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## Elliptic Integral modified with $q$-pochhammer symbol

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### Elliptic Integral modified with $q$-pochhammer symbol

Wed Jan 07, 2015 2:30 pm

Posts: 1
I was looking at the Complete Elliptic Integral of the First Kind a few days ago:

$K(k) = \int_{0}^{1} \frac{dx}{\sqrt{(1-t^{2})(1-k^{2}t^{2})}}$

And it occurred to me that the denominator kind of resembles the first two terms of the product defined by a $q$-Pochhammer symbol. The natural thing to do would then seem to be:

$V(k)= \int_{0}^{1} \frac{dt}{\sqrt{(t^{2} ; k^{2})_{\infty}}}$

I'd like to be able to extend results to ordinary elliptic integrals to this beast.

Preliminary experiments with elliptic integral singular values are promising:

$k_{3} = (\sqrt{6}-\sqrt{2})/4$:

I'd like to be able to evaluate $V(k)$ in terms of known special functions. But that square root complicates