96 - Conformal Mappings

Abstract

This chapter describes conformal mappings applicable to Laplace's equation in two dimensions. It yields a reformulation of the original problem. Laplace's equation in two dimensions with a given boundary can be transformed to Laplace's equation with a different boundary by a conformal map. The idea is to choose the conformal map in such a way that the new boundary makes the problem easy to solve. In this method, for Laplace's equation in the variables {x,y} the complex variable z = x + iy is defined, where i = √−1. All of the boundaries of the original problem can now be described by values of z. The chapter highlights that a commonly used conformal map is the Schwartz–Christoffel transformation. This maps a closed polygonal figure into a half plane.