Barnes integral

Remarks

Example: take $m = 1$, $a_1 = 0$, no other factors, $x > 0$, and $L$ the line $\operatorname{Re} s = c$ with $c > 0$. All poles $s = 0, -1, -2, \dots$ of $\Gamma(s)$ lie to the left of $L$, so $L$ is admissible. Shifting the line to the left and summing the residues $(-1)^\ell x^\ell / \ell!$ at $s = -\ell$ gives $$\frac{1}{2\pi i} \int_{c - i\infty}^{c + i\infty} \Gamma(s)\, x^{-s}\, ds = e^{-x}.$$ Non-example: the same integrand on the line $\operatorname{Re} s = -\tfrac12$ is not a Barnes integral in this sense, since the pole $s = 0$ lies on the wrong side. Its value is $e^{-x} - 1$, because the residue $1$ at $s = 0$ is lost. Barnes' representation of the hypergeometric function reads, for $\abs{\arg(-z)} < \pi$ and $a, b$ not in $\set{0, -1, -2, \dots}$, $$\frac{\Gamma(a)\Gamma(b)}{\Gamma(c)}\, {}_2F_1(a, b; c; z) = \frac{1}{2\pi i} \int_{-i\infty}^{i\infty} \frac{\Gamma(a + s)\Gamma(b + s)\Gamma(-s)}{\Gamma(c + s)} (-z)^s\, ds,$$ with the path separating the poles of $\Gamma(a + s)\Gamma(b + s)$ from those of $\Gamma(-s)$. The Meijer G-function and the Fox H-function are defined by integrals of this type.