Rapidly convergent representations for Green's functions for Laplace's equation

Representations for Green's functions for Laplace's equation in domains with infinite boundaries are obtained by integrating solutions to appropriate heat-conduction problems with respect to time. By using different representations for these heat-equation solutions for small and large times, the changeover being determined by an arbitrary positive parameter a, a one-parameter family of formulae for the required Green's function is derived and by varying a the convergence characteristics of this new representation can be altered. Letting a --> 0 results in known eigenfunction expansions and, in those situations in which they exist, letting a --> 0 recovers known image-series representations. The method, which is essentially equivalent to Ewald summation, is applied to two types of problem. First, it is applied to potential flow between parallel planes and in a rectangular channel, and, second, to two- and three-dimensional water-wave problems in which the depth is constant. In all cases the results of computations are presented showing the accuracy and efficiency of the resulting formulae.