Closed form for ∫∞0(∫101√1−y2√1+x2y2dy)3dx.\int_0^\infty\left(\int_0^1\frac1{\sqrt{1-y^2}\sqrt{1+x^2\,y^2}}\mathrm dy\right)^3\mathrm dx.

I need to find a closed form for these nested definite integrals:
I=0(1011y21+x2y2dy)3dx.
The inner integral can be represented using the hypergeometric function 2F1 or the complete elliptic integral of the 1st kind K with an imaginary argument:
1011y21+x2y2dy=π22F1(12,12;1;x2)=K(x1).
My conjecture is the integral I has a closed-form representation:I?=3Γ(14)81280π2=7.09022700484626946098980237…,
but I was neither able to find a proof of it, nor disprove the equality using numerical integration. Could you please help me with resolving this question?

Answer

Using
K(ik)=11+k2K(k2k2+1)
and a substitution t2=k21+k2, rewrite the integral as
0K(ik)3dk=10K(t)3dt.

There is a paper “Moments of elliptic integrals and critical L-values”
by Rogers, Wan and Zucker (http://arxiv.org/abs/1303.2259; also one of
the authors’ earlier papers: http://arxiv.org/abs/1101.1132), and the
authors, by relating this integral to an L-series of a modular form (their theorems 1 and 2),
show that
10K(k)3dk=35K(1/2)4=3Γ(14)81280π2,
using K(1/2)=14π1/2Γ(14)2.

Attribution
Source : Link , Question Author : Vladimir Reshetnikov , Answer Author : Kirill

Leave a Comment