Complex Variables: A Physical Approach with Applications and MATLAB

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Overview

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.

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Product Details

  • ISBN-13: 9781584885801
  • Publisher: Taylor & Francis
  • Publication date: 8/31/2007
  • Series: Textbooks in Mathematics Series , #1
  • Edition description: New Edition
  • Pages: 440
  • Product dimensions: 7.20 (w) x 10.00 (h) x 1.10 (d)

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

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