The asymptotic behaviour of Ramanujan's integral and its application to two-dimensional diffusion-like equations

The large-time behaviour of a large class of solutions to the two-dimension al linear diffusion equation in situations with radial symmetry is governed by the function known as Ramanujan's integral. This is also true when the diffusion coefficient is complex, which corresponds to Schrodinger's equati on. We examine the asymptotic expansion of Ramanujan's integral for large v alues of its argument over the whole complex plane by considering the analy tic continuation of Ramanujan's integral to the left half-plane. The result ing expansions are compared to accurate numerical computations of the integ ral. The large-time behaviour derived from Ramanujan's integral of the solu tion to the diffusion equation outside a cylinder is not valid far from the domain boundary. A simple method based on matched asymptotic expansions is outlined to calculate the solution at large times and distances: the resul ting form of the solution combines the inverse logarithmic decay in time ty pical of Ramanujan's integral with spatial dependence on the usual similari ty variable for the diffusion equation.