Formula:KLS:01.06:10

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