# Applied Complex Variables

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ISBN-13: 9780486158280 Dover Publications 04/09/2012 Dover Books on Mathematics Barnes & Noble NOOK Book 512 46 MB This product may take a few minutes to download.

#### Applied Complex Variables

By John W. Dettman

#### Dover Publications, Inc.

ISBN: 978-0-486-15828-0

CHAPTER 1

The Complex Number Plane

1.1. INTRODUCTION

Probably the first time a school child comes across the need for any kind of number other than the real numbers is in an algebra course where he studies the quadratic equation ax2 + bx + c = 0. He is shown by quite elementary methods that the roots of this equation must be of the form

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the usual quadratic formula. He is then told that if the discriminant b2 - 4ac ≥ 0, then the roots of the quadratic equation are real. Otherwise, they are complex. It must seem very strange to the student, who has known only the real numbers and is told that a, b, c, and x in the equation are real, that he must be confronted with this strange new beast, the complex number. Actually, the difficulty could be avoided if the problem were stated in the following way: For what real values of x does the polynomial P(x) = ax2 + bx + c, with a, b, and c real, and a ≠ 0, take on the value zero? Then P(x) can be written as follows:

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Obviously, since (x + b/2a)2 and 4a2 are nonnegative, if b2 - 4ac< 0, then P(x) is never zero, and the answer, in this case, is that there are no real values of x for which P(x) = 0. The notion of a complex number need not enter the discussion. Furthermore, if the student has any notion of "the graph of a function," then there is an obvious geometrical picture which makes the situation quite clear. Let us assume, without loss of generality, that a > 0. Then, clearly, P(x) takes on a minimum value of (4ac - b2)/4a when x = -(b/2a). If b2 - 4ac > 0, there are two zeros of P(x), if b2 - 4ac = 0, there is one zero, and if b2 - 4ac< 0, there are no zeros. See Figure 1.1.1.

Actually, the discussion of complex roots of the quadratic equation need not come up at all unless we are trying to answer the more difficult question: For what values of the complex variable z does the polynomial P(z) = az2 + bz + c take on the value zero? Here, a, b, and c may be complex numbers. But this plunges us into the heart of our subject and raises many interesting questions: What is a complex number? What is a function of a complex variable? What are some of the more interesting properties of functions of a complex variable? Do polynomial functions have zeros? (Fundamental Theorem of Algebra), and so on. In trying to answer these questions and others which will most certainly come up, we shall uncover a very rich branch of mathematics, which has and will have far reaching implications and great applicability to other branches of science.

The first notion of a complex number was discovered in connection with solving quadratic equations. Consider, for example, the equation z2 + 1 = 0. Obviously, this has no real solutions, since for any real x, x2 ≥ 0 and x2 + 1 > 0. We are tempted to write z = ± √-1 but there is no real number which when squared yields -1. If the equation is to have a solution, it must be in some system of numbers larger than the set of reals. This then was the problem faced by seventeenth-century mathematicians: To extend the reals to a larger system of numbers in which the equation z2 + 1 = 0 would have a solution. In the next section, we begin a systematic discussion of the complex numbers which is the result of this extension of the reals.

1.2. COMPLEX NUMBERS

Consider the collection (set) of all ordered pairs of real numbers (x, y). A particular member of the collection (x1, y1) we shall refer to as z1. If z1 = (x1, y1) and z2 = (x2, y2), then we say that z1 = z2 · if and only if x1 = x2and y1 = y2. We can prove immediately that

1. z1 = z1 for all z1,

2. if z1 = z2, then z2 = z1,

3. if z1 = z2 and z2 = z3, then z1 = z3.

When a relation satisfies these three properties we say that it is an equivalence relation.

Next, we define addition. We say that z1 + z2 = z3 if and only if x1 + x2 = x3 and y1 + y2 = y3. It is not difficult to prove the following properties:

1. z1 + z2 = z2 + z1 (commutativity),

2. z1 + (z2 + z3) = (z1 + z2) + z3 (associativity),

3. there exists a zero, 0, such that for every z, z + 0 = z,

4. for every z there exists a negative - z such that z + (-z) = 0.

