Math 113 (Spring 2009) Yum-Tong Siu 1
Homework Assigned on February 17, 2009
due February 24, 2009
Problem 1 (from Stein & Shakarchi, p.67, #14). Let R > 1 and z0∈Cwith
|z0|= 1. Let h(z) be a holomorphic function on { |z|< R }with h(z0)6= 0.
Let mbe a positive integer and
f(z) = h(z)
(z−z0)m.
Show that if ∞
X
n=0
anzn
denotes the power series expansion of fon { |z|<1}, then
lim
n→∞
an
an+1
=z0.
Problem 2 (from Stein & Shakarchi, p.67, #15). Suppose fis a nowhere
vanishing continuous function on the closure Dof the open unit disk Dand
fis holomorphic in D. Prove that if
|f(z)|= 1 whenver |z|= 1,
then fis constant.
Hint: Extend fto all of Cby
f(z) = 1
f¡1
¯z¢
whenever |z|>1, and argue as in the Schwarz reflection principle.
Problem 3 (from Stein & Shakarchi, p.67-68, Problem 1). For a function f
defined on the open unit disk D, a point wof the boundary Cof Dis said to
be regular for fif there is an open neighborhood Uof wand a holomorphic
function gon U, so that f=gon D∩U. A function fdefined on Dcannot
be continued analytically past the unit disk if no point of Cis regular for f.
