\HyperpFq 21 @ @ a , b c z = Γ ( c ) Γ ( b ) Γ ( c - b ) ∫ 0 1 t b - 1 ( 1 - t ) c - b - 1 ( 1 - z t ) - a 𝑑 t \HyperpFq 21 @ @ 𝑎 𝑏 𝑐 𝑧 Euler-Gamma 𝑐 Euler-Gamma 𝑏 Euler-Gamma 𝑐 𝑏 superscript subscript 0 1 superscript 𝑡 𝑏 1 superscript 1 𝑡 𝑐 𝑏 1 superscript 1 𝑧 𝑡 𝑎 differential-d 𝑡 {\displaystyle{\displaystyle{\displaystyle{}{}\HyperpFq{2}{1}@@{a,b}{c}{z}=% \frac{\Gamma\left(c\right)}{\Gamma\left(b\right)\Gamma\left(c-b\right)}\int_{0% }^{1}t^{b-1}(1-t)^{c-b-1}(1-zt)^{-a}\,dt}}} {\displaystyle \index{Euler's integral representation for a $\HyperpFq{2}{1}$}\index{Hypergeometric function!Euler's integral representation} \HyperpFq{2}{1}@@{a,b}{c}{z}=\frac{\EulerGamma@{c}}{\EulerGamma@{b}\EulerGamma@{c-b}} \int_0^1t^{b-1}(1-t)^{c-b-1}(1-zt)^{-a}\,dt } Γ ( a ) Γ ( b ) Γ ( c ) \HyperpFq 21 @ @ a , b c z = 1 2 π i ∫ - i ∞ i ∞ Γ ( a + s ) Γ ( b + s ) Γ ( - s ) Γ ( c + s ) ( - z ) s 𝑑 s Euler-Gamma 𝑎 Euler-Gamma 𝑏 Euler-Gamma 𝑐 \HyperpFq 21 @ @ 𝑎 𝑏 𝑐 𝑧 1 2 imaginary-unit superscript subscript imaginary-unit imaginary-unit Euler-Gamma 𝑎 𝑠 Euler-Gamma 𝑏 𝑠 Euler-Gamma 𝑠 Euler-Gamma 𝑐 𝑠 superscript 𝑧 𝑠 differential-d 𝑠 {\displaystyle{\displaystyle{\displaystyle{}{}\frac{\Gamma\left(a\right)\Gamma% \left(b\right)}{\Gamma\left(c\right)}\,\HyperpFq{2}{1}@@{a,b}{c}{z}=\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(-s\right)}{\Gamma\left(c+s\right)}(-z% )^{s}\,ds}}} {\displaystyle \index{Barnes' integral representation}\index{Hypergeometric function!Barnes' integral representation} \frac{\EulerGamma@{a}\EulerGamma@{b}}{\EulerGamma@{c}}\,\HyperpFq{2}{1}@@{a,b}{c}{z}=\frac{1}{2\cpi \iunit} \int_{-\iunit\infty}^{\iunit\infty}\frac{\EulerGamma@{a+s}\EulerGamma@{b+s}\EulerGamma@{-s}}{\EulerGamma@{c+s}}(-z)^s\,ds } 1 2 π i ∫ - i ∞ i ∞ Γ ( a + s ) Γ ( b + s ) Γ ( c - s ) Γ ( d - s ) 𝑑 s = Γ ( a + c ) Γ ( a + d ) Γ ( b + c ) Γ ( b + 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 differential-d 𝑠 Euler-Gamma 𝑎 𝑐 Euler-Gamma 𝑎 𝑑 Euler-Gamma 𝑏 𝑐 Euler-Gamma 𝑏 𝑑 Euler-Gamma 𝑎 𝑏 𝑐 𝑑 {\displaystyle{\displaystyle{\displaystyle{}{}\frac{1}{2\pi\mathrm{i}}\int_{-% \mathrm{i}\infty}^{\mathrm{i}\infty}\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+% c\right)\Gamma\left(a+d\right)\Gamma\left(b+c\right)\Gamma\left(b+d\right)}{% \Gamma\left(a+b+c+d\right)}}}} {\displaystyle \index{Mellin-Barnes integral}\index{Barnes' first lemma} \frac{1}{2\cpi \iunit}\int_{-\iunit\infty}^{\iunit\infty}\EulerGamma@{a+s}\EulerGamma@{b+s}\EulerGamma@{c-s}\EulerGamma@{d-s}\,ds {}=\frac{\EulerGamma@{a+c}\EulerGamma@{a+d}\EulerGamma@{b+c}\EulerGamma@{b+d}}{\EulerGamma@{a+b+c+d}} } 1 2 π i ∫ - i ∞ i ∞ Γ ( a + s ) Γ ( b + s ) Γ ( c + s ) Γ ( 1 - d - s ) Γ ( - s ) Γ ( e + s ) 𝑑 s = Γ ( a ) Γ ( b ) Γ ( c ) Γ ( 1 + a - d ) Γ ( 1 + b - d ) Γ ( 1 + c - d ) Γ ( e - a ) Γ ( e - b ) Γ ( e - c ) 1 2 imaginary-unit superscript subscript imaginary-unit imaginary-unit