: ack-nhfb
journalURL: https://dl.acm.org/loi/toms
rawauthor: Zhonggang Zeng
rawtitle: Algorithm 835: {MultRoot}---a {Matlab} package for computing polynomial roots and multiplicities
abstract:
MultRoot is a collection of Matlab modules for accurate computation of polynomial roots, especially roots with non-trivial multiplicities. As a blackbox-type software, MultRoot requires the polynomial coefficients as the only input, and outputs the computed roots, multiplicities, backward error, estimated forward error, and the structure-preserving condition number. The most significant features of MultRoot are the multiplicity identification capability and high accuracy on multiple roots without using multiprecision arithmetic, even if the polynomial coefficients are inexact. A comprehensive test suite of polynomials that are collected from the literature is included for numerical experiments and performance comparison.

rawabstract:
MultRoot is a collection of Matlab modules for accurate computation of polynomial roots, especially roots with non-trivial multiplicities. As a blackbox-type software, MultRoot requires the polynomial coefficients as the only input, and outputs the computed roots, multiplicities, backward error, estimated forward error, and the structure-preserving condition number. The most significant features of MultRoot are the multiplicity identification capability and high accuracy on multiple roots without using multiprecision arithmetic, even if the polynomial coefficients are inexact. A comprehensive test suite of polynomials that are collected from the literature is included for numerical experiments and performance comparison.

entry:
@Article{Zeng:2004:AMM,
  author =       "Zhonggang Zeng",
  title =        "Algorithm 835: {MultRoot}---a {Matlab} package for
                 computing polynomial roots and multiplicities",
  journal =      j-TOMS,
  volume =       "30",
  number =       "2",
  pages =        "218--236",
  month =        jun,
  year =         "2004",
  CODEN =        "ACMSCU",
  DOI =          "https://doi.org/10.1145/992200.992209",
  ISSN =         "0098-3500 (print), 1557-7295 (electronic)",
  ISSN-L =       "0098-3500",
  MRclass =      "65H05",
  MRnumber =     "MR2075984 (2005c:65041)",
  bibdate =      "Tue Mar 30 16:16:28 2010",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  abstract =     "MultRoot is a collection of Matlab modules for
                 accurate computation of polynomial roots, especially
                 roots with non-trivial multiplicities. As a
                 blackbox-type software, MultRoot requires the
                 polynomial coefficients as the only input, and outputs
                 the computed roots, multiplicities, backward error,
                 estimated forward error, and the structure-preserving
                 condition number. The most significant features of
                 MultRoot are the multiplicity identification capability
                 and high accuracy on multiple roots without using
                 multiprecision arithmetic, even if the polynomial
                 coefficients are inexact. A comprehensive test suite of
                 polynomials that are collected from the literature is
                 included for numerical experiments and performance
                 comparison.",
  acknowledgement = ack-nhfb,
  fjournal =     "Association for Computing Machinery. Transactions on
                 Mathematical Software",
  journal-URL =  "https://dl.acm.org/loi/toms",
}


bibtype: article
filename: elefunt.bib
label: Zhu:2004:ISR
author: Hufei Zhu and Zhongding Lei and F. P. S. Chin
title: An improved square-root algorithm for BLAST
journal: j-IEEE-SIGNAL-PROCESS-LETT
volume: 11
number: 9
pages: 772--775
month: sep
monthnumber: 9
year: 2004
CODEN: ISPLEM
DOI: https://doi.org/10.1109/LSP.2004.833483
ISSN: 1070-9908 (print), 1558-2361 (electronic) OR 10709908 (print), 15582361 (electronic)
ISSNL: 1070-9908 OR 10709908
bibdate: Sat Jul 16 15:28:13 MDT 2005
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2005.07.16 15:28:13 MDT
fjournal: IEEE Signal Processing Letters
acknowledgement: ack-nhfb
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=97
rawauthor: Hufei Zhu and Zhongding Lei and F. P. S. Chin
rawtitle: An improved square-root algorithm for {BLAST}
entry:
@Article{Zhu:2004:ISR,
  author =       "Hufei Zhu and Zhongding Lei and F. P. S. Chin",
  title =        "An improved square-root algorithm for {BLAST}",
  journal =      j-IEEE-SIGNAL-PROCESS-LETT,
  volume =       "11",
  number =       "9",
  pages =        "772--775",
  month =        sep,
  year =         "2004",
  CODEN =        "ISPLEM",
  DOI =          "https://doi.org/10.1109/LSP.2004.833483",
  ISSN =         "1070-9908 (print), 1558-2361 (electronic)",
  ISSN-L =       "1070-9908",
  bibdate =      "Sat Jul 16 15:28:13 MDT 2005",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "IEEE Signal Processing Letters",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=97",
  summary =      "In this letter, an improved square-root algorithm for
                 Bell Labs Layered Space-Time (BLAST) system is
                 proposed. It speeds up the original square-root
                 algorithm by 36\% in terms of the number of
                 multiplications and additions. Compared with the
                 \ldots{}",
}


