Introduction to Complex Analysis, Fall 2013
Homework 4
Due: Tuesday, October 8th
Problem 1: a) Show that the following functions are harmonic, and find
harmonic conjugates:
xy + 3x2 y − y 3
ex
2 −y 2
cos(2xy)
b) Show that log|z| has no harmonic conjugate on C \ {0}.
c) Derive the polar form of Laplace’s equation. Use this to show that
u(reiθ ) = θlogr is harmonic.
Problem 2:
a) Gamelin p.83 III.2 #7
b) Gamelin p.84 III.3 #4
Problem 3:
Gamelin p.87 III.4 #4
Problem 4: The Wirtinger derivatives are defined to be the following differential operators:
1 ∂
∂
∂
1 ∂
∂
∂
= (
−i·
)
= (
+i·
)
∂z
2 ∂x
∂y
∂z
2 ∂x
∂y
a) Gamelin p.128 IV.8 #2
b) Gamelin p.129 IV.8 #7
Problem 5: Let D be a domain, and let u : D → [−∞, ∞). We say that u
is subharmonic if for each z0 ∈ D there is > 0 so that u satisfies the mean
value inequality:
Z 2π
dθ
u(z0 ) ≤
u(z0 + reiθ ) , 0 < r < 2π
0
a) Gamelin p.397 XV.2 #5
b) Gamelin p.397 XV.2 #6
Problem 6:
a) Let D ⊂ C represent the closed unit disk in the complex plane, and
find where the maximum value of f (z) = z 2 + iz is attained on D.
b) Gamelin p.89 III.5 #3
1