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. 1.4.20.
Exercise 3. Let y0 R and let arg z denote the branch
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 = (1/2) log 2 and arg(1 + i) cfw_/4 + 2n | n Z. Hence,
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 = rei with r 0 and 0 < . Then z 2 = r2 ei2 and 0 2 < 2
so
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 + 3)4 is a composition of polynomials so is analytic eve
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 that
2u
4x3 12xy 2 = 12x2 12y 2
=
2
x
x
and
2u
=
12x2 y +
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 1 about 0, the Cauchy Integral
Formula immediately giv
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 series
n2
n=0
converges if and only if
n=1
1
+ z2
1
n2
con
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 and reindexing we
have
1
(1)n
=
z (z + 1) n=2 z n
for |
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:/www.trinity.edu/rdaileda
Oce Hours: Consult the course we
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 0, and yet it is not the zero function.
n=1
Why doesnt
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 two nonzero terms of this series.
Exercise 2. Prove tha
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 : |z | > R.
a. Show that if f (z ) is dened on a neighbor
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? What about (z 2 1)1/2 ?
Exercise 2. Show that f (z ) = |
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 log z is analytic on A and that
d
1
log z = .
dz
z
Exerc
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 A then there is a purely imaginary
number ik so that f
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 you evaluate
f (z ) dz .
R
Exercise 2. Let G be the reg
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 where is any simple closed curve containing the origin.
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 where is any simple closed curve containing the origin.
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 ) = then f is surjective.
z
Exercise 3. Let A C be a sim
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 expression yields f (z ) = k (z z0 )k1 (z ) +
(z z0 )k (z ). Th