integration of jacobian elliptical functions

On Aug 3, 9:41 pm, Badger wrote:
On Fri, 03 Aug 2007 08:11:51 -0700, wrote:
Hello,

I have a question about integrating Jacobian elliptical functions,
specifically,
\int_0^2K cn^2(u|k) du,
where K is the complete Jacobian elliptical integral of the first
kind. I am interested in finding the average value of cn^2(u,k) over
its cycle.

My interpretation of Abramowitz and Stegun suggests that this integral
should evaluate to
(2/k) [E + (1-k) K]

[snip]

Looking at A&S, I don't get your result for the integral. Assuming
your k is the m used in A&S, I got

2/k [ E - (1-k) K ]

int_0^x cn^2 x dx = (1/k^2) [E(am x, k) - (k')^2 x]

This is the correct intgral function in question.
Calculate rest by yourself.

Page 77, the formula 585, in the book:

Peirce,B.O., Foster, R.M., 1956.
A Short Table of Integrals.
4.th edition, Ginn and Company,
Printed in the United States of America. 189 pages.
Pages 89-91, pages 71-77 and pages 133-135 (more details).

I hope I could help.

Hannu