Math 417, Fall 2011 1. Evaluate orientation.
Exercise Set #6
Turn in: October 28
C
z4
2z + 1 dz , where C is the circle |z| = 10 with the positive - 2z 2 + 1
n=1
2. Show that if
n=1
zn = S, then
zn = S.
3. What is the radius of convergence of the Taylo
Math 417, Fall 2011
Exercise Set #9 corrected
Turn in: November 28
For problems 1 to 5 below, use the method of residues to calculate the indicated integrals. Show all of your work. 1. Compute
0 0
x2
dx . +9 dx 3 = . 3 + 4) 512
R -R
2. Show that
(x2
3.
Math 417, Fall 2011
Exercise Set #8
Turn in: November 18
1. For each of the following functions, find all of the isolated singularities. Determine whether they are removable singularities, poles or essential singularities. If the function has a pole, dete
Math 417, Fall 2011
Exercise Set #7
Turn in: November 7
1. Find the maximum and minimum of |f (z)| on the unit disk cfw_z C | |z| 1, where f (z) = z 2 - 2. 2. What is the radius of convergence for the Taylor series of the function 1 f (z) = 2 z - 3z + 2 a
Math 417, Fall 2011
Exercise Set #5 = i to z = 1. Show that 4 2.
Turn in: October 5
1. Let C denote the line segment from z dz 4 C z
2. Let CR denote the upper half circle for |z| = R, for R > 1, parametrized in the counter-clockwise direction. Show that
Math 417, Fall 2011
Exercise Set #4
Turn in: September 26
1. Log is the Principal Part of log, so Log(z) = ln(r) + , for - < < . Calculate: a) Log(1 + i) b) Log(-ei) c) log(1 - i) 2. Find all values of: a) 2i b) (1 + i 3)3/2 3. Prove the following formula
Math 417, Fall 2011
Exercise Set #3
Turn in: September 19
1. Does the limit lim ez exists? Discuss what happens to the values of ez as z .
z
2. Find an analytic function f (z) = u(z) + i v(z) such that u(x, y) = xy. Is there an analytic function g(z) whos
Math 417, Fall 2011
Exercise Set #2
Turn in: September 12
1. Find the real and imaginary parts of the functions: z-1 a) f (z) = z-2 b) f (z) = exp(2z + 5i) c) f (z) = 3z 2 + 7z - i 2. Show (using the definition and properties of limits) that a) lim
z2
z2
Math 417, Fall 2011 1. Solve a + i b =
Exercise Set #1
Turn in: September 2
(2 + i)(3 + 2i) for a and b. (1 - i)
2. Draw the graph in the complex plane of all z C satisfying z - 1 a) z + 2 1. b) |z|2 = im(z). 3. a) Express the complex number w = -1 + i 3
Chapter 10
Discrete Applications of the Residue Theorem
All means (even continuous) sanctify the discrete end. Doron Zeilberger
On the surface, this chapter is just a collection of exercises. They are more involved than any of the ones we've given so far
Chapter 9
Isolated Singularities and the Residue Theorem
1/r2 has a nasty singularity at r = 0, but it did not bother Newton-the moon is far enough. Edward Witten
9.1
Classification of Singularities
1 What is the difference between the functions sin z , z
Chapter 8
Taylor and Laurent Series
We think in generalities, but we live in details. A. N. Whitehead
8.1
Power Series and Holomorphic Functions
We will see in this section that power series and holomorphic functions are intimately related. In fact, the t
Chapter 7
Power Series
It is a pain to think about convergence but sometimes you really have to. Sinai Robins
7.1
Sequences and Completeness
As in the real case (and there will be no surprises in this chapter of the nature `real versus complex'), a (compl
Chapter 6
Harmonic Functions
The shortest route between two truths in the real domain passes through the complex domain. J. Hadamard
6.1
Definition and Basic Properties
We will now spend a chapter on certain functions defined on subsets of the complex pla
Chapter 5
Consequences of Cauchy's Theorem
If things are nice there is probably a good reason why they are nice: and if you do not know at least one reason for this good fortune, then you still have work to do. Richard Askey
5.1
Extensions of Cauchy's For
Chapter 4
Integration
Everybody knows that mathematics is about miracles, only mathematicians have a name for them: theorems. Roger Howe
4.1
Definition and Basic Properties
At first sight, complex integration is not really anything different from real int
Chapter 3
Examples of Functions
Obvious is the most dangerous word in mathematics. E. T. Bell
3.1
Mbius Transformations o
The first class of functions that we will discuss in some detail are built from linear polynomials. Definition 3.1. A linear fraction
Chapter 2
Differentiation
Mathematical study and research are very suggestive of mountaineering. Whymper made several efforts before he climbed the Matterhorn in the 1860's and even then it cost the life of four of his party. Now, however, any tourist can
Chapter 1
Complex Numbers
Die ganzen Zahlen hat der liebe Gott geschaffen, alles andere ist Menschenwerk. (God created the integers, everything else is made by humans.) Leopold Kronecker (18231891)
1.0
Introduction
The real numbers have nice properties. T
A First Course in Complex Analysis
Version 1.2c
Matthias Beck, Gerald Marchesi, and Dennis Pixton
Department of Mathematics San Francisco State University San Francisco, CA 94132 [email protected]
Department of Mathematical Sciences Binghamton University