Homework 3
Due: Friday, February 15
1. [SS] 2.1. This is exercise 1, not problem 1.
2. [SS]2.5.
3. Let f be a function on a domain . Let be a closed contour.
Suppose that, for each
contour , one has
> 0, there exists a polynomial P such that, for each poi
Homework 6
Due: Friday, March 15
Some of these will require Theorem 3.4 and related ideas, to be covered on Monday.
1. Consider the unit sphere X = cfw_( a, b, c) : a2 + b2 + c2 = 1 R3 . Let N = (0, 0, 1), S =
(0, 0, 1), UN = X cfw_ N , US = X cfw_S. Cons
Homework 5
Due: Friday, March 1
1. Consider the function
f (z) =
2z 3
.
z2 4
(a) Find the residue res( f ; 2). (H INT: Try and expand f (z) in terms of powers of (z 2).)
(b) Find the residue res( f ; 2).
(c) Let C = C5 (0), the circle of radius 5 centered
Homework 4
Due: Friday, February 22
1. For 0 r n, the binomial coefcient (n) is dened by
r
n
r
=
n!
.
r !(n r )!
(a) Let be any simple closed contour around 0. Show that
n
r
=
1
2 i
(1 + z ) n
dz.
z r +1
(b) Now let = C1 (0), the unit circle. Use (a) to s
Homework 7
Due: Friday, March 29
1.
(a) [SS]3.15(a)
(b) [SS]3.15(c)
(c) [SS]3.15(d)
2. Use Rouch s theorem to give another proof of the fundamental theorem of algebra, as fole
lows. Let p(z) = d=0 a j z j be a polynomial, where d 1 and ad = 0.
j
(a) Show
Homework 8
Due: Friday, April 5
1. Let be a domain with 0 .
(a) Suppose that f (z) and g(z) are continuous branches of the logarithm on . Show that
there is some integer n such that g(z) = f (z) + 2 in. (H INT: is connected.)
(b) Suppose that f (z) is a c
Homework 9
Due: Friday, April 19
1. [SS]5.10. (H INT: Use the rst couple terms in a series expansion to calculate the polynomial P(z).)
2. [SS]5.11.
3. [SS]5.14. (H INT: In the context of Hadamards theorem, what is the order of growth of exp( P(z)?)
4. Do
Homework 1
Due: Friday, February 1
1. [SS]1.1. In other words, do problem 1 from Chapter 1 of Stein and Shakarchi.
2.
(a) Show that the complex conjugation map
C
-C
-z
z
is an involution, i.e., a ring homomorphism such that = id.
(b) Suppose a R and z C.
Homework 2
Due: Friday, February 8
1. [SS]1.19(a),(b).
N
2. [SS]1.19(c). (H INT: You may use the result of [SS]1.14. If z = 1, what is n=1 zn ?)
3. This is another way to derive the Cauchy-Riemann equations Suppose S C is a domain, and
that f : S C is a f
Homework 12
Due: Friday, 10
1. [SS]8.1.
2. [SS] 8.4. It turns out there is no such holomorphic bijection.
3. [SS]8.11.
4. [SS]8.14.
Professor Jeff Achter
Colorado State University
MATH 519: Complex Variables
Spring 2013
Homework 10
Due: Friday, April 26
1. [SS]6.14.
2.
(a) Suppose n 1. Calculate
x s1 enx dx
0
in terms of (s). (H INT: Try a change of variables y = nx.)
(b) [SS]6.15. Read problem 16, too.
3. In class, we will dene (s) and explain how to meromorphically con
Homework 11
Due: Friday, May 3
1. [SS]7.8.
2. Read [SS]7.6, assume its result, and proceed
[SS]7.6:
1
( a) = 1
2
0
as follows. Let be the function dened in
1<a
a=1
0a<1
(a) Show that
( X ) =
X
( n ) ( n ).
n 1
(b) Consider the function
G (s) =
Show that