Formula:KLS:01.06:11

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1 2 ⁢ π ⁢ ∫ 0 ∞ | Γ ⁡ ( a + i ⁢ x ) ⁢ Γ ⁡ ( b + i ⁢ x ) ⁢ Γ ⁡ ( c + i ⁢ x ) ⁢ Γ ⁡ ( d + i ⁢ x ) Γ ⁡ ( 2 ⁢ i ⁢ x ) | 2 ⁢ 𝑑 x = 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 superscript subscript 0 superscript Euler-Gamma 𝑎 imaginary-unit 𝑥 Euler-Gamma 𝑏 imaginary-unit 𝑥 Euler-Gamma 𝑐 imaginary-unit 𝑥 Euler-Gamma 𝑑 imaginary-unit 𝑥 Euler-Gamma 2 imaginary-unit 𝑥 2 differential-d 𝑥 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}\int_{0}^{\infty}% \left|\frac{\Gamma\left(a+\mathrm{i}x\right)\Gamma\left(b+\mathrm{i}x\right)% \Gamma\left(c+\mathrm{i}x\right)\Gamma\left(d+\mathrm{i}x\right)}{\Gamma\left(% 2\mathrm{i}x\right)}\right|^{2}\,dx{}=\frac{\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

∫ {\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
i i {\displaystyle{\displaystyle{\displaystyle\mathrm{i}}}}  : imaginary unit : http://dlmf.nist.gov/1.9.i

Bibliography

Equation in Section 1.6 of KLS.

URL links

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