Handbook of Complex Variables

This book is written to be a convenient reference for the working scientist, student, or engineer who needs to know and use basic concepts in complex analysis. It is not a book of mathematical theory. It is instead a book of mathematical practice. All the basic ideas of complex analysis, as well as many typical applica­ tions, are treated. Since we are not developing theory and proofs, we have not been obliged to conform to a strict logical ordering of topics. Instead, topics have been organized for ease of reference, so that cognate topics appear in one place. Required background for reading the text is minimal: a good ground­ ing in (real variable) calculus will suffice. However, the reader who gets maximum utility from the book will be that reader who has had a course in complex analysis at some time in his life. This book is a handy com­ pendium of all basic facts about complex variable theory. But it is not a textbook, and a person would be hard put to endeavor to learn the subject by reading this book.

Table of Contents:
1 The Complex Plane.- 1.1 Complex Arithmetic.- 1.2 The Exponential and Applications.- 1.3 Holomorphic Functions.- 1.4 The Relationship of Holomorphic and Harmonic Functions.- 2 Complex Line Integrals.- 2.1 Real and Complex Line Integrals.- 2.2 Complex Differentiability and Conformality.- 2.3 The Cauchy Integral Theorem and Formula.- 2.4 A Coda on the Limitations of the Cauchy Integral Formula.- 3 Applications of the Cauchy Theory.- 3.1 The Derivatives of a Holomorphic Function.- 3.2 The Zeros of a Holomorphic Function.- 4 Isolated Singularities and Laurent Series.- 4.1 The Behavior of a Holomorphic Function near an Isolated Singularity.- 4.2 Expansion around Singular Points.- 4.3 Examples of Laurent Expansions.- 4.4 The Calculus of Residues.- 4.5 Applications to the Calculation of Definite Integrals and Sums.- 4.6 Meromorphic Functions and Singularities at Infinity.- 5 The Argument Principle.- 5.1 Counting Zeros and Poles.- 5.2 The Local Geometry of Holomorphic Functions.- 5.3 Further Results on the Zeros of Holomorphic Functions.- 5.4 The Maximum Principle.- 5.5 The Schwarz Lemma.- 6 The Geometric Theory of Holomorphic Functions.- 6.1 The Idea of a Conformal Mapping.- 6.2 Conformal Mappings of the Unit Disc.- 6.3 Linear Fractional Transformations.- 6.4 The Riemann Mapping Theorem.- 6.5 Conformal Mappings of Annuli.- 7 Harmonic Functions.- 7.1 Basic Properties of Harmonic Functions.- 7.2 The Maximum Principle and the Mean Value Property.- 7.3 The Poisson Integral Formula.- 7.4 Regularity of Harmonic Functions.- 7.5 The Schwarz Reflection Principle.- 7.6 Harnack’s Principle.- 7.7 The Dirichlet Problem and Subharmonic Functions.- 7.8 The General Solution of the Dirichlet Problem.- 8 Infinite Series and Products.- 8.1 Basic Concepts Concerning Infinite Sums andProducts.- 8.2 The Weierstrass Factorization Theorem.- 8.3 The Theorems of Weierstrass and Mittag-Leffler.- 8.4 Normal Families.- 9 Applications of Infinite Sums and Products.- 9.1 Jensen’s Formula and an Introduction to Blaschke Products.- 9.2 The Hadamard Gap Theorem.- 9.3 Entire Functions of Finite Order.- 10 Analytic Continuation.- 10.1 Definition of an Analytic Function Element.- 10.2 Analytic Continuation along a Curve.- 10.3 The Monodromy Theorem.- 10.4 The Idea of a Riemann Surface.- 10.5 Picard’s Theorems.- 11 Rational Approximation Theory.- 11.1 Runge’s Theorem.- 11.2 Mergelyan’s Theorem.- 12 Special Classes of Holomorphic Functions.- 12.1 Schlicht Functions and the Bieberbach Conjecture.- 12.2 Extension to the Boundary of Conformal Mappings.- 12.3 Hardy Spaces.- 13 Special Functions.- 13.0 Introduction.- 13.1 The Gamma and Beta Functions.- 13.2 Riemann’s Zeta Function.- 13.3 Some Counting Functions and a Few Technical Lemmas.- 14 Applications that Depend on Conformal Mapping.- 14.1 Conformal Mapping.- 14.2 Application of Conformal Mapping to the Dirichlet Problem.- 14.3 Physical Examples Solved by Means of Conformal Mapping.- 14.4 Numerical Techniques of Conformal Mapping.- Appendix to Chapter 14: A Pictorial Catalog of Conformal Maps.- 15 Transform Theory.- 15.0 Introductory Remarks.- 15.1 Fourier Series.- 15.2 The Fourier Transform.- 15.3 The Laplace Transform.- 15.4 The z-Transform.- 16 Computer Packages for Studying Complex Variables.- 16.0 Introductory Remarks.- 16.1 The Software Packages.- Glossary of Terms from Complex Variable Theory and Analysis.- List of Notation.- Table of Laplace Transforms.- A Guide to the Literature.- References.