Complex analysis / for mathematics and engineering

Contents
Preface ix
Chapter 1
Complex Numbers
1
1.1 The Origin of Complex Numbers 1
1.2 The Algebra of Complex Numbers 5
1.3 The Geometry of Complex Numbers 12
1.4 The Geometry of Complex Numbers, Continued 18
1.5 The Algebra of Complex Numbers, Revisited 24
1.6 The Topology of Complex Numbers 30
Chapter 2
Complex Functions
38
2.1 Functions of a Complex Variable 38
2.2 Transformations and Linear Mappings 41
2.3 The Mappings w = zn and w = z1/n 47
2.4 Limits and Continuity 53
2.5 Branches of Functions 60
2.6 The Reciprocal Transformation w = 1/z (Prerequisite for Section 9.2) 64
Chapter 3
Analytic and Harmonic Functions
71
3.1 Differentiable Functions 71
3.2 The Cauchy-Riemann Equations 76
3.3 Analytic Functions and Harmonic Functions 84
Chapter 4
Sequences, Series, and Julia and Mandelbrot Sets
95
4.1 Definitions and Basic Theorems for Sequences and Series 95
4.2 Power Series Functions 109
4.3 Julia and Mandelbrot Sets 116
Page vi
Chapter 5
Elementary Functions
125
5.1 The Complex Exponential Function 125
5.2 Branches of the Complex Logarithm Function 132
5.3 Complex Exponents 138
5.4 Trigonometric and Hyperbolic Functions 143
5.5 Inverse Trigonometric and Hyperbolic Functions 152
Chapter 6
Complex Integration
157
6.1 Complex Integrals 157
6.2 Contours and Contour Integrals 160
6.3 The Cauchy-Goursat Theorem 175
6.4 The Fundamental Theorems of Integration 189
6.5 Integral Representations for Analytic Functions 195
6.6 The Theorems of Morera and Liouville and Some
Applications 201
Chapter 7
Taylor and Laurent Series
208
7.1 Uniform Convergence 208
7.2 Taylor Series Representations 214
7.3 Laurent Series Representations 223
7.4 Singularities, Zeros, and Poles 232
7.5 Applications of Taylor and Laurent Series 239
Chapter 8
Residue Theory
244
8.1 The Residue Theorem 244
8.2 Calculation of Residues 246
8.3 Trigonometric Integrals 252
8.4 Improper Integrals of Rational Functions 256
8.5 Improper Integrals Involving Trigonometric Functions 260
8.6 Indented Contour Integrals 264
8.7 Integrands with Branch Points 270
8.8 The Argument Principle and Rouché's Theorem 274
Chapter 9
Conformal Mapping
281
9.1 Basic Properties of Conformal Mappings 281
9.2 Bilinear Transformations 287
9.3 Mappings Involving Elementary Functions 294
9.4 Mapping by Trigonometric Functions 303
Page vii
Chapter 10
Applications of Harmonic Functions
310
10.1 Preliminaries 310
10.2 Invariance of Laplace's Equation and the Dirichlet Problem 312
10.3 Poisson's Integral Formula for the Upper Half Plane 323
10.4 Two-Dimensional Mathematical Models 327
10.5 Steady State Temperatures 329
10.6 Two-Dimensional Electrostatics 342
10.7 Two-Dimensional Fluid Flow 349
10.8 The Joukowski Airfoil 360
10.9 The Schwarz-Christoffel Transformation 369
10.10 Image of a Fluid Flow 380
10.11 Sources and Sinks 384
Chapter 11
Fourier Series and the Laplace Transform
397
11.1 Fourier Series 397
11.2 The Dirichlet Problem for the Unit Disk 406
11.3 Vibrations in Mechanical Systems 412
11.4 The Fourier Transform 418
11.5 The Laplace Transform 422
11.6 Laplace Transforms of Derivatives and Integrals 430
11.7 Shifting Theorems and the Step Function 433
11.8 Multiplication and Division by t 438
11.9 Inverting the Laplace Transform 441
11.10 Convolution 448
Appendix A
Undergraduate Student Research Projects
456
Bibliography 458
Answers to Selected Problems 466
Index 477

Dirigida a estudiantes de pregrado de matemáticas, física o ingeniería, la sexta edición de Análisis Complejo para Matemáticas e Ingeniería ofrece una presentación completa y práctica de esta interesante área de las matemáticas. Los autores logran un equilibrio entre los aspectos puros y aplicados de la materia, presentando los conceptos con un estilo claro y apropiado para estudiantes de penúltimo y último año de carrera. Gracias a su presentación completa y accesible, y a sus numerosas aplicaciones, la sexta edición de este clásico permite a los estudiantes resolver con facilidad incluso las demostraciones más complejas. Los nuevos ejercicios ayudan a los estudiantes a evaluar su comprensión del material y a evaluar su progreso en el curso. También se encuentran disponibles en línea ejercicios adicionales de Mathematica y Maple, así como una guía de estudio para estudiantes.

Ingeniería en Tecnologías de la Información y Comunicación