Real and Complex Analysis (Rudin)

CONFORMAL MAPPING 281
Let us discuss one such mapping more explicitly, namely,
1 + z
cp(z) = --.
1-z
(6)
This cp maps { -1, 0, 1} to {O, 1, oo}; the segment (-1, 1) maps onto the positive
real axis. The unit circle T passes through -1 and 1; hence cp(T) is a straight line
through cp( -1) = O. Since T makes a right angle with the real axis at -1, cp(T)
makes a right angle with the real axis at O. Thus cp(T) is the imaginary axis. Since
cp(O) = 1, it follows that cp is a conformal one-to-one mapping of the open unit disc
onto the open right half plane.
The role of linear fractional transformations in the theory of conformal
mapping is also well illustrated by Theorem 12.6.
14.4 Linear fractional transformations make it possible to transfer theorems concerning
the behavior of holomorphic functions near straight lines to situations
where circular arcs occur instead. It will be enough to illustrate the method with
an informal discussion of the reflection principle.
Suppose Q is a region in U, bounded in part by an arc L on the unit circle,
andfis continuous on n, holomorphic in Q, and real on L. The function
z-i
I/I(z)=-.
Z+I
(1)
maps the upper half plane onto U. If g = f 0 1/1, Theorem 11.14 gives us a holomorphic
extension G of g, and then F = G 0 1/1 - 1 gives a hoI om orphic extension
F off which satisfies the relation
f(z*) = F(z), (2),
where z* = liz.
The last assertion follows from a property of 1/1: If w = I/I(z) and W1 = I/I(z),
then W1 = w*, as is easily verified by computation.
Exercises 2 to 5 furnish other applications of this technique.
Normal Families
The Riemann mapping theorem will be proved by exhibiting the mapping function
as the solution of a certain extremum problem. The existence of this solution
depends on a very useful compactness property of certain families of holomorphic
functions which we now formulate.
14.5 Definition Suppose fF c H(Q), for some region Q. We call fF a normal
family if every sequence of members of fF contains a subsequence which converges
uniformly on compact subsets of Q. The limit function is not required
t'O belong to fF.