Real and Complex Analysis (Rudin)

296 REAL AND COMPLEX ANALYSIS
27 Prove that L,:",.I (1 - r.J2 < 00, where {r.} is the sequence which occurs in Exercise 26. Hint:
28 Suppose that in Exercise 26 we choose ex. E U - 0._ 1 without insisting that I ex. I = r •. For
example, insist only that
1 + r.
lex.1 :5- 2 -.
Will the resulting sequen~ {!/t.} still converge to the desired mapping function?
29 Suppose 0 is a bounded region, a E O,f E H(O),f(O) c: n, andf(a) = a.
(a) Putf l = fandf. = f 0 f.-I' computef~(a), and conclude that I/,(a) I :5 1.
(b) If/,(a) = 1, prove thatf(z) = z for all z E O. Hint: If
f(z) = z + c,"(z - a)'" + ... ,
compute the coefficient of (z - a)'" in the expansion of f.(z).
(c) If I /,(a) I = 1, prove thatfis one-to-one and thatf(O) = O.
Hint: If y = /'(a), there are integers n.-+ 00 such that Y··-+ 1 and f •• -+ g. Then g'(a) = 1,
g(O) c: 0 (by Exercise 20, Chap. 10), hence g(z) = z, by part (b). Use 9 to draw the desired conclusions
about!
30 Let A be the set of all linear fractional transformations.
If {ex, p, y, a} is an ordered quadruple of distinct complex numbers, its cross ratio is defined to be
(ex - P)(y - 0)
[ex, P, y, 0] = (ex _ oXy - p).
If one of these numbers is 00, the definition is modified in the obvious way, by continuity. The same
applies if ex coincides with P or y or o.
(a) If cp(z) = [z, ex, p, y], show that cp E A and cp maps {ex, p, y} to {O, 1, oo}.
(b) Show that the equation [w, a, b, c] = [z, ex, p, y] can be solved in the form w = cp(z); then
cp E A maps {ex, p, y} to {a, b, c}.
(c) If cp E A, show that
[cp(ex), cp(P), cp(y), cp(o)] = [ex, p, y, 0].
(d) Show that [ex, p, y, 0] is real if and only if the four points lie on the same circle or straight
line.
(e) Two points z and z* are said to be symmetric with respect to the circle (or straight line) C
through ex, p, and y if [z*, ex, p, y] is the complex conjugate of [z, ex, p, y]. If C is the unit circle, find a
simple geometric relation between z and z*. Do the same if C is a straight line.
(f) Suppose z and z* are symmetric with respect to C. Show that cp(z) and cp(z*) are symmetric
with respect to cp( C), for every cp E A.
31 (a) Show that A (see Exercise 30) is a group, with composition as group operation. That is, if
cp E A and !/t E A, show that cp 0 !/t E A and that the inverse cp-I of cp is in A. Show that A is not
commutative.
(b) Show that each member of A (other than the identity mapping) has either one or two fixed
points on 8 2 • [A fixed point of cp is a point ex such that cp(ex) = ex.]
(c) Call two mappings cp and CPI E A conjugate if there exists a!/t E A such that CPI = !/t-I 0 cp o!/t.
Prove that every cp E A with a unique fixed point is conjugate to the mapping z-+ z + 1. Prove that
every cp E A with two distinct fixed points is conjugate to the mapping z-+ exz, where ex is a complex
number; to what extent is ex determined by cp?