Cain - Complex Analysis.pdf

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Complex Analysis
Complex Analysis
George Cain
(c)Copyright 1999 by George Cain.
All rights reserved.
Table of Contents
1.1 Introduction
1.2 Geometry
1.3 Polar coordinates
2.1 Functions of a real variable
2.2 Functions of a complex variable
2.3 Derivatives
3.1 Introduction
3.2 The exponential function
3.3 Trigonometric functions
3.4 Logarithms and complex exponents
4.1 Introduction
4.2 Evaluating integrals
4.3 Antiderivatives
5.1 Homotopy
5.2 Cauchy's Theorem
Chapter Six - More Integration
6.1 Cauchy's Integral Formula
6.2 Functions defined by integrals
6.3 Liouville's Theorem
6.4 Maximum moduli
7.1 The Laplace equation
7.2 Harmonic functions
7.3 Poisson's integral formula
Chapter Eight - Series
8.1 Sequences
8.2 Series
8.3 Power series
8.4 Integration of power series
8.5 Differentiation of power series
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9.1 Taylor series
9.2 Laurent series
10.1 Residues
10.2 Poles and other singularities
11.1 Argument principle
11.2 Rouche's Theorem
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George Cain
School of Mathematics
Georgia Institute of Technology
Atlanta, Georgia 0332-0160
cain@math.gatech.edu
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Chapter One
Complex Numbers
1.1 Introduction. Let us hark back to the first grade when the only numbers you knew
were the ordinary everyday integers. You had no trouble solving problems in which you
were, for instance, asked to find a number x such that 3 x 6. You were quick to answer
”2”. Then, in the second grade, Miss Holt asked you to find a number x such that 3 x 8.
You were stumped—there was no such ”number”! You perhaps explained to Miss Holt that
3 2   6 and 3 3   9, and since 8 is between 6 and 9, you would somehow need a number
between 2 and 3, but there isn’t any such number. Thus were you introduced to ”fractions.”
These fractions, or rational numbers, were defined by Miss Holt to be ordered pairs of
integers—thus, for instance, 8, 3 is a rational number. Two rational numbers n , m and
p , q were defined to be equal whenever nq pm . (More precisely, in other words, a
rational number is an equivalence class of ordered pairs, etc. ) Recall that the arithmetic of
these pairs was then introduced: the sum of n , m and p , q was defined by
n , m  p , q    nq pm , mq ,
and the product by
n , m  p , q    np , mq .
Subtraction and division were defined, as usual, simply as the inverses of the two
operations.
In the second grade, you probably felt at first like you had thrown away the familiar
integers and were starting over. But no. You noticed that n ,1  p ,1    n p ,1 and
also n ,1  p ,1    np ,1 . Thus the set of all rational numbers whose second coordinate is
one behave just like the integers. If we simply abbreviate the rational number n ,1 by n ,
there is absolutely no danger of confusion: 2 3 5 stands for 2, 1  3, 1    5, 1 . The
equation 3 x 8 that started this all may then be interpreted as shorthand for the equation
3, 1  u , v    8, 1 , and one easily verifies that x   u , v    8, 3 is a solution. Now, if
someone runs at you in the night and hands you a note with 5 written on it, you do not
know whether this is simply the integer 5 or whether it is shorthand for the rational number
5, 1 . What we see is that it really doesn’t matter. What we have ”really” done is
expanded the collection of integers to the collection of rational numbers. In other words,
we can think of the set of all rational numbers as including the integers–they are simply the
rationals with second coordinate 1.
One last observation about rational numbers. It is, as everyone must know, traditional to
1.1
write the ordered pair n , m as
m . Thus n stands simply for the rational number
1 , etc.
Now why have we spent this time on something everyone learned in the second grade?
Because this is almost a paradigm for what we do in constructing or defining the so-called
complex numbers. Watch.
Euclid showed us there is no rational solution to the equation x 2 2. We were thus led to
defining even more new numbers, the so-called real numbers, which, of course, include the
rationals. This is hard, and you likely did not see it done in elementary school, but we shall
assume you know all about it and move along to the equation x 2   1. Now we define
complex numbers . These are simply ordered pairs x , y of real numbers, just as the
rationals are ordered pairs of integers. Two complex numbers are equal only when there
are actually the same–that is x , y    u , v precisely when x u and y v . We define the
sum and product of two complex numbers:
x , y  u , v    x u , y v
and
x , y  u , v    xu yv , xv yu
As always, subtraction and division are the inverses of these operations.
Now let’s consider the arithmetic of the complex numbers with second coordinate 0:
x ,0  u ,0    x u ,0 ,
and
x ,0  u ,0    xu ,0 .
Note that what happens is completely analogous to what happens with rationals with
second coordinate 1. We simply use x as an abbreviation for x ,0 and there is no danger of
confusion: x u is short-hand for x ,0  u ,0    x u ,0 and xu is short-hand for
x ,0  u ,0 . We see that our new complex numbers include a copy of the real numbers, just
as the rational numbers include a copy of the integers.
Next, notice that x u , v    u , v x   x ,0  u , v    xu , xv . Now then, any complex number
z   x , y may be written
1.2
n
n
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