bibtype: article
filename: elefunt.bib
label: Abad:2005:TNA
author: Julio Abad and Javier Sesma
title: Two new asymptotic expansions of the ratio of two gamma functions
journal: j-J-COMPUT-APPL-MATH
volume: 173
number: 2
pages: 359--363
day: 15
month: jan
monthnumber: 1
year: 2005
CODEN: JCAMDI
ISSN: 0377-0427 (print), 1879-1778 (electronic) OR 03770427 (print), 18791778 (electronic)
ISSNL: 0377-0427 OR 03770427
bibdate: Sat Feb 25 13:00:02 MST 2017
bibtimestamp: 2017.02.25 13:00:02 MST
URL: http://www.sciencedirect.com/science/article/pii/S0377042704001669
fjournal: Journal of Computational and Applied Mathematics
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/03770427
rawauthor: Julio Abad and Javier Sesma
rawtitle: Two new asymptotic expansions of the ratio of two gamma functions
bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2005.bib

entry:
@Article{Abad:2005:TNA,
  author =       "Julio Abad and Javier Sesma",
  title =        "Two new asymptotic expansions of the ratio of two
                 gamma functions",
  journal =      j-J-COMPUT-APPL-MATH,
  volume =       "173",
  number =       "2",
  pages =        "359--363",
  day =          "15",
  month =        jan,
  year =         "2005",
  CODEN =        "JCAMDI",
  ISSN =         "0377-0427 (print), 1879-1778 (electronic)",
  ISSN-L =       "0377-0427",
  bibdate =      "Sat Feb 25 13:00:02 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/S0377042704001669",
  acknowledgement = ack-nhfb,
  fjournal =     "Journal of Computational and Applied Mathematics",
  journal-URL =  "http://www.sciencedirect.com/science/journal/03770427",
}


bibtype: inproceedings
filename: elefunt.bib
label: Antelo:2005:LLD
author: Elisardo Antelo and Tomas Lang and Paolo Montuschi and Alberto Nannarelli
title: Low Latency Digit-Recurrence Reciprocal and Square-Root Reciprocal Algorithm and Architecture
crossref: IEEE:2005:PIS
pages: ??--??
year: 2005
bibdate: Wed Jun 22 07:02:55 2005
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2005.06.22 07:02:55 ???
URL: http://arith17.polito.it/final/paper-116.pdf
acknowledgement: ack-nhfb
rawauthor: Elisardo Antelo and Tom{\'a}s Lang and Paolo Montuschi and Alberto Nannarelli
rawtitle: Low Latency Digit-Recurrence Reciprocal and Square-Root Reciprocal Algorithm and Architecture
abstract:
The reciprocal and square-root reciprocal operations are important in several applications. For the operations, we present algorithms that combine a digit-by-digit module and one iteration of a quadratic-convergence approximation. The latter is implemented by a digit-recurrence, which uses the digits produced by the digit-by-digit part. In this way, both parts execute in an overlapped manner, so that the total number of cycles is about half the number that would be required by the digit-by-digit part alone. Because of the approximation, correct rounding of the result cannot be obtained directly in all cases; we propose a variable-time implementation that produces the correctly rounded result with a small average overhead. Radix-4 implementations are described and have been synthesized. They achieve the same cycle time as the standard digit-by-digit implementation, resulting in a speed-up of about 2 and, because of the approximation part, the area factor is also about 2. We also show a combined implementation for both operations that has essentially the same complexity as that for square-root reciprocal alone.

rawabstract:
The reciprocal and square-root reciprocal operations are important in several applications. For the operations, we present algorithms that combine a digit-by-digit module and one iteration of a quadratic-convergence approximation. The latter is implemented by a digit-recurrence, which uses the digits produced by the digit-by-digit part. In this way, both parts execute in an overlapped manner, so that the total number of cycles is about half the number that would be required by the digit-by-digit part alone. Because of the approximation, correct rounding of the result cannot be obtained directly in all cases; we propose a variable-time implementation that produces the correctly rounded result with a small average overhead. Radix-4 implementations are described and have been synthesized. They achieve the same cycle time as the standard digit-by-digit implementation, resulting in a speed-up of about 2 and, because of the approximation part, the area factor is also about 2. We also show a combined implementation for both operations that has essentially the same complexity as that for square-root reciprocal alone.

entry:
@InProceedings{Antelo:2005:LLD,
  author =       "Elisardo Antelo and Tom{\'a}s Lang and Paolo Montuschi
                 and Alberto Nannarelli",
  title =        "Low Latency Digit-Recurrence Reciprocal and
                 Square-Root Reciprocal Algorithm and Architecture",
  crossref =     "IEEE:2005:PIS",
  pages =        "??--??",
  year =         "2005",
  bibdate =      "Wed Jun 22 07:02:55 2005",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://arith17.polito.it/final/paper-116.pdf",
  abstract =     "The reciprocal and square-root reciprocal operations
                 are important in several applications. For the
                 operations, we present algorithms that combine a
                 digit-by-digit module and one iteration of a
                 quadratic-convergence approximation. The latter is
                 implemented by a digit-recurrence, which uses the
                 digits produced by the digit-by-digit part. In this
                 way, both parts execute in an overlapped manner, so
                 that the total number of cycles is about half the
                 number that would be required by the digit-by-digit
                 part alone. Because of the approximation, correct
                 rounding of the result cannot be obtained directly in
                 all cases; we propose a variable-time implementation
                 that produces the correctly rounded result with a small
                 average overhead. Radix-4 implementations are described
                 and have been synthesized. They achieve the same cycle
                 time as the standard digit-by-digit implementation,
                 resulting in a speed-up of about 2 and, because of the
                 approximation part, the area factor is also about 2. We
                 also show a combined implementation for both operations
                 that has essentially the same complexity as that for
                 square-root reciprocal alone.",
  acknowledgement = ack-nhfb,
  pagecount =    "8",
}


