Complex Analysis With Applications

This volume contains the basics of what every scientist and engineer should know about complex analysis. A lively style combined with a simple, direct approach helps readers grasp the fundamentals, from complex numbers, limits in the complex plane, and complex functions to Cauchy's theory, power series, and applications of residues. 1974 edition.

Preface
1. Complex Numbers
  1.1. Basic Concepts
  1.2. The Complex Plane
  1.3. The Modulus and Argument
  1.4. Inversion
    Comments
    Problems
2. Limits in the Complex Plane
  2.1. The Principle of Nested Rectangles
  2.2. Limit Points
  2.3. Convergent Complex Sequences
  2.4. The Riemann Sphere and the Extended Complex Plane
    Comments
    Problems
3. Complex Functions
  3.1. Basic Concepts
  3.2. Curves and Domains
  3.3. Continuity of a Complex Function
  3.4. Uniform Continuity
    Comments
    Problems
4. Differentiation in the Complex Plane
  4.1. The Derivative of a Complex Function
  4.2. The Cauchy-Riemann Equations
  4.3. Conformal Mapping
    Comments
    Problems
5. Integration in the Complex Plane
  5.1. The Integral of a Complex Function
  5.2. Basic Properties of the Integral
  5.3. Integrals along Polygonal Curves
  5.4. Cauchy's Integral Theorem
  5.5. Indefinite Complex Integrals
  5.6. Cauchy's Integral Formula
  5.7. Infinite Differentiability of Analytic Functions
  5.8. Harmonic Functions
    Comments
    Problems
6. Complex Series
  6.1. Convergence vs. Divergence
  6.2. Absolute vs. Conditional Convergence
  6.3. Uniform Convergence
    Comments
    Problems
7. Power Series
  7.1. Basic Theory
  7.2. Determination of the Radius of Convergence
    Comments
    Problems
8. Some Special Mappings
  8.1. The Exponential and Related Functions
  8.2. Fractional Linear Transformations
    Comments
    Problems
9. Multiple-Valued Functions
  9.1. Domains of Univalence
  9.2. Branches and Branch Points
  9.3. Riemann Surfaces
    Comments
    Problems
10. Taylor Series
  10.1. The Taylor Expansion of an Analytic Function
  10.2. Uniqueness Theorems
  10.3. The Maximum Modulus Principle and Its Implications
    Comments
    Problems
11. Laurent Series
  11.1. The Laurent Expansion of an Analytic Function
  11.2. Isolated Singular Points
  11.3. Residues
    Comments
    Problems
12. Applications of Residues
  12.1. Logarithmic Residues and the Argument Principle
  12.2. Rouché's Theorem and Its Implications
  12.3. Evaluation of Improper Real Integrals
  12.4. Integrals Involving Multiple-Valued Functions
    Comments
    Problems
13. Further Theory
  13.1. More on Harmonic Functions
  13.2. The Dirichlet Problem
  13.3. More Conformal Mapping
  13.4. Analytic Continuation
  13.5. The Symmetry Principle
    Comments
    Problems
14. Mapping of Polygonal Domains
  14.1. The Schwarz-Christoffel Transformation
  14.2. Examples
    Comments
    Examples
15. Some Physical Applications
  15.1. Fluid Dynamics
  15.2. Examples
  15.3. Electrostatics
    Comments
    Problems
Selcted Hints and Answers
Bibliography
Index