The Stokes Phenomenon

The Stokes phenomenon plays a central role in the asymptotic description of the special functions of analysis and in the important class of asymptotic phenomena in mathematical physics known as ‘discontinuities’. Such discontinuities, along with those of a different origin arising in boundary layer theory and shocks in gas dynamics, are not true discontinuities in the mathematical sense but are found to depend on the scale on which the phenomenon in question is considered. When viewed under an appropriate magnification these discontinuities appear as a smooth albeit rapid transition from one form of description to another. An interesting discussion of such asymptotic phenomena is given in Friedrichs (1955).

A typical example related to the Stokes phenomenon is the boundary of the shadow which appears when a light wave passes an object. As a first approximation the shadow boundary is a sharp discontinuity, but on a smaller scale there is a transition from light to darkness which takes place in a narrow region along the shadow boundary. This corresponds to the asymptotic expansion of the solution of the wave equation taking on different forms across this boundary. Another example is the change in asymptotic form of a wave function at a caustic surface where the eikonal in the WKBJ description of the wave equation develops a singularity. Indeed, the first account of this phenomenon by Stokes in 1857 arose in this last connection in the theory of the rainbow and the formation of supernumerary arcs.

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.