Euler-Gamma 𝑎 𝑠 Euler-Gamma 𝑏 𝑠 Euler-Gamma 𝑐 𝑠 Euler-Gamma 1 𝑑 𝑠 Euler-Gamma 𝑠 Euler-Gamma 𝑒 𝑠 differential-d 𝑠 Euler-Gamma 𝑎 Euler-Gamma 𝑏 Euler-Gamma 𝑐 Euler-Gamma 1 𝑎 𝑑 Euler-Gamma 1 𝑏 𝑑 Euler-Gamma 1 𝑐 𝑑 Euler-Gamma 𝑒 𝑎 Euler-Gamma 𝑒 𝑏 Euler-Gamma 𝑒 𝑐 {\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(1-d-s\right)\Gamma\left(-s\right)}{% \Gamma\left(e+s\right)}\,ds{}=\frac{\Gamma\left(a\right)\Gamma\left(b\right)% \Gamma\left(c\right)\Gamma\left(1+a-d\right)\Gamma\left(1+b-d\right)\Gamma% \left(1+c-d\right)}{\Gamma\left(e-a\right)\Gamma\left(e-b\right)\Gamma\left(e-% c\right)}}}} {\displaystyle \frac{1}{2\cpi \iunit}\int_{-\iunit\infty}^{\iunit\infty} \frac{\EulerGamma@{a+s}\EulerGamma@{b+s}\EulerGamma@{c+s}\EulerGamma@{1-d-s}\EulerGamma@{-s}}{\EulerGamma@{e+s}}\,ds {}=\frac{\EulerGamma@{a}\EulerGamma@{b}\EulerGamma@{c}\EulerGamma@{1+a-d}\EulerGamma@{1+b-d}\EulerGamma@{1+c-d}} {\EulerGamma@{e-a}\EulerGamma@{e-b}\EulerGamma@{e-c}} } 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)}}}} {\displaystyle \index{Bailey's integral} \frac{1}{2\cpi \iunit}\int_{-\iunit\infty}^{\iunit\infty} \frac{\EulerGamma@{1+a/2+s}\EulerGamma@{a+s}\EulerGamma@{b+s}\EulerGamma@{c+s}\EulerGamma@{d+s}\EulerGamma@{b-a-s}\EulerGamma@{-s}} {\EulerGamma@{a/2+s}\EulerGamma@{1+a-c+s}\EulerGamma@{1+a-d+s}}\,ds {}=\frac{\EulerGamma@{b}\EulerGamma@{c}\EulerGamma@{d}\EulerGamma@{b+c-a}\EulerGamma@{b+d-a}} {2\,\EulerGamma@{1+a-c-d}\EulerGamma@{b+c+d-a}} } 1 2 π i ∫ - i ∞ i ∞ Γ ( a + s ) Γ ( b + s ) Γ ( c + s ) Γ ( d + s ) Γ ( a + 2 s ) Γ ( - s ) Γ ( b - a - s ) Γ ( c - a - s ) Γ ( d - a - s ) Γ ( - a - 2 s ) 𝑑 s = 2 Γ ( b ) Γ ( c ) Γ ( d ) Γ ( b + c - a ) Γ ( b + d - a ) Γ ( c + d - a ) Γ ( b + c + d - 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 𝑑 𝑠 Euler-Gamma 𝑎 2 𝑠 Euler-Gamma 𝑠 Euler-Gamma 𝑏 𝑎 𝑠 Euler-Gamma 𝑐 𝑎 𝑠 Euler-Gamma 𝑑 𝑎 𝑠 Euler-Gamma 𝑎 2 𝑠 differential-d 𝑠 2 Euler-Gamma 𝑏 Euler-Gamma 𝑐 Euler-Gamma 𝑑 Euler-Gamma 𝑏 𝑐 𝑎 Euler-Gamma 𝑏 𝑑 𝑎 Euler-Gamma 𝑐 𝑑 𝑎 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(a+s\right)\Gamma\left(b+% s\right)\Gamma\left(c+s\right)\Gamma\left(d+s\right)}{\Gamma\left(a+2s\right)}% {}\frac{\Gamma\left(-s\right)\Gamma\left(b-a-s\right)\Gamma\left(c-a-s\right)% \Gamma\left(d-a-s\right)}{\Gamma\left(-a-2s\right)}\,ds{}=\frac{2\,\Gamma\left% (b\right)\Gamma\left(c\right)\Gamma\left(d\right)\Gamma\left(b+c-a\right)% \Gamma\left(b+d-a\right)\Gamma\left(c+d-a\right)}{\Gamma\left(b+c+d-a\right)}}}} {\displaystyle \frac{1}{2\cpi \iunit}\int_{-\iunit\infty}^{\iunit\infty} \frac{\EulerGamma@{a+s}\EulerGamma@{b+s}\EulerGamma@{c+s}\EulerGamma@{d+s}}{\EulerGamma@{a+2s}} {}\frac{\EulerGamma@{-s}\EulerGamma@{b-a-s}\EulerGamma@{c-a-s}\EulerGamma@{d-a-s}}{\EulerGamma@{-a-2s}}\,ds {}=\frac{2\,\EulerGamma@{b}\EulerGamma@{c}\EulerGamma@{d}\EulerGamma@{b+c-a}\EulerGamma@{b+d-a}\EulerGamma@{c+d-a}}{\EulerGamma@{b+c+d-a}} } 