Complex Variables: A Physical Approach with Applications and MATLAB, 1st Edition (Hardback) book cover

Complex Variables

A Physical Approach with Applications and MATLAB, 1st Edition

By Steven G. Krantz

Chapman and Hall/CRC

358 pages | 115 B/W Illus.

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pub: 2007-09-19
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Description

From the algebraic properties of a complete number field, to the analytic properties imposed by the Cauchy integral formula, to the geometric qualities originating from conformality, Complex Variables: A Physical Approach with Applications and MATLAB explores all facets of this subject, with particular emphasis on using theory in practice.

The first five chapters encompass the core material of the book. These chapters cover fundamental concepts, holomorphic and harmonic functions, Cauchy theory and its applications, and isolated singularities. Subsequent chapters discuss the argument principle, geometric theory, and conformal mapping, followed by a more advanced discussion of harmonic functions. The author also presents a detailed glimpse of how complex variables are used in the real world, with chapters on Fourier and Laplace transforms as well as partial differential equations and boundary value problems. The final chapter explores computer tools, including Mathematica®, Maple™, and MATLAB®, that can be employed to study complex variables. Each chapter contains physical applications drawing from the areas of physics and engineering.

Offering new directions for further learning, this text provides modern students with a powerful toolkit for future work in the mathematical sciences.

Reviews

". . . presents various topics which rarely appear in a textbook form . . . proofs are carefully chosen . . . has many illustrations which clarify key concepts from complex variable theory . . . strongly recommended to everybody interested in modern complex analysis."

– Marek Jarnicki, in Zentralblatt Math, 2008/2009, Vol. 1131

Table of Contents

PREFACE

BASIC IDEAS

Complex Arithmetic

Algebraic and Geometric Properties

The Exponential and Applications

HOLOMORPHIC AND HARMONIC FUNCTIONS

Holomorphic Functions

Holomorphic and Harmonic Functions

Real and Complex Line Integrals

Complex Differentiability

The Logarithm

THE CAUCHY THEORY

The Cauchy Integral Theorem

Variants of the Cauchy Formula

The Limitations of the Cauchy Formula

APPLICATIONS OF THE CAUCHY THEORY

The Derivatives of a Holomorphic Function

The Zeros of a Holomorphic Function

ISOLATED SINGULARITIES

Behavior near an Isolated Singularity

Expansion around Singular Points

Examples of Laurent Expansions

The Calculus of Residues

Applications to the Calculation of Integrals

Meromorphic Functions

THE ARGUMENT PRINCIPLE

Counting Zeros and Poles

Local Geometry of Functions

Further Results on Zeros

The Maximum Principle

The Schwarz Lemma

THE GEOMETRIC THEORY

The Idea of a Conformal Mapping

Mappings of the Disc

Linear Fractional Transformations

The Riemann Mapping Theorem

Conformal Mappings of Annuli

A Compendium of Useful Conformal Mappings

APPLICATIONS OF CONFORMAL MAPPING

Conformal Mapping

The Dirichlet Problem

Physical Examples

Numerical Techniques

HARMONIC FUNCTIONS

Basic Properties of Harmonic Functions

The Mean Value Property

The Poisson Integral Formula

TRANSFORM THEORY

Introductory Remarks

Fourier Series

The Fourier Transform

The Laplace Transform

A Table of Laplace Transforms

The z-Transform

PDES AND BOUNDARY VALUE PROBLEMS

Fourier Methods

COMPUTER PACKAGES

Introductory Remarks

The Software Packages

APPENDICES

Solutions to Odd-Numbered Exercises

Glossary of Terms

List of Notation

A Guide to the Literature

BIBLIOGRAPHY

INDEX

About the Series

Textbooks in Mathematics

Learn more…

Subject Categories

BISAC Subject Codes/Headings:
MAT003000
MATHEMATICS / Applied
MAT007000
MATHEMATICS / Differential Equations
MAT037000
MATHEMATICS / Functional Analysis