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BibTeX entry
@Article{Abbott:2006:PE,
author = "Paul Abbott",
title = "On the Perimeter of an Ellipse",
journal = j-MATHEMATICA-J,
volume = "11",
number = "2",
pages = "172--185",
month = "????",
year = "2006",
CODEN = "????",
ISSN = "1047-5974 (print), 1097-1610 (electronic)",
ISSN-L = "1047-5974",
bibdate = "Sat Nov 6 13:34:51 MDT 2010",
bibsource = "http://www.math.utah.edu/pub/tex/bib/mathematicaj.bib;
http://www.mathematica-journal.com/issue/v11i2/",
URL = "http://www.mathematica-journal.com/issue/v11i2/contents/Abbott/Abbott.nb;
http://www.mathematica-journal.com/issue/v11i2/contents/Abbott/Abbott.pdf",
abstract = "Computing accurate approximations to the perimeter of
an ellipse is a favorite problem of mathematicians,
attracting luminaries such as Ramanujan [1, 2, 3]. As
is well known, the perimeter {$P$} of an ellipse with
semimajor axis $a$ and semiminor axis $b$ can be
expressed exactly as a complete elliptic integral of
the second kind.\par
What is less well known is that the various exact forms
attributed to Maclaurin, Gauss--Kummer, and Euler are
related via quadratic hypergeometric transformations.
These transformations lead to additional identities,
including a particularly elegant formula symmetric in
$a$ and $b$.\par
Approximate formulas can, of course, be obtained by
truncating the series representations of exact
formulas. For example, Kepler used the geometric mean,
{$ P \approx 2 \pi \sqrt {ab} $}, as a lower bound for
the perimeter. In this article, we examine the
properties of a number of approximate formulas, using
series methods, polynomial interpolation, rational
polynomial approximants, and minimax methods.",
acknowledgement = ack-nhfb,
journal-URL = "http://www.mathematica-journal.com/",
remark = "Proceedings of the Eighth International Mathematica
Symposium (Avignon, France, June 19 -23, 2006).",
}
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15(z)z-4
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16(z)z-6
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14(z)z-7,
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2(2)58,
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