Some applications of the repeated integrals of the error function
The functions (1) arise naturally in the theory of conduction of heat in the semiinfinite solid (or the sphere or slab) with prescribed surface temperature or flow of heat, since the Laplace transforms of the solutions of many such problems involve the functions on the right hand side of (4). The objects of this note are, firstly, to indicate an extension of (4) which applies in the same way to problems with heat transfer at the surface, and, secondly, to give solutions in terms of the functions (1) of a number of problems of practical interest which involve heat generation in the solid. 2. Problems involving heat transfer at a surface at a rate proportional to its temperature difference from its surroundings. The required extension of (4) is that, if n is a positive integer, and a, h, and x are positive,