Barnes’ integral representation of the hypergeometric function

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

$${\int }_C\frac{\mathrm{\Gamma }(a+s)\mathrm{\Gamma }(b+s)}{\mathrm{\Gamma }(c+s)}\mathrm{\Gamma }(-s){(-z)}^s𝑑s=2\pi i\frac{\mathrm{\Gamma }(a)\mathrm{\Gamma }(b)}{\mathrm{\Gamma }(c)}F(a,b;c;z)$$

Title	Barnes’ integral representation of the hypergeometric function
Canonical name	BarnesIntegralRepresentationOfTheHypergeometricFunction
Date of creation	2013-03-22 17:36:15
Last modified on	2013-03-22 17:36:15
Owner	rspuzio (6075)
Last modified by	rspuzio (6075)
Numerical id	4
Author	rspuzio (6075)
Entry type	Theorem
Classification	msc 33C05