"Complex Analysis
George Cain
(c)Copyright 1999 by George Cain. All rights reserved.
Table of Contents
Chapter One - Complex Numbers 1.1 Introduction 1.2 Geometry 1.3 Polar coordinates Chapter Two - Complex Functions 2.1 Functions of a real variable 2.2 Functions of a complex variable 2.3 Derivatives Chapter Three - Elementary Functions 3.1 Introduction 3.2 The exponential function 3.3 Trigonometric functions 3.4 Logarithms and complex exponents Chapter Four - Integration 4.1 Introduction 4.2 Evaluating integrals 4.3 Antiderivatives Chapter Five - Cauchy's Theorem 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 Chapter Seven - Harmonic Functions 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
Chapter Nine - Taylor and Laurent Series 9.1 Taylor series 9.2 Laurent series Chapter Ten - Poles, Residues, and All That 10.1 Residues 10.2 Poles and other singularities Chapter Eleven - Argument Principle 11.1 Argument principle 11.2 Rouche's Theorem
---------------------------------------------------------------------------George Cain School of Mathematics Georgia Institute of Technology Atlanta, Georgia 0332-0160 cain@math.gatech.edu
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 3x 6. You were quick to answer ”2”. Then, in the second grade, Miss Holt asked you to find a number x such that 3x 8. You were stumped—there was no such ”number”! You perhaps explained to Miss Holt that 32 6 and 33 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, mp, 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, 1p, 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 3x 8 that started this all may then be interpreted as shorthand for the equation 3, 1u, 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
n m
. Thus n stands simply for the rational number
n 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, yu, 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, 0u, 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, 0u, 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 xu, v u, vx x, 0u, v xu, xv. Now then, any complex number z x, y may be written
1.2
z x, y x, 0 0, y x y0, 1 When we let 0, 1, then we have z x, y x y Now, suppose z x, y x y and w u, v u v. Then we have zw x yu v xu xv yu 2 yv We need only see what 2 is: 2 0, 10, 1 1, 0, and we have agreed that we can safely abbreviate 1, 0 as 1. Thus, 2 1, and so zw xu yv xv yu and we have reduced the fairly complicated definition of complex arithmetic simply to ordinary real arithmetic together with the fact that 2 1. Let’s take a look at division–the inverse of multiplication. Thus number you must multiply w by in order to get z . An example: x y x y u v z w u v u v u v xu yv yu xv u2 v2 xu yv yu xv 2 2 2 u v u v2 Note this is just fine except when u 2 v 2 0; that is, when u v 0. We may thus divide by any complex number except 0 0, 0. One final note in all this. Almost everyone in the world except an electrical engineer uses the letter i to denote the complex number we have called . We shall accordingly use i rather than to stand for the number 0, 1. Exercises
z w
stands for that complex
1.3
1. Find the following complex numbers in the form x iy: a) 4 7i2 3i b) 1 i 3 b) 52i c) 1 i 1i 2. Find all complex z x, y such that z2 z 1 0 3. Prove that if wz 0, then w 0 or z 0.
1.2. Geometry. We now have this collection of all ordered pairs of real numbers, and so there is an uncontrollable urge to plot them on the usual coordinate axes. We see at once then there is a one-to-one correspondence between the complex numbers and the points in the plane. In the usual way, we can think of the sum of two complex numbers, the point in the plane corresponding to z w is the diagonal of the parallelogram having z and w as sides:
We shall postpone until the next section the geometric interpretation of the product of two complex numbers. The modulus of a complex number z x iy is defined to be the nonnegative real number x 2 y 2 , which is, of course, the length of the vector interpretation of z. This modulus is traditionally denoted |z|, and is sometimes called the length of z. Note that |x, 0| x 2 |x|, and so || is an excellent choice of notation for the modulus. The conjugate z of a complex number z x iy is defined by z x iy. Thus |z| 2 z z . Geometrically, the conjugate of z is simply the reflection of z in the horizontal axis:
1.4
Observe that if z x iy and w u iv, then z w x u iy v x iy u iv z w. In other words, the conjugate of the sum is the sum of the conjugates. It is also true that zw z w. If z x iy, then x is called the real part of z, and y is called the imaginary part of z. These are usually denoted Re z and Im z, respectively. Observe then that z z 2 Re z and z z 2 Im z. Now, for any two complex numbers z and w consider |z w| 2 z wz w z w z w z z w z wz ww |z| 2 2 Rew z |w| 2 |z| 2 2|z||w| |w| 2 |z| |w| 2 In other words, |z w| |z| |w| the so-called triangle inequality. (This inequality is an obvious geometric fact–can you guess why it is called the triangle inequality?) Exercises 4. a)Prove that for any two complex numbers, zw z w. z z b)Prove that w w . c)Prove that ||z| |w|| |z w|.
z 5. Prove that |zw| |z||w| and that | w | |z| |w|
.
