Potential functions with periodicity in one coordinate

Abstract

The general conception underlying the following analysis is that of a field of potential which is invariant under a group of geometrical transformations. The boundary is to consist of a number of parts which transform into each other and the boundary values also transform into each other. To satisfy the conditions we build up functions which are invariant under the same group of transformations and combine them to give the prescribed boundary values over one section of the boundary. The boundary conditions on the other sections are then automatically satisfied. When the groups are simply translations or rotations we get functions periodic in either a Cartesian or an angular coordinate.