In proving Property 3, we must find some member of the collection 0 = (x0, y0) such that z + 0 = z, or x + x0 = x and y + y0 = y. An obvious solution is x0 = 0, y0 = 0. The zero of the system is unique, for if z + 0 = z and z + 0' = z, then

0' = 0' + 0 = 0 + 0' = 0.

The existence of a negative for each z is easily established. As a matter of fact, -z = (-x, - y), since

z + (-z) = (x, y) + (-x, - y) = (x - x, y - y) = (0, 0) = 0.

We define subtraction in terms of addition of the negative, that is,

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Next, we define multiplication. We say that z1z2 = z3 if and only if x3 = x1x2 - y1y2,y3 = x1y2 + x2y1. Again, it is easy to prove the following properties:

1. z1z2 = z2z1 (commutativity),

2. z1(z2z3) = (z1z2)z3 (associativity),

3. there exists an identity 1 such that for every z, 1z = z,

4. for every z, other than zero, there exists an inverse z-1 such that z-1z = 1.

If an identity 1 = (a, b) exists, then 1z = (ax - by, ay + bx) = (x, y). Hence, x = ax - by and y = ay + bx. One solution of this is a = 1 and b = 0. This is the only solution since the identity is unique. Indeed, if 1'z = z and 1z = z, then

1' = 11' = 1'1 = 1.

For z-1 we must have, if z-1 = ([xi], η),

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These equations have the solution [xi] = x/(x2 + y2), η = -y/(x2 + y2). It is obvious that we must exclude zero as a solution, for x2 + y2 = 0 if and only if x = 0 and y = 0, which is the only case where the above equations do not have a unique solution. We define division in terms of multiplication by the inverse, that is,

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Hence, division by any element other than zero is defined.

Definition 1.2.1. The complex number system C is the collection of all ordered pairs of real numbers along with the equality, addition operation, and multiplication operation defined above.

We see immediately that the algebraic structure of C is much like that of the real numbers. For example, Properties 1-4 for addition and multiplication hold also for the real number system. Both the reals and C are examples of an algebraic field. This can be easily verified by establishing the distributive law,

z1(z2 + z3) = z1z2 + z1z3,

in addition to the above listed properties. To show that we have not merely reproduced the reals, we should display at least one element of C which is not a real number. Consider the number i = (0, 1); then i2 = (-1, 0) =

-1. Hence, the equation z2 = -1 has at least one solution, namely i, in C, and we have already verified that this equation has no solution in the system of reals. Actually, z2 = -1 has two solutions ±.

The real numbers behave as a proper subset of C. Consider the subset R in C consisting of all ordered pairs of the form (x, 0), where x is real. There is a one-to-one correspondence between the reals and the elements of R, which preserves the operations of addition and multiplication,

(x1, 0) + (x2, 0) = (x1 + x2, 0), (x1, 0)(x2, 0) = (x1x2,0).

We say that R is isomorphic to the system of reals. In other words, there are just as many elements in R as there are real numbers and the two systems behave algebraically alike. We shall henceforth simply refer to R as the system of reals.

There is another very important difference between C and R. R is an ordered field, whereas C is not. We know, for example, that for every pair of reals a and b either a < b, a = b, or a > b. Furthermore, if x1 ≠ 0 then either -x1 > 0 or -x1 > 0. Now consider i in C. If C were ordered like R either i or -i would be positive. But i2 = (-i)2 = -1 and this would imply that -1 is positive. However -1 and 1 cannot both be positive. Hence, it is meaningless to write z1< z2 when z1 and z2 are complex numbers unless both z1 and z2 are in R.

As a matter of definition, we distinguish between the two numbers in our ordered pairs by calling the first the real part and the second the imaginary part; that is, if z = (x, y) then x = Re(z) and y = Im(z). It should be noted that Im(z) is a real number.

In C we also have the notion of multiplication by a real scalar. Let a be real; then

az = (a, 0)(x, y) = (ax, ay).