1.5
6. Sketch the set of points satisfying a) |z 2 3i| 2 c) Re z i 4 e)|z 1| |z 1| 4
b)|z 2i| 1 d) |z 1 2i| |z 3 i| f) |z 1| |z 1| 4
1.3. Polar coordinates. Now let’s look at polar coordinates r, of complex numbers. Then we may write z rcos i sin . In complex analysis, we do not allow r to be negative; thus r is simply the modulus of z. The number is called an argument of z, and there are, of course, many different possibilities for . Thus a complex numbers has an infinite number of arguments, any two of which differ by an integral multiple of 2. We usually write arg z. The principal argument of z is the unique argument that lies on the interval , . Example. For 1 i, we have 1i 2 cos 7 i sin 7 4 4 2 cos i sin 4 4 399 i sin 399 2 cos 4 4
7 4
etc., etc., etc. Each of the numbers principal argument is . 4
, , and 4
399 4
is an argument of 1 i, but the
Suppose z rcos i sin and w scos i sin . Then zw rcos i sin scos i sin rscos cos sin sin isin cos sin cos rscos i sin We have the nice result that the product of two complex numbers is the complex number whose modulus is the product of the moduli of the two factors and an argument is the sum of arguments of the factors. A picture:
1.6
We now define expi, or e i by e i cos i sin We shall see later as the drama of the term unfolds that this very suggestive notation is an excellent choice. Now, we have in polar form z re i , where r |z| and is any argument of z. Observe we have just shown that e i e i e i . It follows from this that e i e i 1. Thus 1 e i e i It is easy to see that re i r z w se i s cos i sin Exercises 7. Write in polar form re i : a) i c) 2 e) 3 3i
b) 1 i d) 3i
8. Write in rectangular form—no decimal approximations, no trig functions: a) 2e i3 b) e i100 c) 10e i/6 d) 2 e i5/4 9. a) Find a polar form of 1 i1 i 3 . b) Use the result of a) to find cos 7 and sin 12 10. Find the rectangular form of 1 i 100 .
7 12
.
1.7
11. Find all z such that z 3 1. (Again, rectangular form, no trig functions.) 12. Find all z such that z 4 16i. (Rectangular form, etc.)
1.8
Chapter Two
Complex Functions
2.1. Functions of a real variable. A function : I C from a set I of reals into the complex numbers C is actually a familiar concept from elementary calculus. It is simply a function from a subset of the reals into the plane, what we sometimes call a vector-valued function. Assuming the function is nice, it provides a vector, or parametric, description of a curve. Thus, the set of all t : t e it cos t i sin t cos t, sin t, 0 t 2 is the circle of radius one, centered at the origin. We also already know about the derivatives of such functions. If t xt iyt, then the derivative of is simply t x t iy t, interpreted as a vector in the plane, it is tangent to the curve described by at the point t. Example. Let t t it 2 , 1 t 1. One easily sees that this function describes that part of the curve y x 2 between x 1 and x 1:
1
-1
-0.5
0
0.5 x
1
Another example. Suppose there is a body of mass M ”fixed” at the origin–perhaps the sun–and there is a body of mass m which is free to move–perhaps a planet. Let the location of this second body at time t be given by the complex-valued function zt. We assume the only force on this mass is the gravitational force of the fixed body. This force f is thus zt f GMm 2 |zt| |zt| where G is the universal gravitational constant. Sir Isaac Newton tells us that zt mz t f GMm 2 |zt| |zt|
2.1
Hence, z GM z |z| 3 Next, let’s write this in polar form, z re i : d 2 re i k e i r2 dt 2 where we have written GM k. Now, let’s see what we have. d re i r d e i dr e i dt dt dt Now, d e i d cos i sin dt dt sin i cos d dt icos i sin d dt d e i . i dt (Additional evidence that our notation e i cos i sin is reasonable.) Thus, d re i r d e i dr e i dt dt dt r i d e i dr e i dt dt dr ir d e i . dt dt Now,
2.2
d 2 re i dt 2
Now, the equation
d2 dt 2
d 2 r i dr dt dt 2 dr ir d dt dt d2r r dt 2
d ir d 2 e i dt dt 2 i d e i dt d 2 i r d 2 2 dr d dt dt dt dt 2
e i
re i rk2 e i becomes d 2 r r d dt dt 2
2 2 i r d 2 2 dr d dt dt dt
k2 . r
This gives us the two equations d 2 r r d dt dt 2 and,
2 r d 2 2 dr d 0. dt dt dt 2
k2 , r
Multiply by r and this second equation becomes d r 2 d dt dt This tells us that r 2 d dt is a constant. (This constant is called the angular momentum.) This result allows us to get rid of d in the first of the two differential equations above: dt d2r r dt 2 r2 or, d2r 2 k . r2 dt 2 r3 2.3
2
0.
