Math 246a, Homework 4
Christoph Thiele
due Thursday May 4
Problem 1, Morera
From a qualifying exam:
Let I= [−1,1] be the closed unit interval in RLet U=C\I.
a) prove that there is a non-constant bounded holomorphic function on U.
b) prove that if fis bounded and holomorphic on U, and fhas a continuous
extension to C, then fis constant.
Hint for a): show that z→(z2−1)/z maps the punctured unit disc bijectively
and holomorphically to U. This is an important map, up to linear fractional trans-
formations it is the unique one that can be used to collapse a disc to a line segment.)
Problem 2
From a qualifying exam:
Calculate the integral
Z∞
0
x
1 + x3dx
Hint: Consider the integral
Zγ
z
1 + z3dz
over a large triangle with corners 0, one side on the positive real axis, and the proper
angle at 0. (Other contours work as well, but we are having fun with triangles these
days in the lectures.)
Problem 3. Schwarz reflection principle
Let Ω be an open subset of C, which is symmetric about the real axis, i.e., z∈Ω if
and only if z∈Ω. Let =(z) denote the imaginary part of z.
Assume f:{z∈Ω : =(z)≥0} → Cis continuous and holomorphic in {z∈Ω :
=(z)>0}, and f(z)∈Rfor all z∈Ω∩R. Prove that there is a holomorphic
g: Ω →C
which coincides with fon the domain of f.
(Hint: use symmetry considersations to define gin the lower half of Ω. Then show
holomorphy of g.)
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