Maybe this will be useful:

http://mathworld.wolfram.com/EllipticIntegralSingularValue.html

http://mathworld.wolfram.com/EllipticLambdaFunction.html

Maybe this will be useful:

http://mathworld.wolfram.com/EllipticIntegralSingularValue.html

http://mathworld.wolfram.com/EllipticLambdaFunction.html

Thanks Maksim for your help.

I indeed was aware of these sites that partially address my topic. I need to know how to perform the transformation. I am speculating that the work of Selberg in which he generalized Euler Beta function for multi variables is the starting point to evaluate the integral. The result is always a product of several Gamma functions, quite similar to the results of the elliptic integrals. Any clue about that topic??

Kind Regards

Hi Hady,

you are familiar with the work of Glasser and Wood (1971)? See the link below.

Best regards

Herbert

http://www.ams.org/journals/mcom/1971-25-115/S0025-5718-71-99714-6/S0025-5718-71-99714-6.pdf

Hi Herbert,

 I was definitely aware of the Work of Glasser and Wood (1971). It is actually what you first get on Google when you type the question I asked. However I have the following comments on this paper:

1 - If you read the part just after equation (7), it says: "...appear to represent the ONLY real cases where the elliptic integral has been expressed explicitly in terms of gamma functions" . They give only three examples however, there are many more (infinitely many?) cases where elliptic integrals can be expressed in terms of Gamma Function.

See for instance the mathworld site suggested by Maksim (answer #1).

It appears to me that this paper was written in 1971. Thus the general procedure to obtain the integral was discovered later.

2 - This paper finds the relation for the case k = sqrt(2) -1, by the use of Hypergeometric function. I doubt if this procedure can be applied to find the results for other values of "k" whise K(k) can be expressed in terms of Gamma function.

3 - The paper states: "The list can be extended somewhat by use of Landen's transformation." No justification is given regarding this statement.

All in all, the paper addresses the question but it does not target the general case.  I am speculating that the "Selberg Integral Formula" along with Dedekind Eta function could form a solution to my question.

Do you, by chance, have a solution to Legendre proof for the case k = (sqrt(6) - sqrt(2))/4? I would like to have a look at it.

Thanks for your input to my question 

In addition to the excellent suggestions already made, perhaps you will find the following helpful.

The evaluation of several elliptic integrals is presented in a detailed fashion in Section 7.8, starting on page 132 (see evaluation in terms of the Gamma function, starting on page 134) in

J.G. feng Wan, Random walks, elliptic integrals and related constants, Ph.D. thesis, University of New Castle, 2013:

http://docserver.carma.newcastle.edu.au/1487/1/JWthesis_rev.pdf

Answer to Elliptic Integral: If you look to the book : Special functions by E. D. Rainville, then on p.71 these integrals are stated as exercises whose out come (evaluation) is shown in the form of hypergeometric series 2F1 with argument k^2. If this series, either by truncating or by any other way, is summed up then one would find the evaluation in 'k'.  I think there one need not evaluate Gamma function separately. But the value of to be taken should be such that |k|