when i'm using Euler's reflection formula $\Gamma(1/4)$ appears which i'm unable to solve again
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2Have a look at https://oeis.org/A068466. It is not rational. – Claude Leibovici Apr 15 '17 at 08:15
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so it's not possible to have any explicit formula ? @ClaudeLeibovici – Siddhartha Apr 15 '17 at 08:53
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1The comment in OEIS says its transcendental which would rule out an explicit formula. – Ian Miller Apr 15 '17 at 09:21
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@IanMiller At least not the normal kind of formula. – Simply Beautiful Art Apr 15 '17 at 11:11
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@ClaudeLeibovici The Gamma function of rational numbers tend to include $\pi$ when known, so it should be expected to be irrational either way. – Simply Beautiful Art Apr 15 '17 at 11:18
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This PDF is great for finding these values real quick. Taking a quick look, we have:
$$\Gamma\left(\frac14\right)=2\sqrt[4]\pi\sqrt{K\left(\frac1{\sqrt2}\right)}$$
where $K()$ is an elliptic integral of the first kind.
Simply Beautiful Art
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1Thanks for the link ! But take care about how is defined the elliptic integral. Depending on convention it could also be $$\Gamma \left(\frac{1}{4}\right)=2 \sqrt[4]{\pi } \sqrt{K\left(\frac{1}{2}\right)}$$ – Claude Leibovici Apr 15 '17 at 14:12