Complex Analysis for Mathematics and Engineering

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