bibtype: book
filename: elefunt.bib
label: Arfken:2005:MMP
author: George B. Arfken and Hans-Jurgen Weber
title: Mathematical Methods for Physicists
publisher: pub-ELSEVIER
address: pub-ELSEVIER:adr
edition: Sixth
pages: xii + 1182
year: 2005
ISBN: 0-12-059876-0, 0-12-088584-0 (paperback) OR 0120598760, 0120885840 (paperback)
ISBN13: 978-0-12-059876-2, 978-0-12-088584-8 (paperback) OR 9780120598762, 9780120885848 (paperback)
LCCN: QA37.3 .A74 2005
bibdate: Tue Feb 17 18:23:45 MST 2009
bibtimestamp: 2009.02.17 18:23:45 MST
subject: Mathematics; Mathematical physics
acknowledgement: ack-nhfb
rawauthor: George B. Arfken and Hans-J{\"u}rgen Weber
rawtitle: Mathematical Methods for Physicists
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

TOC:
1. Vector Analysis 2. Vector Analysis in Curved Coordinates and Tensors 3. Determinants and Matrices 4. Group Theory 5. Infinite Series 6. Functions of a Complex Variable I: Analytic Properties, Mapping 7. Functions of a Complex Variable II 8. The Gamma Function (Factorial Function) 9. Differential Equations 10. Sturm--Liouville Theory-Orthogonal Functions 11. Bessel Functions 12. Legendre Functions 13. More Special Functions 14. Fourier Series 15. Integral Transforms 16. Integral Equations 17. Calculus of Variations 18. Nonlinear Methods and Chaos 19. Probability

rawTOC:
1. Vector Analysis \\ 2. Vector Analysis in Curved Coordinates and Tensors \\ 3. Determinants and Matrices \\ 4. Group Theory \\ 5. Infinite Series \\ 6. Functions of a Complex Variable I: Analytic Properties, Mapping \\ 7. Functions of a Complex Variable II \\ 8. The Gamma Function (Factorial Function) \\ 9. Differential Equations \\ 10. Sturm--Liouville Theory-Orthogonal Functions \\ 11. Bessel Functions \\ 12. Legendre Functions \\ 13. More Special Functions \\ 14. Fourier Series \\ 15. Integral Transforms \\ 16. Integral Equations \\ 17. Calculus of Variations \\ 18. Nonlinear Methods and Chaos \\ 19. Probability

entry:
@Book{Arfken:2005:MMP,
  author =       "George B. Arfken and Hans-J{\"u}rgen Weber",
  title =        "Mathematical Methods for Physicists",
  publisher =    pub-ELSEVIER,
  address =      pub-ELSEVIER:adr,
  edition =      "Sixth",
  pages =        "xii + 1182",
  year =         "2005",
  ISBN =         "0-12-059876-0, 0-12-088584-0 (paperback)",
  ISBN-13 =      "978-0-12-059876-2, 978-0-12-088584-8 (paperback)",
  LCCN =         "QA37.3 .A74 2005",
  bibdate =      "Tue Feb 17 18:23:45 MST 2009",
  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,
  subject =      "Mathematics; Mathematical physics",
  tableofcontents = "1. Vector Analysis \\
                 2. Vector Analysis in Curved Coordinates and Tensors
                 \\
                 3. Determinants and Matrices \\
                 4. Group Theory \\
                 5. Infinite Series \\
                 6. Functions of a Complex Variable I: Analytic
                 Properties, Mapping \\
                 7. Functions of a Complex Variable II \\
                 8. The Gamma Function (Factorial Function) \\
                 9. Differential Equations \\
                 10. Sturm--Liouville Theory-Orthogonal Functions \\
                 11. Bessel Functions \\
                 12. Legendre Functions \\
                 13. More Special Functions \\
                 14. Fourier Series \\
                 15. Integral Transforms \\
                 16. Integral Equations \\
                 17. Calculus of Variations \\
                 18. Nonlinear Methods and Chaos \\
                 19. Probability",
  xxauthor =     "George B. (George Brown) Arfken and Hans-J{\"u}rgen
                 Weber",
  xxURL =        "http://www.loc.gov/catdir/enhancements/fy0625/2005049844-d.html;
                 http://www.loc.gov/catdir/enhancements/fy0625/2005049844-t.html",
}


