Complex Analysis Midterm Exam, Spring 2012 - Problem Solutions, Exams of Mathematics

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Math 246C, Complex Analysis

Spring 2012

Midterm

Name:

Problem 2:

(a) Let a ∈ C, R > 0, and B = B(a, R). Show that if f : B → C is holomor- phic, and u := Ref has a continuous extension to B, then

f (z) = iImf (0) +

2 π

∫ (^2) π

0

Reit^ + (z − a) Reit^ − (z − a)

u(a + Reit)dt

for all z ∈ B. Hint: Your proof should be fairly short. (4 pts) (b) Prove the following statement: if u and un for n ∈ N are harmonic functions on an open set U ⊆ C and un → u locally uniformly on U , then ∂un ∂xk

∂u ∂xk locally uniformly on U for k = 1, 2. Here x 1 and x 2 stand for the standard coordinates on R^2. Hint: Use an integral as in (a). (8 pts)

Problem 3: Let u : D → R be a continuous function and assume that u|D is harmonic. The Fourier coefficients of u|∂D are defined as

an :=

2 π

∫ (^2) π

0

u(eit) e−int^ dt

for n ∈ Z (since u is real-valued we have a−n = an for n ∈ Z). Prove the following relation between the Dirichlet energy of u and these Fourier coefficients: ∫

D

|∇u|^2 dA = 4π

∑^ ∞

n=

n|an|^2.

This identity is supposed to include the statement that the Dirichlet energy of u is infinite if and only if the sum on the left hand side of this equation is infinite as well. (12 pts)