Complex Variables
Spring 2011
Assignment 3.1
Due September 19
Exercise 1. 1.4.13 and 1.4.14. I didnt dene it in class, but a set is closed if and only if its
complement in C is open.
Exercise 2. 1.4.2
Complex Analysis
Fall 2007
Homework 2: Solutions
1.3.6.
(a) We have log | i| = log 1 = 0 and arg(i) cfw_/2 + 2n | n Z. Hence, the values
of log(i) are
i
+ 2n
2
for n Z.
(b) We have log |1 + i| = log 2
Complex Analysis
Fall 2007
Homework 3: Solutions
1.3.10. Claim: Using the branch of the square root function given in the problem,
i z = 0 or z = rei with r > 0 and 0 < .
z2 = z
Proof: () Suppose z =
Complex Analysis
Fall 2007
Homework 4: Solutions
1.5.2.
(a) The function f (z ) = 3z 2 +7z +5 is a polynomial so is analytic everywhere with derivative
f (z ) = 6z + 7.
(b) The function f (z ) = (2z +
Complex Analysis
Fall 2007
Homework 5: Solutions
1.5.22 If z = x + iy then
z 4 = (x4 6x2 y 2 + y 4 ) + i(4x3 y 4xy 3 )
so that u = Re(z 4 ) = x4 6x2 y 2 + y 4 and v = Im(z 4 ) = 4x3 y 4xy 3 . We nd th
Complex Analysis
Fall 2007
Homework 8: Solutions
2.R.1
(a) Since sin z is entire, its integral around any closed curve is zero by Cauchys Theorem.
(b) Since sin z is entire and has a winding number of
Complex Analysis
Fall 2007
Homework 9: Solutions
3.1.4
(a) Let z C \ cfw_ni : n Z. Then
lim
n
n2
1/(n2 + z 2 )
= lim 2
= 1.
n n + z 2
1/n2
According to the limit comparison test from calculus, the ser
Complex Analysis
Fall 2007
Homework 10: Solutions
3.3.2 We have
11
1
1
1
1
=
=2
=2
z (z + 1)
zz+1
z 1 + 1/z
z
n=0
(1)n
zn
since |z | > 1 implies that |1/z | < 1. Multiplying the 1/z 2 into the series
Math 4364
Theory of Complex Variables
Instructor:
Oce:
e-mail:
Telephone:
URL:
Fall 2011
Dr. Ryan C. Daileda
Marrs McLean Science Building (MMS), Room 115h
[email protected]
(210) 999-8265
http:/ww
Complex Variables
Spring 2011
Assignment 11.2
Due November 16
Exercise 1. Consider the function f (z ) = e1/z 1. We have seen that it has a sequence
of distinct zeros cfw_zn with the property that zn
Complex Variables
Spring 2011
Assignment 10.2
Due November 7
Exercise 1. What is the largest circle within which the Taylor series at z0 = 0 for the
function tanh z converges to tanh z ? Find the rst
Complex Variables
Spring 2011
Assignment 3.2
Due September 19
Exercise 1. Recall that a function f (z ) is dened on a neighborhood of if there exists an
R > 0 so that the domain of f contains cfw_z :
Complex Variables
Spring 2011
Assignment 3.3
Due September 19
Exercise 1. Let log w denote the branch of the logarithm with arg w (0, 2 ] and let
1
w1/2 = e 2 log w . Where is (z 2 + 4)2 continuous? W
Complex Variables
Spring 2011
Assignment 4.1
Due September 26
Exercise 1. Let z = x + iy and A = cfw_z : x > 0. For z A let arg z = arctan(y/x) and
use this to dene a branch of log z on A. Show that l
Complex Variables
Spring 2011
Assignment 5.1
Due October 3
Exercise 1. Let A be a connected open set and suppose that f and g are both analytic
functions on A. Prove that if Re f = Re g everywhere on
Complex Variables
Spring 2011
Exercise 1. Dene f (z ) =
Assignment 8.1
Due October 24
rei/2 where z = rei with < . Let R denote the
rectangle with vertices i, i, 2 i and 2+ i. Use Lemma 2.3.4 to help
Complex Variables
Spring 2011
Assignment 9.1
Due October 31
Exercise 1. Use Cauchys Integral Formula to evaluate the following denite integrals. All
contours are positively oriented.
a.
b.
sin ez
dz w
Complex Variables
Spring 2011
Assignment 9.1
Due October 31
Exercise 1. Use Cauchys Integral Formula to evaluate the following denite integrals. All
contours are positively oriented.
a.
b.
sin ez
dz w
Complex Variables
Spring 2011
Assignment 10.1
Due November 7
f (z )
= 0. Prove that f is constant.
z z
Exercise 1. Suppose that f is entire and lim
Exercise 2. Show that if f is entire and lim f (z )
Complex Analysis
Fall 2007
Homework 11: Solutions
3.3.16 If f has a zero of multiplicity k at z0 then we can write f (z ) = (z z0 )k (z ), where
is analytic and (z0 ) = 0. Dierentiating this expressi