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CONFORMAL MAPPINGS CONSTRUCTION
OF CONFORMAL MAPPINGS CONFORMAL
CHAPTER 23 MAPPING SOLUTIONS OF DIRICHLET
PROBLEMS MODELS OF PLANE FLUID
Conformal

Mappings and

Applications

In this chapter, we will discuss conformal mappings and applications of complex functions to the
solution of Dirichlet problems and the analysis of fluid flow.

23.1 Conformal Mappings

It is sometimes useful to think of a complex function as a mapping.If f (z) is defined for all z in
some set S of complex numbers and each f (z) is in some set K if z is in S, we write f : S → K
and say that f maps S into K. If every point in K is the image of some point in S, then f maps
S onto K. This is the same notion of onto encountered in Chapter 7 for linear transformations.
f (S) denotes the set of numbers f (z) for z in S. Then f : S → K is an onto mapping exactly
when f (S) = K.

EXAMPLE 23.1
Let w = f (z) = z/|z| for z = 0. Then f (z) is defined on the set S consisting of the plane with the
origin removed. If z = 0, then | f (z)|= 1 because

z
= 1.
|z|

If K is the set of all points of magnitude 1, then f maps S into K. In this case, f maps S onto
K because every number in K is the image of some number under this mapping. Indeed, if z is
in K, then |z|= 1 and f (z) = z. This mapping contracts the entire plane (with origin removed)
onto the unit circle.

In visualizing the action of a mapping f it is convenient to make two copies of the complex
plane, the z-copy and the w-copy, as in Figure 23.1. Picture S in the z-plane, and K in the
w-plane.

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