Math 621 Homework Assignment 7 Spring 2006
Due: Wednessday, May 3
1. Let P(z) and Q(z) be polynomials and suppose that deg Q≥deg P+ 2. Show
that X
ζ
Resζ(P/Q) = 0,
where the sum runs over all singularities of the rational function P /Q. Do this
problem in two ways: (1) Directly, without considering the residue at ∞. (2) As
a special case of problem 3.
2. Lang Problem 37 page 190: Let fbe analytic on Cwith exception of a finite
number of isolated singularities {z1,...zn}. Define the residue at infinity
Res∞f(z)dz := −1
2πi Z|z|=R
f(z)dz
for R > max{|z1|,...|zn|}.
(a) Show that Res∞f(z)dz is independent of R.
(b) Show that the sum of the residues of fin the extended complex plane CP 1is
equal to zero. (This result is often refered to as The residue Theorem.)
3. Lang Problem 38 page 190 (Cauchy’s Residue Formula on the Riemann Sphere).
4. Basic Exam, September 1998 Problem 5: Compute ZC
z4e1/z
1−z4dz where Cdenotes
the circle {|z|= 2}transversed counterclockwise. Hint: Use the residue at infinity
(problem 3) to save computations.
5. Basic Exam, January 2000 Problem 7: Let fbe a one-to-one holomorphic map
from a region D1onto a region D2. Suppose that D1contains the closure of the
disk ∆ := {|z−z0|< ρ}. Prove that for w∈f(∆) the inverse function f−1(w) is
given by
f−1(w) = 1
2πi Z{|z−z0|=ρ}
f0(z)
f(z)−w·zdz
6. Let P(z) be a polynomial of degree dand assume that the roots ζ1, . . . , ζdof Pare
simple. Show that for Rsufficiently large:
Z{|z|=R}
zkP0(z)
P(z)dz =
d
X
i=1
ζk
i.
7. Lang page 189 Problem 29: Let Ube a connected open set, and let Dbe an open
disk whose closure is contained in U. Let fbe analytic on Uand not constant.
Assume that the absolute value |f|is constant on the boundary of D. Prove that
fhas at least one zero in D.Hint: Consider g(z) := f(z)−f(z0)with z0∈D.
8. Basic Exam January 2000 Problem 9: Prove that the equation
20 z3
z2+ 4 =ez
has 3 roots in the unit disk {|z|<1}.
1