k2 r
Although this now involves only the one unknown function r, as it stands it is tough to solve. Let’s change variables and think of r as a function of . Let’s also write things in terms of the function s 1 . Then, r d d d d . dt dt d r 2 d Hence, dr dr ds , d dt r 2 d and so d 2 r d ds dt d dt 2 2 2 s 2 d s , d 2 and our differential equation looks like d 2 r 2 2 s 2 d 2 s 2 s 3 ks 2 , dt 2 d 2 r3 or, d2s s k . 2 d 2 This one is easy. From high school differential equations class, we remember that s 1 A cos k2 , r where A and are constants which depend on the initial conditions. At long last, r 2 /k , 1 cos s 2 d ds d d
where we have set A 2 /k. The graph of this equation is, of course, a conic section of eccentricity .
Exercises 2.4
1. a)What curve is described by the function t 3t 4 it 6, 0 t 1 ? b)Suppose z and w are complex numbers. What is the curve described by t 1 tw tz, 0 t 1 ? 2. Find a function that describes that part of the curve y 4x 3 1 between x 0 and x 10. 3. Find a function that describes the circle of radius 2 centered at z 3 2i . 4. Note that in the discussion of the motion of a body in a central gravitational force field, it was assumed that the angular momentum is nonzero. Explain what happens in case 0.
2.2 Functions of a complex variable. The real excitement begins when we consider function f : D C in which the domain D is a subset of the complex numbers. In some sense, these too are familiar to us from elementary calculus—they are simply functions from a subset of the plane into the plane: fz fx, y ux, y ivx, y ux, y, vx, y Thus fz z 2 looks like fz z 2 x iy 2 x 2 y 2 2xyi. In other words, ux, y x 2 y 2 and vx, y 2xy. The complex perspective, as we shall see, generally provides richer and more profitable insights into these functions. The definition of the limit of a function f at a point z z 0 is essentially the same as that which we learned in elementary calculus: lim fz L
zz 0
means that given an 0, there is a so that |fz L| whenever 0 |z z 0 | . As you could guess, we say that f is continuous at z 0 if it is true that lim fz fz 0 . If f is continuous at each point of its domain, we say simply that f is continuous. Suppose both lim fz and lim gz exist. Then the following properties are easy to establish:
zz 0 zz 0 zz 0
2.5
lim fz gz lim fz lim gz
zz 0 zz 0 zz 0
lim fzgz lim fz lim gz
zz 0 zz 0 zz 0
and, lim
zz 0
lim fz fz zz 0 lim gz gz
zz 0
provided, of course, that lim gz 0.
zz 0
It now follows at once from these properties that the sum, difference, product, and quotient of two functions continuous at z 0 are also continuous at z 0 . (We must, as usual, except the dreaded 0 in the denominator.) It should not be too difficult to convince yourself that if z x, y, z 0 x 0 , y 0 , and fz ux, y ivx, y, then lim fz
zz 0
x,yx 0 ,y 0
lim
ux, y i
x,yx 0 ,y 0
lim
vx, y
Thus f is continuous at z 0 x 0 , y 0 precisely when u and v are. Our next step is the definition of the derivative of a complex function f. It is the obvious thing. Suppose f is a function and z 0 is an interior point of the domain of f . The derivative f z 0 of f is f z 0 lim
zz 0
fz fz 0 z z0
Example Suppose f..."
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