Formula:KLS:01.06:07

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1 2 ⁢ π ⁢ i ⁢ ∫ - i ⁢ ∞ i ⁢ ∞ Γ ⁡ ( a + s ) ⁢ Γ ⁡ ( b + s ) ⁢ Γ ⁡ ( c + s ) ⁢ Γ ⁡ ( d + s ) ⁢ Γ ⁡ ( a - s ) ⁢ Γ ⁡ ( b - s ) ⁢ Γ ⁡ ( c - s ) ⁢ Γ ⁡ ( d - s ) Γ ⁡ ( 2 ⁢ s ) ⁢ Γ ⁡ ( - 2 ⁢ s ) ⁢ 𝑑 s = 2 ⁢ Γ ⁡ ( a + b ) ⁢ Γ ⁡ ( a + c ) ⁢ Γ ⁡ ( a + d ) ⁢ Γ ⁡ ( b + c ) ⁢ Γ ⁡ ( b + d ) ⁢ Γ ⁡ ( c + d ) Γ ⁡ ( a + b + c + d ) 1 2 imaginary-unit superscript subscript imaginary-unit imaginary-unit Euler-Gamma 𝑎 𝑠 Euler-Gamma 𝑏 𝑠 Euler-Gamma 𝑐 𝑠 Euler-Gamma 𝑑 𝑠 Euler-Gamma 𝑎 𝑠 Euler-Gamma 𝑏 𝑠 Euler-Gamma 𝑐 𝑠 Euler-Gamma 𝑑 𝑠 Euler-Gamma 2 𝑠 Euler-Gamma 2 𝑠 differential-d 𝑠 2 Euler-Gamma 𝑎 𝑏 Euler-Gamma 𝑎 𝑐 Euler-Gamma 𝑎 𝑑 Euler-Gamma 𝑏 𝑐 Euler-Gamma 𝑏 𝑑 Euler-Gamma 𝑐 𝑑 Euler-Gamma 𝑎 𝑏 𝑐 𝑑 {\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-s\right)% \Gamma\left(b-s\right)\Gamma\left(c-s\right)\Gamma\left(d-s\right)}{\Gamma% \left(2s\right)\Gamma\left(-2s\right)}\,ds{}=\frac{2\,\Gamma\left(a+b\right)% \Gamma\left(a+c\right)\Gamma\left(a+d\right)\Gamma\left(b+c\right)\Gamma\left(% b+d\right)\Gamma\left(c+d\right)}{\Gamma\left(a+b+c+d\right)}}}}

Proof

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Symbols List

π 𝜋 {\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
Γ Γ {\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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