MATH 4356-001
Exam III
November 30, 1999
Answer the problems on separate paper. You do not need to rewrite the problem
statements on your answer sheets. Do your own work. Show all relevant steps which lead
to your solutions. Retain this question sheet for your records.
Part I. Do 5 (five) of the following problems.
k
1− i 
[12 pts] Find the sum of the series ∑ 
 .
k =2  2 
∞
1.
k
∞
2.
i 
[12 pts] Determine whether the series ∑ k 
 converges.
1−i 
k =1
3.
[12 pts] Find the Taylor series centered at i for
2
1
and determine its (the Taylor
1− z
series) radius of convergence.
(2 + i ) k ( z − 1) k
.
k2
k =1
∞
∑
4.
[12 pts] Find the radius of convergence of the power series
5.
[12 pts] Find and classify each of the finite isolated singularities for each of the
following functions (classify as either removable, pole [and order], or essential):
(a)
z3 − 1
z3 + 1
b)
1
sin z
c)
1
1
( + 1)sin
z
z
6.
[12 pts] For each of the functions in Problem 5., if the singularity at the point at
infinity is isolated, classify the singularity at the point of infinity.
7.
[12 pts] Let f and g be analytic on a neighborhood of 0. Suppose that f has a zero at
0 of order m ≥ 1 and that g has a zero at 0 of order n ≥ 1. Determine whether each
of the following statements is true or false. Supply a counterexample for each
statement that is false.
(a)
(b)
(c)
(d)
(e)
(f)
f + g has a zero at 0 of order m + n
f + g has a zero at 0 of order min(m,n)
f g has a zero at 0 of order m + n
f g has a zero at 0 of order mn
f o g has a zero at 0 of order mn
f o g has a zero at 0 of order max(m,n)
Part II.
1.
[12 pts] Find the residue at each of the isolated singularities for each of the
following functions:
(a)
z −1
z + 2z + 2
(b)
2
1 − cos z
z3
(c)
z 2e
− 2z
Part III. Do 1 (one) of the following problems.
1.
[12 pts] Find the integral .
2.
[12 pts] Find the integral.
sin2z
dz
|z |= 2 1 − cos z
∫
∫
|z − 2| =2
z − 2i
dz
z 2 − 2i
[Note: z2 - 2i = z2 - (1+i)2]
Part IV. Do 2 (two) of the following problems.
Use residues to evaluate the following integrals. If you introduce a contour integral,
clearly describe the contour. [If the contour is composed of subcontours, clearly describe
each of the subcontours.] If your argument introduces a limit process for the contour
integral, clearly detail the limit behavior of your integral over the contour [or subcontours,
as the case maybe].
2π
1.
[12 pts]
∫
0
∞
2.
[12 pts]
∫
0
∞
3.
[12 pts]
∫
0
dθ
5 + 4sin θ
dx
( x + 1) 2
2
x sin2x
dx
x2 + 3