Formula:KLS:01.06:05: Difference between revisions

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