Handbook of complex variables

Handbook of Complex Variables is a comprehensive reference work for scientists and engineers who need to know and use essential information and methods involving complex variables and analysis. Its focus is on basic concepts and informational tools for mathematical 'practice': solving problems in applied mathematics, science and engineering. The information is self-contained and accessible to a broad readership. All the indispensable ideas are presented, as well as applications topics and a brief survey of available computer software. The material has been carefully organized for quick, convenient reference by specialists and non-specialists alike. Features and Topics: Comprehensive table of notation Extensive glossary of key terms Detailed subject index A catalog of conformal maps Extensive examples of evaluating indefinite integrals using the calculus of residues Generously illustrated with helpful figures and graphs Brief survey of available computer software Carefully worked examples for all key concepts Tables and charts to summarize information for ease of use, i.e., conformal mappings, equivalent definitions and equivalent concepts Conformal mapping applications Coverage of basic transform theory This handbook is an essential reference and authoritative resource for all professionals, practitioners, and researchers in mathematics, physical sciences and engineering. Specialists and non-specialists will find its practical, problem-solving style both accessible and useful for their work. 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 and 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 Residue 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 Holomorphic Functions as Geometric Mappings 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 and Products 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 Theorem 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 13.1 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 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 [f(z), Mathematica, Maple, Matlab, Ricci] Glossary of Terms from Complex Variable Theory and Analysis List of Notation Table of Laplace Transforms A Guide to the Literature Bibliography Index