Advanced Engineering Mathematics, 10th Edition

CHAPTER 17

Conformal Mapping

Conformal mappings are invaluable to the engineer and physicist as an aid in solving problems in potential theory. They are a standard method for solving boundary value problems in two-dimensional potential theory and yield rich applications in electrostatics, heat flow, and fluid flow, as we shall see in Chapter 18.

The main feature of conformal mappings is that they are angle-preserving (except at some critical points) and allow a geometric approach to complex analysis. More details are as follows. Consider a complex w = f(z) function defined in a domain D of the z–plane; then to each point in D there corresponds a point in the w-plane. In this way we obtain a mapping of D onto the range of values of f (z) in the w- plane. In Sec. 17.1 we show that if f(z) is an analytic function, then the mapping given by w = f(z) is a conformal mapping, that is, it preserves angles, except at points where the derivative f′(z) is zero. (Such points are called critical points.)

Conformality appeared early in the history of construction of maps of the globe. Such maps can be either “conformal,” that is, give directions correctly, or “equiareal,” that is, give areas correctly except for a scale factor. However, the maps will always be distorted because they cannot have both properties, as can be proven, see [GenRef8] in App. 1. The designer of accurate maps then has ...