# Mellin–Barnes integrals

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##### 1: 5.19 Mathematical Applications
###### §5.19(ii) Mellin–BarnesIntegrals
Many special functions $f(z)$ can be represented as a MellinBarnes integral, that is, an integral of a product of gamma functions, reciprocals of gamma functions, and a power of $z$, the integration contour being doubly-infinite and eventually parallel to the imaginary axis at both ends. …
##### 2: Richard B. Paris
Wood), published by Longman Scientific and Technical in 1986, and Asymptotics and Mellin-Barnes Integrals (with D. …
##### 3: 8.6 Integral Representations
###### Mellin–BarnesIntegrals
8.6.10 $\gamma\left(a,z\right)=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\frac{% \Gamma\left(s\right)}{a-s}z^{a-s}\mathrm{d}s,$ $|\operatorname{ph}z|<\tfrac{1}{2}\pi$, $a\neq 0,-1,-2,\dots$,
8.6.11 $\Gamma\left(a,z\right)=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\Gamma\left% (s+a\right)\frac{z^{-s}}{s}\mathrm{d}s,$ $|\operatorname{ph}z|<\tfrac{1}{2}\pi$,
8.6.12 $\Gamma\left(a,z\right)=-\frac{z^{a-1}e^{-z}}{\Gamma\left(1-a\right)}\*\frac{1}% {2\pi i}\int_{c-i\infty}^{c+i\infty}\Gamma\left(s+1-a\right)\frac{\pi z^{-s}}{% \sin\left(\pi s\right)}\mathrm{d}s,$ $|\operatorname{ph}z|<\tfrac{3}{2}\pi$, $a\neq 1,2,3,\dots$.
##### 5: 35.8 Generalized Hypergeometric Functions of Matrix Argument
###### §35.8(v) Mellin–BarnesIntegrals
Multidimensional MellinBarnes integrals are established in Ding et al. (1996) for the functions ${{}_{p}F_{q}}$ and ${{}_{p+1}F_{p}}$ of matrix argument. …These multidimensional integrals reduce to the classical MellinBarnes integrals5.19(ii)) in the special case $m=1$. …
##### 7: Bibliography P
• R. B. Paris and D. Kaminski (2001) Asymptotics and Mellin-Barnes Integrals. Cambridge University Press, Cambridge.
• R. B. Paris (1992b) Smoothing of the Stokes phenomenon using Mellin-Barnes integrals. J. Comput. Appl. Math. 41 (1-2), pp. 117–133.