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Is there any reference showing how to do definite integrals involving the complete elliptic integral of the first kind K(k)? Something like

  1. $\int_0^1 K(k) dk $
  2. $\int_0^1 k^nK(k) dk$
  3. $\int_0^1 \frac{K(k)}{1+k} dk $

etc...Thanks a lot.

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This is Chapter 6.1 of Gradshteyn, I.S. & Ryzhik, I.M. (2007) "Table of Integrals, Series and Products", 7th edn., Amsterdam: Academic Press. – Aleksey Pichugin Aug 30 '10 at 21:24
Anyway... it's good you (seem to have) specified that you were using the modulus $k$ instead of the parameter $m=k^2$ as the argument for your elliptic integrals. – J. M. Aug 30 '10 at 22:24
The first and second one can be handled by the techniques in this paper:… ; the last one does not seem to have a nice closed form. – J. M. Aug 31 '10 at 7:42
J. Mangaldan, all of these integrals are available in Gradshteyn & Ryzhik. In particular, the last one is given by eq. (6.144): $$ \int_0^1\frac{K(k)}{1+k}dk=\frac{\pi^2}{8}. $$ – Aleksey Pichugin Aug 31 '10 at 8:43
Hmm, too bad I didn't have my copy of G&R nearby to check the third one, but the first and second one are easily handled as integrals of hypergeometric functions. Thanks for the pointer Aleksey! – J. M. Aug 31 '10 at 12:10

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