Complex Variables for Scientists and Engineers · 2nd Edition by Paliouras, John D. -- Read -- Imperial Library of Trantor

Index

Cover Title Page Copyright Page Dedication Preface Contents PART I: FOUNDATIONS OF COMPLEX VARIABLES

Chapter 1 Complex Numbers

Section 1 Complex Numbers and Their Algebra 2 Geometry of Complex Numbers Appendix 1 Part A: A Formal Look at Complex Numbers Part

Part B: Stereographic Projection

Chapter 2 Complex Functions

Section 3 Preliminaries 4 Definition and Elementary Geometry of a Complex Function 5 Limits, Continuity 6 Differentiation 7 The Cauchy-Riemann Equations 8 Elementary Complex Functions: Definitions and Basic Properties 9 Analytic Functions; Domains of Analyticity Appendix 2 Proofs of Theorems

Chapter 3 Harmonic Functions with Applications

Section 10 Harmonic Functions 11 Applications to Fluid Flow 12 Applications to Electrostatics Appendix 3 Part A: The Equations of Fluid Flow

Part B: Basic Laws of Electrostatics

Chapter 4 Complex Integration

Section 13 Paths; Connectedness 14 Line Integrals 15 The Complex Integral Appendix 4 Proofs of Theorems

Chapter 5 Cauchy Theory of Integration

Section 16 Integrals of Analytic Functions; Cauchy’s Theorem 17 The Annulus Theorem and Its Extension 18 The Cauchy Integral Formulas; Morera’s Theorem Appendix 5 Part A: Proofs of Theorems

Chapter 6 Complex Power Series

Section 19 Sequences and Series of Complex Numbers 20 Power Series 21 Power Series as Analytic Functions 22 Analytic Functions as Power Series Appendix 6 Part A: Proofs of Theorems

Chapter 7 Laurent Series; Residues

PART II: FURTHER THEORY AND APPLICATIONS OF COMPLEX VARIABLES

Chapter 8 Mapping Properties of Analytic Functions

Section 27 Algebraic Functions 28 Transcendental Functions 29 Behavior of Functions at Infinity Appendix 8 Part A: Riemann Surfaces of Multivalued Functions

Part B: Integration Involving Branch Points

Chapter 9 Conformai Mapping with Applications

Section 30 Conformality and Analytic Functions 31 Laplace’s Equation 32 Applications to Boundary Value Problems 33 Applications to Aerodynamics 34 The Schwarz-Christoffel Integral Appendix 9 Univalent Functions

Chapter 10 Further Theoretical Results

Bibliography Answers to Selected Exercises Index