Math 534 HOMEWORK 4 (due Wednesday, Oct. 24) Autumn 2007
1(4 pts) #3 (a,b,c; 2 pts for c) in Gamelin IV.3 (p. 112)
2(3 pts) (a) Calculate using the Cauchy integral formula:
Z|z+1|=1
dz
(z+ 1)(z−1)3
and Z|z+1|=1
dz
(z+ 1)3(z−1) .
(b) Let f:C→Cbe analytic and z1, z2∈Care distinct. Show that
Z|z|=r|
f(z)
(z−z1)(z−z2)dz = 2πi f(z1)−f(z2)
z1−z2
,
if r > max{|z1|,|z2|}.
3(2 pts) #2 in Gamelin IV.4 (p. 117)
4(2 pts) #4 in Gamelin IV.4 (p. 117)
5(3 pts) Let fbe a non-constant entire function. Show that f(C) is dense in C.
6(2 pts) #3 in Gamelin IV.5 (p. 119)
7(4 pts) Suppose that fis a continuously differentiable function in a domain Dsuch that
Rγf(z)dz = 0 for every circle γcontained in D. Prove that fis analytic in D.Hint: use
Green’s Theorem.
8. Challenge Problem (4 pts of extra credit) Let K⊂Cbe a compact set and µa finite
Borel measure supported on K. The Cauchy transform is the function
f(z) = ZK
dµ(ζ)
ζ−z,
defined for z6∈ K.
(a) (0 pts, but need to start with this) Show that fis analytic in C\K.
(b) (4 pts) Suppose that there are constants α > 1 and C > 0 such that µ(B(z , r)) < Crα
for each z∈Kand each r > 0. Show that fhas a continuous extension to C.
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