Ma 416: Complex Variables Final Examination
Prof. Wickerhauser Wednesday, December 21st, 2005
You may use your textbook and graded homework sets as well as any calculator. Please write your complete answers in the bluebook. 1. Use Rouch's theorem to deter
Math 416 Complex variables
Solutions to Problem Set 6
1. Let C be the positively oriented circle |z| = 1, with parametric representation
z = ei (0 2). and let m and n be integers. Then
2
z m z n dz =
C
2
(ei )m (ei )n iei d = i
0
ei(m+1) ein d.
0
But we k
Math 416 Complex variables
Solutions to Problem Set 7
2
z
1. (a) The function f (z) = z3 is analytic everywhere except at z = 3 which is outside
the unit circle |z| = 1, so we can apply the Cauchy-Goursat theorem to conclude that
the integral is zero.
(b)
Math 416 Complex variables
Solutions to Problem Set 8
1.
(i) We have
cos(z) =
n=0
so we get
z cos(z 2 ) =
n=0
(1)n 2n
z ,
(2n)!
(1)n 4n+1
z
.
(2n)!
(ii) Since
1
=
1+z
(1)n z n
when |z| < 1
n=0
we get
z
z
1
z
z2
(1)n 2n+1
=
(1)n ( )n =
z
2 =
z2 + 9
9 1+ z
Math 416 Complex variables
Solutions to Problem Set 9
1. You can show that the given function is entire by nding the Laurent series of f (z)
at z = /2.
You can also argue as follows: First we show that the function
g(z) =
sin z
z
when z = 0
when z = 0
1
i
Math 416 Complex variables
Solutions to Problem Set 10
1. (i) The point z = 0 is the isolated singular point of
sin z
z
and we can write
1
z3 z5
z2 z4
sin z
= (z
+
.) = 1
+
.
z
z
3!
5!
3!
5!
when 0 < |z| < . So z = 0 is a removable singular point.
(ii
Magi 4J6
50/0411}?ng +0 prob/am Set e: 1!
Fail 9.0093
Astor the pomts z = Inn: tn = 1,4, ,N), write
1 =£S£l
zzsinz q(z)
, where 19(2) ml and 9(2) = 22 sinz.
Since
Minx) :1 i 0, q(imr) : 0, and q(:tmr) = n2: cos ma: = (1)"n27t2 at 0,
it follows tha
Ma 416: Complex Variables Solutions to Homework Assignment 12
Prof. Wickerhauser Due Thursday, December 8th, 2005
Read R. P. Boas, Invitation to Complex Analysis, Chapter 4, sections 21A23B and 25A-25E. 1. Suppose that u = u(x, y) is continuous on the clo
Ma 416: Complex Variables Solutions to Homework Assignment 7
Prof. Wickerhauser Due Thursday, October 27, 2005
Read R. P. Boas, Invitation to Complex Analysis, Chapter 2, sections 12A13C. 1. Use the argument principle to count the zeros of P (z) = z 4 + z
Ma 416: Complex Variables Solutions to Homework Assignment 2
Prof. Wickerhauser Due Thursday, September 15th, 2005
1. Prove or nd a counterexample to the following statements: (a) If f (x) = O(g(x) as x 0, then f (x)/g(x) 0 as x 0. (b) If f (x) = o(g(x) a
Ma 416: Complex Variables Solutions to Homework Assignment 11
Prof. Wickerhauser Due Thursday, December 1st, 2005
Read R. P. Boas, Invitation to Complex Analysis, Chapter 4, sections 19A20F. 1. Find an analytic function f whose real part is Solution: Use
Ma 416: Complex Variables Solutions to Homework Assignment 9
Prof. Wickerhauser Due Thursday, November 10th, 2005
Read R. P. Boas, Invitation to Complex Analysis, Chapter 2, sections 16A16C. 1. Suppose f is analytic on the closed unit disk, f (0) = 0, and
Ma 416: Complex Variables Solutions to Homework Assignment 4
Prof. Wickerhauser Due Thursday, September 29th, 2005
1. Let fn (x) = [xn (1 xn )] for n = 1, 2, 3, . . . Does the sequence cfw_fn (x) converge uniformly on 0 < x < 1? Solution: No. For any xed
Ma 416: Complex Variables Solutions to Homework Assignment 3
Prof. Wickerhauser Due Thursday, September 22nd, 2005
1. Find the Maclaurin series of sinh z = 1 (ez ez ). 2 Solution: The even-power terms of the exponential series cancel, leaving sinh z = z 2
Ma 416: Complex Variables Solutions to Homework Assignment 1
Prof. Wickerhauser Due Thursday, September 8th, 2005
1. Find the real parts, imaginary parts, and absolute values of the complex numbers (a) i+1 i-1 (b) 1 (1 + 2i)(3i - 4)
(a) real part 0, imagi
Ma 416: Complex Variables Solutions to Homework Assignment 6
Prof. Wickerhauser Due Thursday, October 13th, 2005
Read R. P. Boas, Invitation to Complex Analysis, Chapter 2, sections 9A11C. 1. Evaluate the denite integral Solution: get
2 0 2 0 (2
+ sin )2
Ma 416: Complex Variables Solutions to Homework Assignment 8
Prof. Wickerhauser Due Thursday, November 3, 2005
Read R. P. Boas, Invitation to Complex Analysis, Chapter 2, sections 14A15F. 1. Find the Laurent series (in powers of (z 0) in the punctured dis
Ma 416: Complex Variables Solutions to Homework Assignment 10
Prof. Wickerhauser Due Thursday, November 17th, 2005
Read R. P. Boas, Invitation to Complex Analysis, Chapter 3, sections 17A18C. 1. Verify that 1/(1 z) can be continued outside the unit disk b
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