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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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    $\begingroup$ This is Chapter 6.1 of Gradshteyn, I.S. & Ryzhik, I.M. (2007) "Table of Integrals, Series and Products", 7th edn., Amsterdam: Academic Press. $\endgroup$
    – Aleksey Pichugin
    Commented Aug 30, 2010 at 21:24
  • $\begingroup$ 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. $\endgroup$
    – J. M. isn't a mathematician
    Commented Aug 30, 2010 at 22:24
  • $\begingroup$ The first and second one can be handled by the techniques in this paper: repository.cmu.edu/cgi/… ; the last one does not seem to have a nice closed form. $\endgroup$
    – J. M. isn't a mathematician
    Commented Aug 31, 2010 at 7:42
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    $\begingroup$ J. Mangaldan, all of these integrals are available in Gradshteyn & Ryzhik. In particular, the last one is given by eq. (6.144): $$ \int_0^1\frac{K(k)}{1+k}dk=\frac{\pi^2}{8}. $$ $\endgroup$
    – Aleksey Pichugin
    Commented Aug 31, 2010 at 8:43
  • $\begingroup$ 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! $\endgroup$
    – J. M. isn't a mathematician
    Commented Aug 31, 2010 at 12:10

2 Answers 2

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Question bumped to the home page by Community ...
No answer yet, I guess. (Dan has not been here for over 6 years, so I doubt he will ever choose an answer.)

Gradshteyn & Ryzhik

6.141.1: $$ \int_0^1 \mathbf{K}(k)\;dk = 2G $$ where $G$ is Catalan's constant.

6.144: $$ \int_0^1 \frac{\mathbf{K}(k)}{1+k}\;dk = \frac{\pi^2}{8} $$

6.146 is a reduction formula, reducing the exponent by $2$: $$ n^2 \int_0^1 k^n \mathbf{K}(k)\;dk = (n-1)^2\int_0^1 k^{n-2} \mathbf{K}(k)\;dk + 1 $$

I don't see the odd powers of $k$ directly. Maybe you can get them using

6.142.1: $$ \int_0^1 \left(\mathbf{K}(k) - \frac{\pi}{2}\right)\;\frac{dk}{k} = \pi\log 2 - 2 G, $$ 6.142.2: $$ \int_0^1 \left(\mathbf{K}(k) - \frac{\pi}{2}\right)\;\frac{dk}{k^2} = \frac{\pi}{2} - 1 $$

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A. Dieckmann compiled a big table of indefinite and definite integrals envolving non-elementary functions. See the following links:

http://www-elsa.physik.uni-bonn.de/~dieckman/IntegralsIndefinite/IndefInt.html

and

http://www-elsa.physik.uni-bonn.de/~dieckman/IntegralsDefinite/DefInt.html

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