Real and Complex Analysis (Rudin)

CONFORMAL MAPPING 293
Put y(t) = .JR;. e it
Theorem 10.43, (7) gives
(-n ~ t ~ n); put r = /0 y. As in the proof of
1 r f'(z)
ex = 2ni Jy /(z) dz = Indr (0). (8)
Thus ex is an integer. By (3), ex> O. By (7), the derivative of z-"f(z) is 0 in A 1 •
Thus/(z) = cz". Since/is one-to-one in A 1, ex = 1. Hence R2 = R 1• IIII
Exercises
I Find necessary and sufficjent conditions which the complex numbers a, b, c, and d have to satisfy
so that the linear fractional transformation z --+ (az + b)/(cz + d) maps the upper half plane onto itself.
2 In Theorem 11.14 the hypotheses were, in simplified form, that n c: II +, L is on the real axis, and
ImJ(z)--+ 0 as z--+ L. Use this theorem to establish analogous reflection theorems under the following
hypotheses:
(a) n c: II +, L on real axis, I J(z) 1--+ I as z --+ L.
(b) n c: U, L c: T, I J(z}l--+ I as z--+ L.
(c) n c: U, L c: T, ImJ(z)--+ 0 as z--+ L.
In case (b), if J has a zero at IX E n, show that its extension has a pole at I/a. What are the
analogues of this in cases (a) and (c)?
3 Suppose R is a rational function such that I R(z) I = I if I z I = I. Prove that
k Z - IX
R(z) = cz .. n --_-.
• =1 I - or;.z
where c is a constant, m is an integer, and IX., ... , IXk are complex numbers such that IX. #- 0 and
1 IX. I #- 1. Note that each of the above factors has absolute value 1 if 1 z I = 1.
4 Obtain an analogous description of those rational functions which are positive on T.
Hint: Such a function must have the same number of zeros as poles in U. Consider products of
factors of the form
where IIXI < 1 and IPI < 1.
5 SupposeJis a trigonometric polynomial,
(z - IX)(1 - az)
(z - P)(1 - pz)
.
J(9) = L a k elk',
andJ(9) > 0 for all real 9. Prove that there is a polynomial p(z) = Co + c. z + ... + c. z" such that
II:=-n
J(9) = I p(ei9) 12
(9 real).
Hint: Apply Exercise 4 to the rational function :Ea k zk. Is the result still valid if we assume'J(9) ;;;: 0
instead ofJ(9) > O?
6 Find the fixed points of the mappings CP. (Definition 12.3). Is there a straight line which CP. maps to
itself?
7 Find all complex numbers IX for whichJ. is one-to-one in U, where
Describe f.(U) for all these cases.
z
J.(z) = 1 + 0(Z2 •