Mellin-Barnes integrals are characterised by integrands involving one or more gamma functions (and possibly simple trigonometric or other functions) with integration contours that thread their way around sequences of poles of the integrands. They are a powerful tool in the development of asymptotic expansions of functions defined by integrals, sums or differential equations and, combined with the closely related Mellin transform, form an important part of the toolkit of any practising analyst. The great utility of these integrals resides in the facts that the asymptotic behaviour near the origin and at infinity of the function being represented is related to the singularity structure in the complex plane of the resulting integrand and to the inherent flexibility associated with deformation of the contour of integration over subsets of these singularities.

It is a principal aim of this book to describe the theory of these integrals and to illustrate their power and usefulness in asymptotic analysis. Mellin-Barnes integrals have their early history bound up in the study of hypergeometric functions of the late nineteenth and early twentieth centuries. This association has lent a classical feel to their use and in the domain of asymptotic analysis, the account of their utility in other works has largely been restricted to the analysis of special sums or their role in inversion of Mellin transforms.