Author Profile

Michael Taylor is Professor Emeritus of Mathematics at The University of Michigan-Dearborn. He earned his A.B. in physics from Harvard University and his A.M. and Ph.D. in mathematics from The University of Michigan in Ann Arbor, where he was an Institute of Science and Technology Predoctoral Fellow. He is co-author with Dr. Churchill of Fourier series and Boundary Value Problems, now in its eighth edition. He has received a research grant from the National Science Foundation as well as a Distinguished Faculty Award from the Michigan Association of Governing Boards of Colleges and Universities. Dr. Brown is listed in Who's Who in the World.

About The Book

Complex analysis is one of the most attractive of all the core topics in an undergraduate mathematics course. Its importance to applications means that it can be studied both from a very pure perspective and a very applied perspective. This book takes account of these varying needs and backgrounds and provides a self-study text for students in mathematics, science and engineering. Beginning with a summary of what the student needs to know at the outset, it covers all the topics likely to feature in a first course in the subject, including: complex numbers, differentiation, integration, Cauchy's theorem, and its consequences, Laurent series and the residue theorem, applications of contour integration, conformal mappings, and harmonic functions. A brief final chapter explains the Riemann hypothesis, the most celebrated of all the unsolved problems in mathematics, and ends with a short descriptive account of iteration, Julia sets and the Mandelbrot set. Clear and careful explanations are backed up with worked examples and more than 100 exercises, for which full solutions are provided. This textbook is intended for a one semester course in complex analysis for upper level undergraduates in mathematics. Applications, primary motivations for this text, are presented hand-in-hand with theory enabling this text to serve well in courses for students in engineering or applied sciences. The overall aim in designing this text is to accommodate students of different mathematical backgrounds and to achieve a balance between presentations of rigorous mathematical proofs and applications. Chapter topics include complex numbers and functions, analytic functions, complex integration, complex series, residues: applications and theory, conformal mapping, partial differential equations: methods and applications, transform methods, and partial differential equations in polar and spherical coordinates. The text is adapted to enable maximum flexibility to instructors and to students who may also choose to progress through the material outside of coursework. Detailed examples may be covered in one course, giving the instructor the option to choose those that are best suited for discussion.

Table of Contents

Preface............................................................................................................. v
Chapter 1. Complex Numbers and the Complex Plane............................... 1
€ Review of Real Numbers and Algebra............................................... 4
€ Definition of Complex Numbers....................................................... 14
€ Geometric Interpretation in the Complex Plane.............................. 21
€ Polar and Exponential Forms............................................................. 23
Chapter 2. Analytic Functions........................................................................ 27
€ Definition and Examples of Analytic Functions.............................. 30
€ Limits and Continuity in the Complex Plane.................................. 34
€ Differentiability and the Cauchy-Riemann Equations................... 41
€ Harmonic Functions and Laplace’s Equation.................................. 44
€ Elementary Analytic Functions.......................................................... 48
€ Inverse and Composite Functions..................................................... 52
Chapter 3. Complex Integration..................................................................... 59
€ Contours and Contour Integrals........................................................ 62
€ Properties of Complex Integrals........................................................ 63
€ The Cauchy-Goursat Theorem........................................................... 67
€ Applications to Real Integrals............................................................ 70
€ Independence of Path.......................................................................... 74
€ Evaluation Techniques for Definite Integrals................................... 77
Chapter 4. Series Representations................................................................. 82
€ Sequences ............................................................................................. 84
€ Convergence of Series......................................................................... 92
€ Power Series and Radius of Convergence........................................ 95
€ Taylor Series........................................................................................ 105
€ Laurent Series..................................................................................... 110
€ Classification of Singularities........................................................... 112
Chapter 5. Residues and Poles...................................................................... 116
€ Isolated Singularities......................................................................... 118
Contents
€ Residue Theorem............................................................................... 119
€ Poles and Essential Singularities..................................................... 122
€ Evaluation of Real Integrals via Residues...................................... 124
€ Applications to Physics and Signals................................................ 128
Chapter 6. Conformal Mapping................................................................... 133
€ Concept of Conformal Maps............................................................ 135
€ Linear and Möbius Transformations.............................................. 139
€ Mapping Circles and Lines............................................................... 151
€ Schwarz-Christoffel Transformation............................................... 155
€ Applications in Fluid Dynamics...................................................... 157
€ Use in Electrostatics and Heat Transfer.......................................... 161
Chapter 7. Laplace Transforms and Applications.................................... 173
€ Definition and Properties of Laplace Transforms......................... 175
€ Inverse Laplace Transforms............................................................. 183
€ Inverse Function Theorem................................................................ 184
€ Use of Residues in Inverse Transforms.......................................... 187
€ Differential Equations....................................................................... 190
€ Circuit Analysis Using Laplace Transforms.................................. 195
Chapter 8. Fourier Analysis and Complex Methods................................ 203
€ Fourier Series Overview.................................................................... 205
€ Analytic Signals.................................................................................. 210
€ Complex Representation of Harmonics.......................................... 215
€ Hilbert Transform.............................................................................. 219
€ Signal Processing Applications........................................................ 221
€ Use in Vibrational Analysis.............................................................. 226
Chapter 9. Applications to Partial Differential Equations...................... 230
€ Laplace’s Equation in Complex Form............................................. 232
€ Heat and Wave Equations................................................................. 236
€ Use of Separation of Variables......................................................... 240
€ Complex Potentials in Fluid Flow................................................... 243
€ Method of Images.............................................................................. 247
€ Boundary Value Problems................................................................ 248
Bibliography.............................................................................................. 251