A complex function w = f(z) can be regarded as a mapping from its domain in the z-plane to its range in the w-plane. In this chapter, we go beyond the previous chapters by analyzing in greater depth the geometric properties associated with mappings represented by complex functions. First, we examine the linkage between the analyticity of a complex function and the conformality of a mapping. A mapping is said to be conformal at a point if it preserves the angle of intersection between a pair of smooth arcs through that point. The invariance of the Laplace equation under a conformal mapping is also established. This invariance property allows us to use conformal mappings to solve various types of physical problem, like steady state temperature distribution, electrostatics and fluid flows, where problems with complicated configurations can be transformed into those with simple geometries.
First, we introduce various techniques for effecting the mappings of regions. Two special classes of transformation, the bilinear transformations and the Schwarz–Christoffel transformations, are discussed fully. A bilinear transformation maps the class of circles and lines to the same class, and it is conformal at every point except at its pole. The Schwarz–Christoffel transformations take half-planes onto polygonal regions. These polygonal regions can be unbounded with one or more of their vertices at infinity. We also consider the class of hodograph transformations, where the roles of the dependent and independent variables are reversed.