Complex Variables for Scientists and Engineers: Second Edition

by John D. Paliouras, Douglas S. Meadows

Overview

This outstanding text for undergraduate students of science and engineering requires only a standard course in elementary calculus. Designed to provide a thorough understanding of fundamental concepts and create the basis for higher-level courses, the treatment features numerous examples and extensive exercise sections of varying difficulty, plus answers to selected exercises.
The two-part approach begins with the development of the primary concept of analytic function, advancing to the Cauchy integral theory, the series development of analytic functions through evaluation of integrals by residues, and some elementary applications of harmonic functions. The second part introduces some of the deeper aspects of complex function theory: mapping properties of analytic functions, applications to various vector field problems with boundary conditions, and a collection of further theoretical results. 1990 edition.

Product Details

ISBN-13:	9780486782225
Publisher:	Dover Publications
Publication date:	02/19/2014
Series:	Dover Books on Mathematics
Sold by:	Barnes & Noble
Format:	eBook
Pages:	608
File size:	54 MB
Note:	This product may take a few minutes to download.

About the Author

Douglas S. Meadows is Professor of Mathematics and Head of the School of Mathematical Sciences at the Rochester Institute of Technology.
John S. Paliouras is Professor Emeritus at the Rochester Institute of Technology and the sole author of this text's first edition.

Table of Contents

Preface vii

Part I Foundations of Complex Variables

Chapter 1 Complex Numbers 3

Section 1 Complex Numbers and Their Algebra 4

2 Geometry of Complex Numbers 10

Appendix 1 Part A: A Formal Look at Complex Numbers 26

Part B Stereographic Projection 30

Chapter 2 Complex Functions 33

Section 3 Preliminaries 34

4 Definition and Elementary Geometry of a Complex Function 38

5 Limits, Continuity 47

6 Differentiation 54

7 The Cauchy-Riemann Equations 61

8 Elementary Complex Functions: Definitions and Basic Properties 66

9 Analytic Functions; Domains of Analyticity 85

Appendix 2 Proofs of Theorems 90

Chapter 3 Harmonic Functions with Applications 96

Section 10 Harmonic Functions 97

11 Applications to Fluid Flow 108

12 Applications to Electrostatics 123

Appendix 3 Part A: The Equations of Fluid Flow 136

Part B Basic Laws of Electrostatics 144

Chapter 4 Complex Integration 158

Section 13 Paths; Connectedness 159

14 Line Integrals 166

15 The Complex Integral 177

Appendix 4 Proofs of Theorems 189

Chapter 5 Cauchy Theory of Integration 193

Section 16 Integrals of Analytic Functions; Cauchy's Theorem 194

17 The Annulus Theorem and Its Extension 204

18 The Cauchy Integral Formulas; Morera's Theorem 211

Appendix 5 Part A: Proofs of Theorems 219

Part B Proof of the Cauchy Integral Theorem 224

Part C The Winding Number and the Generalized Cauchy Theorems 241

Chapter 6 Complex Power Series 248

Section 19 Sequences and Series of Complex Numbers 249

20 Power Series 260

21 Power Series as Analytic Functions 266

22 Analytic Functions as Power Series 274

Appendix 6 Part A: Proofs of Theorems 288

Part B More on Sequences and Series; The Cauchy-Hadamard Theorem 291

Chapter 7 Laurent Series; Residues 298

Section 23 Laurent Series 299

24 Singularities and Zeros of an Analytic Function 308

25 Theory of Residues 317

26 Evaluation of Certain Real Integrals by Use of Residues 325

Appendix 7 Proof of Laurent's Theorem; Uniqueness of Taylor and Laurent Expansions 339

Part II Further Theory and Applications of Complex Variables

Chapter 8 Mapping Properties of Analytic Functions 345

Section 27 Algebraic Functions 346

28 Transcendental Functions 380

29 Behavior of Functions at Infinity 403

Appendix 8 Part A: Riemann Surfaces of Multivalued Functions 410

Part B Integration Involving Branch Points 418

Chapter 9 Conformal Mapping with Applications 424

Section 30 Conformality and Analytic Functions 425

31 Laplace's Equation 437

32 Applications to Boundary Value Problems 459

33 Applications to Aerodynamics 484

34 The Schwarz-Christoffel Integral 497

Appendix 9 Univalent Functions 525

Chapter 10 Further Theoretical Results 529

Section 35 The Maximum Modulus Principle 530

36 Liouville's Theorem; The Fundamental Theorem of Algebra 536

37 Behavior of Functions Near Isolated Singularities 540

38 Analytic Continuation and the Schwarz Reflection Principle 544

Bibliography 555

Answers to Selected Exercises 557

Index 577