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)}}}} {\displaystyle \frac{1}{2\cpi \iunit}\int_{-\iunit\infty}^{\iunit\infty} \frac{\EulerGamma@{a+s}\EulerGamma@{b+s}\EulerGamma@{c+s}\EulerGamma@{d+s}\EulerGamma@{a-s}\EulerGamma@{b-s}\EulerGamma@{c-s}\EulerGamma@{d-s}} {\EulerGamma@{2s}\EulerGamma@{-2s}}\,ds {}=\frac{2\,\EulerGamma@{a+b}\EulerGamma@{a+c}\EulerGamma@{a+d}\EulerGamma@{b+c}\EulerGamma@{b+d}\EulerGamma@{c+d}}{\EulerGamma@{a+b+c+d}} } 1 2 π i ∫ - i ∞ i ∞ Γ ( a + s ) Γ ( b - s ) z - s 𝑑 s = Γ ( a + b ) z a ( 1 + z ) a + b 1 2 imaginary-unit superscript subscript imaginary-unit imaginary-unit Euler-Gamma 𝑎 𝑠 Euler-Gamma 𝑏 𝑠 superscript 𝑧 𝑠 differential-d 𝑠 Euler-Gamma 𝑎 𝑏 superscript 𝑧 𝑎 superscript 1 𝑧 𝑎 𝑏 {\displaystyle{\displaystyle{\displaystyle\frac{1}{2\pi\mathrm{i}}\int_{-% \mathrm{i}\infty}^{\mathrm{i}\infty}\Gamma\left(a+s\right)\Gamma\left(b-s% \right)z^{-s}\,ds=\Gamma\left(a+b\right)\frac{z^{a}}{(1+z)^{a+b}}}}} {\displaystyle \frac{1}{2\cpi \iunit}\int_{-\iunit\infty}^{\iunit\infty}\EulerGamma@{a+s}\EulerGamma@{b-s}z^{-s}\,ds =\EulerGamma@{a+b}\frac{z^a}{(1+z)^{a+b}} } 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)}}}} {\displaystyle \frac{1}{2\cpi \iunit}\int_{-\iunit\infty}^{\iunit\infty}\frac{\EulerGamma@{a+s}\EulerGamma@{b-s}}{\EulerGamma@{c+s}\EulerGamma@{d-s}}\,ds =\frac{\EulerGamma@{a+b}\EulerGamma@{c+d-a-b-1}}{\EulerGamma@{c+d-1}\EulerGamma@{c-a}\EulerGamma@{d-b}} } 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)% }}} {\displaystyle \frac{1}{2\cpi \iunit}\int_{-\iunit\infty}^{\iunit\infty} \frac{\EulerGamma@{a+s}\EulerGamma@{b+s}\EulerGamma@{c+s}\EulerGamma@{-s}\EulerGamma@{b-a-s}\EulerGamma@{c-a-s}} {\EulerGamma@{a+2s}\EulerGamma@{-a-2s}}\,ds {}=\frac{1}{2}\EulerGamma@{b}\EulerGamma@{c}\EulerGamma@{b+c-a} } 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)}}}} {\displaystyle \index{Wilson's integral} \frac{1}{2\cpi}\int_0^{\infty} \left|\frac{\EulerGamma@{a+\iunit x}\EulerGamma@{b+\iunit x}\EulerGamma@{c+\iunit x}\EulerGamma@{d+\iunit x}}{\EulerGamma@{2\iunit x}}\right|^2\,dx {}=\frac{\EulerGamma@{a+b}\EulerGamma@{a+c}\EulerGamma@{a+d}\EulerGamma@{b+c}\EulerGamma@{b+d}\EulerGamma@{c+d}}{\EulerGamma@{a+b+c+d}} } 1 2 π ∫ 0 ∞ | Γ ( a + i x ) Γ ( b + i x ) Γ ( c + i x ) Γ ( 2 i x ) | 2 𝑑 x = Γ ( a + b ) Γ ( a + c ) Γ ( b + c ) 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 2 imaginary-unit 𝑥 2 differential-d 𝑥 Euler-Gamma 𝑎 𝑏 Euler-Gamma 𝑎 𝑐 Euler-Gamma 𝑏 𝑐 {\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(2\mathrm{i}x\right)}\right|^{2}\,dx=% \Gamma\left(a+b\right)\Gamma\left(a+c\right)\Gamma\left(b+c\right)}}} {\displaystyle \frac{1}{2\cpi}\int_0^{\infty} \left|\frac{\EulerGamma@{a+\iunit x}\EulerGamma@{b+\iunit x}\EulerGamma@{c+\iunit x}}{\EulerGamma@{2\iunit x}}\right|^2\,dx =\EulerGamma@{a+b}\EulerGamma@{a+c}\EulerGamma@{b+c} }