Complex Variables

Complex Variables : A Physical Approach with Applications and MATLAB

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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(R), Maple(TM), and MATLAB(R), 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 more

Product details

  • Hardback | 358 pages
  • 175.26 x 251.46 x 27.94mm | 975.22g
  • Taylor & Francis Inc
  • Chapman & Hall/CRC
  • Boca Raton, FL, United States
  • English
  • 115 black & white illustrations, 8 black & white tables
  • 1584885807
  • 9781584885801
  • 2,405,248

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 INDEXshow more

About Steven G. Krantz

Washington University, St. Louis, Missouri, USAshow more