Homework 1
Due Monday, Aug. 30
1. Find the cube roots of 8i.
2. If you consider the polynomial ax2 + bx + c, the quadratic formula tells us
that the roots of this polynomial are
b
b2 4ac
.
x=
2a
2a
Note that if a, b, and c are real, and if the polynomial
Homework 2
Due Wednesday, Sept. 8
1. Suppose that f : C C and g : C C and that limzz0 f (z ) = 0. Prove
that if there are positive real numbers M and r such that |g (z )| M
whenever |z z0 | < r, then limzz0 f (z )g (z ) = 0.
(Note that if f (z ) = z and g
Homework 3
Due Monday, Sept. 13
1. Recall that a function h : R2 R is harmonic if it satises Laplaces
equation: hxx + hyy = 0. Suppose that f : C C is dierentiable with
f (z ) = u(x, y ) + iv (x, y ). Prove that if h : R2 R is a harmonic function,
then th
Homework 4
Due Monday, Sept. 20
1. Let h : R2 \ cfw_(0, 0) R be given by h(x, y ) := ln(x2 + y 2 ). Prove that
h is harmonic. Can you nd a harmonic conjugate for h? What is the
domain of this harmonic conjugate? (Hint: Think of h as the real part of
a fam
Homework 5
Due Monday, Sept. 27
1. A closed, bounded interval I is a nonempty subset of R of the form
I := [a, b] = cfw_x R : a x b.
We say that the length of I is b a.
Prove that if
I1 I2 I3 . . .
is a nested sequence of closed, bounded intervals of R wi
Homework 6
Due Monday, Oct. 11
1. Suppose that f is an entire function and let C be a simple closed contour
oriented in the positive direction. Also let z0 C. If z0 is on the interior
of C , then Cauchys Integral Formula implies that
C
f (z )
dz = 2if (z0
Homework 7
Due Monday, Oct. 18
1. Prove that if z C and |z | < 1, then
zn =
n=0
1
.
1z
2. Prove that if cfw_zn is a sequence of complex numbers and if one has
n=1
n=1 zn = S , then it is the case that
n=1 zn = S .
3. Find the Maclaurin series for
f (z )
Homework 8
Due Monday, Oct. 25
1. Prove that if limn zn = z , then limn |zn | = |z |.
2. You probably showed in your Real Analysis class that uniform limits of
functions pass through integrals (i.e., if cfw_fn converges uniformly to f
n=1
b
b
on the clos
Homework 11
Due Monday, Dec. 6
We saw in class that if a function f is conformal at a point z0 (i.e., f is
analytic at z0 and f (z0 ) = 0) then f preserves the angle between curves passing
through z0 . The following problems consider what happens when z0
Test 1
Due Oct. 1 at the beginning of class
This exam must be worked on independently. However, you are allowed to
use your notes, the course textbook, and old homework. You are also allowed to
talk to the instructor. You are not allowed to use any other