Representations for Green's functions for Laplace' 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' function is derived and by varying a the convergence characteristics of this new representation can be altered. Letting a → zero results in known eigenfunction expansions and, in those situations in which they exist, letting a → ∞ 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.