# Barnes’ integral representation of the hypergeometric function

When $a,b,c,d$ are complex numbers and $z$ is a complex number such that $-\pi<\arg(-z)<+\pi$ and $C$ is a contour in the complex $s$-plane which goes from $-i\infty$ to $+i\infty$ chosen such that the poles of $\Gamma(a+s)\Gamma(b+s)$ lie to the left of $C$ and the poles of $\Gamma(-s)$ lie to the right of $C$, then

 $\int_{C}{\Gamma(a+s)\Gamma(b+s)\over\Gamma(c+s)}\Gamma(-s)(-z)^{s}\,ds=2\pi i{% \Gamma(a)\Gamma(b)\over\Gamma(c)}F(a,b;c;z)$
Title Barnes’ integral representation of the hypergeometric function BarnesIntegralRepresentationOfTheHypergeometricFunction 2013-03-22 17:36:15 2013-03-22 17:36:15 rspuzio (6075) rspuzio (6075) 4 rspuzio (6075) Theorem msc 33C05