Formula:KLS:01.06:09

From DRMF
Jump to navigation Jump to search


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

Proof

We ask users to provide proof(s), reference(s) to proof(s), or further clarification on the proof(s) in this space.

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

We ask users to provide relevant URL links in this space.


Retrieved from "https://drmf-beta.wmflabs.org/index.php?title=Formula:KLS:01.06:09&oldid=2979"