Complex variables

Front cover......Page 1
Series......Page 2
Title page......Page 3
Date-line......Page 4
Dedication......Page 5
Preface to the 2nd Edition......Page 7
Preface to the 1st Edition......Page 9
A Note to the Student......Page 11
Contents......Page 13
1.1 The Complex Numbers and the Complex Plane......Page 17
1.1.1* A Formal View of the Complex Numbers......Page 26
1.2 Some Geometry......Page 28
1.3 Subsets of the Plane......Page 38
1.4 Functions and Limits......Page 46
1.5 The Exponential, Logarithm, and Trigonometric Functions......Page 59
1.6 Line Integrals and Green\'s Theorem......Page 72
2.1 Analytic and Harmonic Functions; The Cauchy-Riemann Equations......Page 93
2.1.1* Flows, Fields, and Analytic Functions......Page 102
2.2 Power Series......Page 109
2.3 Cauchy\'s Theorem and Cauchy\'s Formula......Page 122
2.3.1* The Cauchy-Goursat Theorem......Page 135
2.4 Consequences of Cauchy\'s Formula......Page 139
2.5 Isolated Singularities......Page 151
2.6 The Residue Theorem and Its Application to the Evaluation of Definite Integrals......Page 169
3.1 The Zeros of an Analytic Function......Page 187
3.1.1* The Stability of Solutions of a System of Linear Differential Equations......Page 199
3.2 Maximum Modulus and Mean Value......Page 207
3.3 Linear Fractional Transformations......Page 212
3.4 Conformal Mapping......Page 224
3.4.1* Conformal Mapping and Flows......Page 235
3.5 The Riemann Mapping Theorem and Schwarz-Christoffel Transformations......Page 240
4.1 Harmonic Functions......Page 261
4.2 Harmonic Functions as Solutions to Physical Problems......Page 270
4.3 Integral Representations of Harmonic Functions......Page 300
4.4 Boundary-Value Problems......Page 314
4.5 Impulse Functions and the Green\'s Function of a Domain......Page 325
5.1 The Fourier Transform: Basic Properties......Page 334
5.2 Formulas Relating $u$ and $hat{u}$......Page 351
5.3 The Laplace Transform......Page 362
5.4 Application: of the Laplace Transform to Differential Equations......Page 372
5.5 The $Z$-Transform......Page 381
5.5.1* The Stability of a Discrete Linear System......Page 390
Appendix 1 Locating the Zeros of a Polynomial......Page 397
Appendix 2 A Table of Conformal Mappings......Page 405
Appendix 3 A Table of Laplace Transforms......Page 411
Solutions to Odd-Numbered Exercises......Page 413
Index......Page 441
Series......Page 444
Back cover......Page 445