We see that multiplication by the real scalar has the effect of multiplying both the real and imaginary parts by the scalar. It is easy to verify that

1. (a + b) z = az + bz,

2. a(z1 + z2) = az1 + az2,

3. a(bz) = (ab)z,

4. 1z = z.

These properties plus Properties 1-4 for addition imply that C is a linear vector space over the field of reals.

Let a and b be real. Then

(a, b) = a(1, 0) + b(0, 1) = a1 + bi = a + bi.

Hence, in the future, we shall simply write z = x + iy, which is the more familiar notation for the complex number.

The modulus |z| of a complex number z is defined as follows

|z| = √ x2 + y2.

The modulus is obviously a nonnegative real number and |z| = 0 if and only if z = 0.

The conjugate z of a complex number z = x + iy is obtained by changing the sign of the imaginary part,

z = x -iy

(Continues...)

Excerpted from Applied Complex Variables by John W. Dettman. Copyright © 1965 John W. Dettman. Excerpted by permission of Dover Publications, Inc..
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Part I. Analytic Function Theory
Chapter 1. The Complex Number Plane
1.1 Introduction
1.2 Complex Numbers
1.3 The Complex Plane
1.4 Point Sets in the Plane
1.5 Stereographic Projection. The Extended Complex Plane
1.6 Curves and Regions
Chapter 2. Functions of a Complex Variable
2.1 Functions and Limits
2.2 Differentiability and Analyticity
2.3 The Cauchy-Riemann Conditions
2.4 Linear Fractional Transformations
2.5 Transcendental functions
2.6 Riemann Surfaces
Chapter 3. Integration in the Complex Plane
3.1 Line Integrals
3.2 The Definite Integral
3.3 Cauchy's Theorem
3.4 Implications of Cauchy's Theorem
3.5 Functions Defined by Integration
3.6 Cauchy Formulas
3.7 Maximum Modulus Principle
Chapter 4. Sequences and Series
4.1 Sequences of Complex Numbers
4.2 Sequences of Complex Functions
4.3 Infinite Series
4.4 Power Series
4.5 Analytic Continuation
4.6 Laurent Series
4.7 Double Series
4.8 Infinite Products
4.9 Improper Integrals
4.10 The Gamma Function
Chapter 5. Residue Calculus
5.1 The Residue Theorem
5.2 Evaluation of Real Integrals
5.3 The Principle of the Argument
5.4 Meromorphic Functions
5.5 Entire Functions
Part II. Applications of Analytic Function Theory
Chapter 6. Potential Theory
6.1 Laplace's Equation in Physics
6.2 The Dirichlet Problem
6.3 Green's Functions
6.4 Conformal Mapping
6.5 The Schwarz-Christoffel Transformation
6.6 Flows with Sources and Sinks
6.7 Volume and Surface Distributions
6.8 Singular Integral Equations
Chapter 7. Ordinary Differential Equations
7.1 Separation of Variables
7.2 Existence and Uniqueness Theorems
7.3 Solution of a Linear Second-Order Differential Equation Near an Ordinary Point
7.4 Solution of a Linear Second-Order Differential Equation Near a Regular Singular Point
7.5 Bessel Functions
7.6 Legendre Functions
7.7 Sturm-Liouville Problems
7.8 Fredholm Integral Equations
Chapter 8. Fourier Transforms
8.1 Fourier Series
8.2 The Fourier Integral Theorem
8.3 The Complex Fourier Transform
8.4 Properties of the Fourier Transform
8.5 The Solution of Ordinary Differential Equations
8.6 The Solution of Partial Differential Equations
8.7 The Solution of Integral Equations
Chapter 9. Laplace Transforms
9.1 From Fourier to Laplace Transform
9.2 Properties of the Laplace Transform
9.3 Inversion of Laplace Transforms
9.4 The Solution of Ordinary Differential Equations
9.5 Stability
9.6 The Solution of Partial Differential Equations
9.7 The Solution of Integral Equations
Chapter 10. Asymptotic Expansions
10.1 Introduction and Definitions
10.2 Operations on Asymptotic Expansions
10.3 Asymptotic Expansion of Integrals
10.4 Asymptotic Solutions of Ordinary Differential Equations
References; Index

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