where *z* = (*x*, *y*) is the (possibly complex) coordinate in the plane and *h*(*z*) is the distribution of temperature on the curve. The region *R* is simply connected. In some cases, we will consider the exterior Dirichlet problem for conduction in the region exterior to *σ*, in which case we consider the exterior region, *E*, to be the extended complex plane, including the point at infinity. For the exterior problem in two dimensions, in order for a steady-state solution to exist, there can be no heat transfer to the region far from the object. That condition is met if the temperature far from the object is the average temperature on any circle drawn around the outside of the object, by the mean value theorem for harmonic functions.^{1} For objects having one high temperature side and one low temperature side, the far field will resemble that of a dipole.