Formula:KLS:01.06:06

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formula in Some integrals

Formula:KLS:01.06:07 >>

${\displaystyle{\displaystyle{\displaystyle\frac{1}{2\pi\mathrm{i}}\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\frac{\Gamma\left(a+s\right)\Gamma\left(b+s\right)\Gamma\left(c+s\right)\Gamma\left(d+s\right)}{\Gamma\left(a+2s\right)}{}\frac{\Gamma\left(-s\right)\Gamma\left(b-a-s\right)\Gamma\left(c-a-s\right)\Gamma\left(d-a-s\right)}{\Gamma\left(-a-2s\right)}\,ds{}=\frac{2\,\Gamma\left(b\right)\Gamma\left(c\right)\Gamma\left(d\right)\Gamma\left(b+c-a\right)\Gamma\left(b+d-a\right)\Gamma\left(c+d-a\right)}{\Gamma\left(b+c+d-a\right)}}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle\pi}}}$ : ratio of a circle's circumference to its diameter : http://dlmf.nist.gov/5.19.E4
${\displaystyle{\displaystyle{\displaystyle\int}}}$ : integral : http://dlmf.nist.gov/1.4#iv
${\displaystyle{\displaystyle{\displaystyle\Gamma}}}$ : Euler's gamma function : http://dlmf.nist.gov/5.2#E1

Bibliography

Equation in Section 1.6 of KLS.

URL links

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<< Formula:KLS:01.06:05

formula in Some integrals

Formula:KLS:01.06:07 >>