Let $c$ be a constant in ${\bf R}^t$. For a plane algebraic curve $r^{2m-n} = 2c^n \cos
n\theta$, which depends on $m$ and $n$ in ${\bf N}$, we show that the whole length of the
curve are given by a value of a product formula defined by the Beta function and Gauss's
hypergeometric function depending $m$ and $n$ in ${\bf N}$. Besides, we point out the fact
to be a similar model and an expansion for the complete elliptic integral of the second
kind. Last, we give a background for the fact explaining the special case $m = n$.
Publié le : 2010-08-15
Classification:
beta function,
hypergeometric function,
transcendental number and the complete elliptic integral of the second
kind,
11A67,
33C75,
33B15,
33C05
@article{1283967405,
author = {Ogawa, Takuma and Kamata, Yasuo},
title = {A product formula defined by the Beta function and Gauss's
hypergeometric function},
journal = {Tsukuba J. Math.},
volume = {34},
number = {1},
year = {2010},
pages = { 13-30},
language = {en},
url = {http://dml.mathdoc.fr/item/1283967405}
}
Ogawa, Takuma; Kamata, Yasuo. A product formula defined by the Beta function and Gauss's
hypergeometric function. Tsukuba J. Math., Tome 34 (2010) no. 1, pp. 13-30. http://gdmltest.u-ga.fr/item/1283967405/