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| 1 The Complex Plane 1.1 The Complex Numbers and the Complex Plane 1.1.1 A Formal View of the Complex Numbers 1.2 Some Geometry 1.3 Subsets of the Plane 1.4 Functions and Limits 1.5 The Exponential, Logarithm, and Trigonometric Functions 1.6 Line Integrals and Green's Theorem 2 Basic Properties of Analytic Functions 2.1 Analytic and Harmonic Functions; The Cauchv-Riemann Equations 2.1.1 Flows, Fields, and Analytic Functions 2.2 Power Series 2.3 Cauchy's Theorem and Cauchy's Formula 2.3.1 The Cauchy-Goursat Theorem 2.4 Consequences of Cauchy's Formula 2.5 Isolated Singularities 2.6 The Residue Theorem ana its Application to the Evaluation of Definite Integrals 3 Analytic Functions as Mappings 3.1 The Zeros of an Analytic Function 3.1.1 The Stability of Solutions of a System of Linear Differential Equations 3.2 Maximum Modulus and Mean Value 3.3 Linear Fractional Transformations 3.4 Conformal Mapping 3.4.1 Conformal Mapping and Flows 3.5 The Riemann Mapping Theorem and Schwarz-Christoffel Transforma-tions 4 Analytic and Harmonic Functions in Applications 4.1 Harmonic Functions 4.2 Harmonic Functions as Solutions to Physical Problems 4.3 Integral Representations of Harmonic Functions 4.4 Boundary-Value Problems 4.5 Impulse Functions and the Green's Function of a Domain 5 Transform Methods 5.1 The Fourier Transform: Basic Properties 5.2 Formulas Relating u and 5.3 The Laplace Transform 5.4 Application; of the Laplace Transform to Differential Equations 5.5 The Z-Transform 365 5.5.1 The Stability of a Discrete Linear System Appendix 1 Locating the Zeros of a Polynomial Appendix 2 A Table of Conformal Mappings Appendix 3 A Table of Laplace Transforms Solutions to Odd-Numbered Exercises Index |
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