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## Overview

*Applied Complex Variables,*designed for sequential study, is a step-by-step treatment of fundamentals, presenting superior coverage of concepts of complex analysis, including the complex number plane; functions and limits; the Cauchy-Riemann conditions for differentiability; Riemann surfaces; the definite integral; power series; meromorphic functions; and much more. The second half provides lucid exposition of five important applications of analytic function theory, each approachable independently of the others: potential theory; ordinary differential equations; Fourier transforms; Laplace transforms; and asymptotic expansions. Helpful exercises are included at the end of each topic in every chapter.

The two-part structure of

*Applied Complex Variables*affords the college instructor maximum classroom flexibility. Once fundamentals are mastered, applications can be studied in any sequence desired. Depending on how many are selected for study, Professor Dettman's impressive text is ideal for either a one- or two-semester course. And, of course, the ambitious student possessing a knowledge of basic calculus will find its straightforward approach rewarding to his independent study efforts.

*Applied Complex Variables*is a cogent, well-written introduction to an important and exciting branch of advanced mathematics — serving both the theoretical needs of the mathematics specialist and the applied math needs of the physicist and engineer. Students and teachers alike will welcome this timely, moderately priced reissue of a widely respected work.

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## Product Details

ISBN-13: | 9780486158280 |
---|---|

Publisher: | Dover Publications |

Publication date: | 04/09/2012 |

Series: | Dover Books on Mathematics |

Sold by: | Barnes & Noble |

Format: | NOOK Book |

Pages: | 512 |

File size: | 46 MB |

Note: | This product may take a few minutes to download. |

## Read an Excerpt

#### Applied Complex Variables

**By John W. Dettman**

**Dover Publications, Inc.**

**Copyright © 1965 John W. Dettman**

All rights reserved.

ISBN: 978-0-486-15828-0

All rights reserved.

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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

the usual quadratic formula. He is then told that if the discriminant *b*2 - 4*ac* ≥ 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:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Obviously, since *(x + b/2a)*2 and 4*a*2 are nonnegative, if *b*2 - 4*ac*< 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 (4*ac - b*2)/4*a* when *x = -(b/2a)*. If *b*2 - 4*ac* > 0, there are two zeros of *P(x)*, if *b*2 - 4*ac* = 0, there is one zero, and if *b*2 - 4*ac*< 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 *z*2 + 1 = 0. Obviously, this has no real solutions, since for any real *x, x*2 ≥ 0 and *x*2 + 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 *z*2 + 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 (*x*1, *y*1) we shall refer to as *z*1. If *z*1 = (*x*1, *y*1) and *z*2 = (*x*2, *y*2), then we say that *z*1 = *z*2 · if and only if *x*1 = *x*2*and y*1 = *y*2. We can prove immediately that

1. *z*1 = *z*1 for all *z*1,

2. if *z*1 = *z*2, then *z*2 = *z*1,

3. if *z*1 = *z*2 and *z*2 = *z*3, then *z*1 = *z*3.

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

Next, we define *addition*. We say that *z*1 + *z*2 = *z*3 if and only if *x*1 + *x*2 = *x*3 and *y*1 + *y*2 = *y*3. It is not difficult to prove the following properties:

1. *z*1 + *z*2 = *z*2 + *z*1 (commutativity),

2. *z*1 + (*z*2 + *z*3) = (*z*1 + *z*2) + *z*3 (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 = (*x*0, *y*0) such that *z* + 0 = *z*, or *x* + *x*0 = *x* and *y* + *y*0 = *y*. An obvious solution is *x*0 = 0, *y*0 = 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,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Next, we define *multiplication*. We say that *z*1*z*2 = *z*3 if and only if *x*3 = *x*1*x*2 - *y*1*y*2,*y*3 = *x*1*y*2 + *x*2*y*1. Again, it is easy to prove the following properties:

1. *z*1*z*2 = *z*2*z*1 (commutativity),

2. *z*1(*z*2*z*3) = (*z*1*z*2)*z*3 (associativity),

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

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

If an identity 1 = (*a, b*) exists, then 1*z* = *(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 1*z = z*, then

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

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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

These equations have the solution [xi] = *x*/(*x*2 + *y*2), η = *-y*/(*x*2 + *y*2). It is obvious that we must exclude zero as a solution, for *x*2 + *y*2 = 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,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

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*,

*z*1(*z*2 + *z*3) = *z*1*z*2 + *z*1*z*3,

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 *i*2 = (-1, 0) =

-1. Hence, the equation *z*2 = -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, *z*2 = -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 *x*1 ≠ 0 then either *-x*1 > 0 or *-x*1 > 0. Now consider *i* in *C*. If *C* were ordered like *R* either *i* or *-i* would be positive. But *i*2 = (*-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 *z*1< *z*2 when *z*1 and *z*2 are complex numbers unless both *z*1 and *z*2 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*(*z*1 + *z*2) = *az*1 + *az*2,

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

4. 1*z = 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) = *a*1 + *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*| = √ *x*2 + *y*2.

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 fromApplied Complex VariablesbyJohn W. Dettman. Copyright © 1965 John W. Dettman. Excerpted by permission of Dover Publications, Inc..

All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.

Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.

## Table of Contents

Part I. Analytic Function TheoryChapter 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