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.