Some integrals
Some integrals
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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 }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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 }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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}} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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}} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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}} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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}} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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}} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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}} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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}} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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}} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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} }}