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23.2 Construction of Conformal Mappings 765

z 2 → 1, z 3 → 0, z 4 →∞. 28. Show that the cross ratio [z 1 , z 2 , z 3 , z 4 ] is the image of
z 1 under the bilinear transformation
Denote this cross ratio [z 1 , z 2 , z 3 , z 4 ]. Suppose T is (z 3 − z 4 )(z − z 2 )
any bilinear transformation. Show that T preserves w = 1 − .
this cross product. This means that (z 3 − z 2 )(z − z 4 )
29. Show that a cross ratio [z 1 , z 2 , z 3 , z 4 ] is real if and only
[z 1 , z 2 , z 3 , z 4 ]=[T (z 1 ), T (z 2 ), T (z 3 ), T (z 4 )]. if the z j ’s are on a circle or a line.

23.2 Construction of Conformal Mappings

One strategy for solving some kinds of problems (such as Dirichlet problems) is to find the
solution for a “simple” domain (for example, a disk) and then map this domain conformally to
the domain D on which we want to solve the problem. The idea is that this mapping may take
the solution for the simple domain to a solution for D. This requires that we be able to construct
conformal mappings between two domains.
Depending on the domains, this can be a daunting task, and it may not even be obvious that
such a conformal mapping exists. The following theorem settles this issue, with one exception.

THEOREM 23.5 The Riemann Mapping Theorem

∗
Let D be the unit disk |z| < 1. Let D be a domain in the w-plane, and assume that D is not the
∗
entire w-plane. Then there exists a one-to-one conformal mapping f : D → D of D onto D .
∗
∗
This powerful result implies the existence of a conformal mapping between two given
∗
domains. Suppose we want a conformal mapping from a domain D onto a domain D with
neither domain the entire plane. Put the unit disk U in between, as in Figure 23.17. The Riemann
mapping theorem ensures the existence of one-to-one, onto conformal mappings
f : U → D and g : U → D .
∗
Then the inverse mapping of f as

f −1 : D →U
is also conformal, and the composition F = g ◦ f −1 is a one-to-one, conformal mapping of D
onto D .
∗

g f −1
°
z w
f −1 g
i
D *
1
D

f
FIGURE 23.17 Mapping D onto D through the unit disk.
∗

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