Author(s): John H. Mathews and Russell W. Howell
Edition: 5
Publisher: Jones & Bartlet Publishers
Year: 2006
Language: English
Pages: 650
Tags: Математика;Комплексное исчисление;
1 Complex Numbers 1
1.1 The Origin of Complex Numbers 1
1.2 The Algebra of Complex Numbers 7
1.3 The Geometry of Complex Numbers 16
1.4 The Geometry of Complex Numbers, Continued 22
1.5 The Algebra of Complex Numbers, Revisited 31
1.6 The Topology of Complex Numbers 38
2 Complex Functions 49
2.1 Functions and Linear Mappings 49
2.2 The Mappings $w=z^n$ and $w=z^\frac{1}{n}$ 63
2.3 Limits and Continuity 70
2.4 Branches of Functions 70
2.5 The Reciprocal Transformation $w=\frac{1}{z}$ 85
3 Analytic and Harmonic Functions 93
3.1 Differentiable and Analytic Functions 93
3.2 The Cauchy-Riemann Equations 100
3.3 Harmonic Functions 112
4 Sequences, Julia and Mandelbrot Sets, and Power Series 123
4.1 Sequences and Series 123
4.2 Julia and Mandelbrot Sets 132
4.3 Geometric Series and Convergence Theorems 141
4.4 Power Series Functions 147
5 Elementary Functions 155
5.1 The Complex Exponential Function 155
5.2 The Complex Logarithm 163
5.3 Complex Exponents 170
5.4 Trigonometric and Hyperbolic Functions 176
5.5 Inverse Trigonometric and Hyperbolic Functions 188
6 Complex Integration 193
6.1 Complex Integrals 193
6.2 Contours and Contour Integrals 198
6.3 The Cauchy-Goursat Theorem 214
6.4 The Fundamental Theorems of Integration 229
6.5 Integral Representations for Analytic Functions 235
6.6 The Theorems of Morera and Liouville, and Extensions 241
7 Taylor and Laurent Series 249
7 .1 Uniform Convergence 249
7.2 Taylor Series Representations 256
7.3 Laurent Series Representations 267
7.4 Singularities, Zeros, and Poles 276
7.5 Applications of Taylor and Laurent Series 285
8 Residue Theory 291
8.1 The Residue Theorem 291
8.2 Trigonometric Integrals 301
8.3 Improper Integrals of Rational Functions 306
8.4 Improper Integrals Involving Trigonometric Functions 311
8.5 Indented Contour Integrals 316
8.6 Integrands with Branch Points 322
8.7 The Argument Principle and Rouche's Theorem 327
9 $z$-Transforms and Applications 337
9.1 The $z$-Transform 337
9.2 Second-Order Homogeneous Difference Equations 358
9.3 Digital Signal Filters 373
10 Conformal Mapping 395
10.1 Basic Properties of Conformal Mappings 395
10.2 Bilinear Transformations 402
10.3 Mappings Involving Elementary Functions 410
10.4 Mapping by Trigonometric Functions 418
11 Applications of Harmonic Functions 425
11.1 Preliminaries 425
11.2 Invariance of Laplace's Equation and the Dirichlet Problem 427
11.3 Poisson's Integral Formula for the Upper Half-Plane 439
11.4 Two-Dimensional Mathematical Models 444
11.5 Steady State Temperatures 446
11.6 Two-Dimensional Electrostatics 459
11.7 Two-Dimensional Fluid Flow 466
11.8 The Joukowski Airfoil 477
11.9 The Schwarz-Christoffel Transformation 486
11.10 Image of a Fluid Flow 496
11.11 Sources and Sinks 499
12 Fourier Series and the Laplace Transform 513
12.1 Fourier Series 513
12.2 The Dirichlet Problem for the Unit Disk 523
12.3 Vibrations in Mechanical Systems 529
12.4 The Fourier Transform 536
12.5 The Laplace Transform 541
12.6 Laplace Transforms of Derivatives and Integrals 549
12.7 Shifting Theorems and the Step Function 553
12.8 Multiplication and Division by $t$ 559
12.9 Inverting the Laplace Transform 562
12.10 Convolution 571
Answers 581
Index 625
