| Preface Introduction FIRST PART Chapter 1 The Complex Plane and Elementary Functions 1.Complex Numbers 2.Polar Representation 3.Stereographic Projection 4.The Square and Square Root Functions 5.The Exponential Function 6.The Logarithm Function 7.Power Functions and Phase Factors 8.Trigonometric and Hyperbolic Functions Chapter 2 Analytic Functions 1.Review of Basic Analysis 2.Analytic Functions 3.The CauChy-Riemann Equations 4.Inverse Mappings and the Jacobian 5.Harmonic Functions 6.Conformal Mappings 7.Fractional Linear Transformations Chapter 3 Line Integrals and Harmonic Functions 1.Line Integrals and Green's Theorem 2.Independence of Path 3.Harmonic Conjugates 4.The Mean Value Property 5.The Maximum Principle 6.Applications to Fluid Dynamics 7.Other Applications to Physics Chapter 4 Complex Integration and Analyticity 1.Complex Line Integrals 2.Fundamental Theorem of Calculus for Analytic Functions 3.Cauchy's Theorem 4.The Cauchy Integral Formula 5.Liouville's Theorem 6.Morera's Theorem 7.Goursat's Theorem 8.Complex Notation and Pompeiu's Formula Chapter 5 Power Series 1.Infinite Series 2.Sequences and Series of Functions 3.Power Series 4.Power Series Expansion of an Analytic Function 5.Power Series Expansion at Infinity 6.Manipulation of Power Series 7.The Zeros of an Analytic Function 8.Analytic Continuation Chapter 6 Laurent Series and Isolated Singularities 1.The Laurent Decomposition 2.Isolated Singularities of an Analytic Function 3.Isolated Singularity at Infinity 4.Partial Fractions Decomposition 5.Periodic Functions 6.Fourier Series Chapter 7 The Residue Calculus 1.The Residue Theorem 2.Integrals Featuring Rational Functions 3.Integrals of Trigonometric Functions 4.Integrands with Branch Points 5.Fractional Residues 6.Principal Values 7.Jordan's Lemma 8.Exterior Domains SECOND PART Chapter 8 The Logarithmic Integral 1.The Argument Principle 2.Rouche's Theorem 3.Hurwitz's Theorem 4.Open Mapping and Inverse Function Theorems 5.Critical Points 6.Winding Numbers …… THIRD PART Hints and Solutions for Selected Exercises References List of Symbols Index |
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