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Complex Variables, Second Edition

Book Description

Complex variables provide powerful methods for attacking problems that can be very difficult to solve in any other way, and it is the aim of this book to provide a thorough grounding in these methods and their application. Part I of this text provides an introduction to the subject, including analytic functions, integration, series, and residue calculus and also includes transform methods, ODEs in the complex plane, and numerical methods. Part II contains conformal mappings, asymptotic expansions, and the study of Riemann-Hilbert problems. The authors provide an extensive array of applications, illustrative examples and homework exercises. This new edition has been improved throughout and is ideal for use in undergraduate and introductory graduate level courses in complex variables.

Table of Contents

  1. Cover
  2. Half Title
  3. Title Page
  4. Copyright
  5. Contents
  6. Preface
  7. Part I: Fundamentals and Techniques of Complex Function Theory
    1. 1. Complex Numbers and Elementary Functions
      1. 1.1 Complex Numbers and Their Properties
      2. 1.2 Elementary Functions and Stereographic Projections
        1. 1.2.1 Elementary Functions
        2. 1.2.2 Stereographic Projections
      3. 1.3 Limits, Continuity, and Complex Differentiation
        1. 1.3.1 Elementary Applications to Ordinary Differential Equations
    2. 2. Analytic Functions and Integration
      1. 2.1 Analytic Functions
        1. 2.1.1 The Cauchy–Riemann Equations
        2. 2.1.2 Ideal Fluid Flow
      2. 2.2 Multivalued Functions
      3. 2.3 More Complicated Multivalued Functions and Riemann Surfaces
      4. 2.4 Complex Integration
      5. 2.5 Cauchy’s Theorem
      6. 2.6 Cauchy’s Integral Formula, Its ∂[Overbar] Generalization and Consequences
        1. 2.6.1 Cauchy’s Integral Formula and Its Derivatives
        2. 2.6.2 Liouville, Morera, and Maximum-Modulus Theorems
        3. 2.6.3 Generalized Cauchy Formula and ∂[Overbar] Derivatives
      7. 2.7 Theoretical Developments
    3. 3. Sequences, Series, and Singularities of Complex Functions
      1. 3.1 Definitions and Basic Properties of Complex Sequences, Series
      2. 3.2 Taylor Series
      3. 3.3 Laurent Series
      4. 3.4 Theoretical Results for Sequences and Series
      5. 3.5 Singularities of Complex Functions
        1. 3.5.1 Analytic Continuation and Natural Barriers
      6. 3.6 Infinite Products and Mittag–Leffler Expansions
      7. 3.7 Differential Equations in the Complex Plane: Painlevé Equations
      8. 3.8 Computational Methods
        1. 3.8.1 Laurent Series
        2. 3.8.2 Differential Equations
    4. 4. Residue Calculus and Applications of Contour Integration
      1. 4.1 Cauchy Residue Theorem
      2. 4.2 Evaluation of Certain Definite Integrals
      3. 4.3 Principal Value Integrals and Integrals with Branch Points
        1. 4.3.1 Principal Value Integrals
        2. 4.3.2 Integrals with Branch Points
      4. 4.4 The Argument Principle, Rouché’s Theorem
      5. 4.5 Fourier and Laplace Transforms
      6. 4.6 Applications of Transforms to Differential Equations
  8. Part II: Applications of Complex Function Theory
    1. 5. Conformal Mappings and Applications
      1. 5.1 Introduction
      2. 5.2 Conformal Transformations
      3. 5.3 Critical Points and Inverse Mappings
      4. 5.4 Physical Applications
      5. 5.5 Theoretical Considerations – Mapping Theorems
      6. 5.6 The Schwarz–Christoffel Transformation
      7. 5.7 Bilinear Transformations
      8. 5.8 Mappings Involving Circular Arcs
      9. 5.9 Other Considerations
        1. 5.9.1 Rational Functions of the Second Degree
        2. 5.9.2 The Modulus of a Quadrilateral
        3. 5.9.3 Computational Issues
    2. 6. Asymptotic Evaluation of Integrals
      1. 6.1 Introduction
        1. 6.1.1 Fundamental Concepts
        2. 6.1.2 Elementary Examples
      2. 6.2 Laplace Type Integrals
        1. 6.2.1 Integration by Parts
        2. 6.2.2 Watson’s Lemma
        3. 6.2.3 Laplace’s Method
      3. 6.3 Fourier Type Integrals
        1. 6.3.1 Integration by Parts
        2. 6.3.2 Analog of Watson’s Lemma
        3. 6.3.3 The Stationary Phase Method
      4. 6.4 The Method of Steepest Descent
        1. 6.4.1 Laplace’s Method for Complex Contours
      5. 6.5 Applications
      6. 6.6 The Stokes Phenomenon
        1. 6.6.1 Smoothing of Stokes Discontinuities
      7. 6.7 Related Techniques
        1. 6.7.1 WKB Method
        2. 6.7.2 The Mellin Transform Method
    3. 7. Riemann–Hilbert Problems
      1. 7.1 Introduction
      2. 7.2 Cauchy Type Integrals
      3. 7.3 Scalar Riemann–Hilbert Problems
        1. 7.3.1 Closed Contours
        2. 7.3.2 Open Contours
        3. 7.3.3 Singular Integral Equations
      4. 7.4 Applications of Scalar Riemann–Hilbert Problems
        1. 7.4.1 Riemann–Hilbert Problems on the Real Axis
        2. 7.4.2 The Fourier Transform
        3. 7.4.3 The Radon Transform
      5. 7.5 Matrix Riemann–Hilbert Problems
        1. 7.5.1 The Riemann–Hilbert Problem for Rational Matrices
        2. 7.5.2 Inhomogeneous Riemann–Hilbert Problems and Singular Equations
        3. 7.5.3 The Riemann–Hilbert Problem for Triangular Matrices
        4. 7.5.4 Some Results on Zero Indices
      6. 7.6 The DBAR Problem
        1. 7.6.1 Generalized Analytic Functions
      7. 7.7 Applications of Matrix Riemann–Hilbert Problems and ∂[Overbar] Problems
  9. Appendix A: Answers to Odd-Numbered Exercises
  10. Bibliography
  11. Index