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Contents xi
PART 6 Complex Functions 667
CHAPTER 19 Complex Numbers and Functions 669
19.1 Geometry and Arithmetic of Complex Numbers 669
19.2 Complex Functions 676
19.2.1 Limits, Continuity, and Differentiability 677
19.2.2 The Cauchy-Riemann Equations 680
19.3 The Exponential and Trigonometric Functions 684
19.4 The Complex Logarithm 689
19.5 Powers 690
CHAPTER 20 Complex Integration 695
20.1 The Integral of a Complex Function 695
20.2 Cauchy’s Theorem 700
20.3 Consequences of Cauchy’s Theorem 703
20.3.1 Independence of Path 703
20.3.2 The Deformation Theorem 704
20.3.3 Cauchy’s Integral Formula 706
20.3.4 Properties of Harmonic Functions 709
20.3.5 Bounds on Derivatives 710
20.3.6 An Extended Deformation Theorem 711
20.3.7 A Variation on Cauchy’s Integral Formula 713
CHAPTER 21 Series Representations of Functions 715
21.1 Power Series 715
21.2 The Laurent Expansion 725
CHAPTER 22 Singularities and the Residue Theorem 729
22.1 Singularities 729
22.2 The Residue Theorem 733
22.3 Evaluation of Real Integrals 740
22.3.1 Rational Functions 740
22.3.2 Rational Functions Times Cosine or Sine 742
22.3.3 Rational Functions of Cosine and Sine 743
22.4 Residues and the Inverse Laplace Transform 746
22.4.1 Diffusion in a Cylinder 748
CHAPTER 23 Conformal Mappings and Applications 751
23.1 Conformal Mappings 751
23.2 Construction of Conformal Mappings 765
23.2.1 The Schwarz-Christoffel Transformation 773
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