Complex Analysis for Mathematics and Engineering

Revised and updated, the new Fifth Edition of Complex Analysis for Mathematics and Engineering presents a comprehensive, student-friendly introduction to Complex Analysis. It's clear, concise writing style and numerous applications make the foundations of the subject matter easily accessible for students and proofs are presented at an elementary level that is understandable by students with a sophomore calculus background. Believing that mathematicians, engineers, and scientists should be exposed to a careful presentation of mathematics, attention to topics such as ensuring required assumptions are met before the use of a theorem or algebraic operations are applied. A new chapter on Z-Transforms and Applications provides students with a current look at Digital Filter Design and Signal Processing. 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