Introductory Complex and Analysis Applications

Introductory Complex and Analysis Applications

1st Edition - January 1, 1972

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  • Author: William R. Derrick
  • eBook ISBN: 9781483260488

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Description

Introductory Complex and Analysis Applications provides an introduction to the functions of a complex variable, emphasizing applications. This book covers a variety of topics, including integral transforms, asymptotic expansions, harmonic functions, Fourier transformation, and infinite series. Organized into eight chapters, this book begins with an overview of the theory of functions of a complex variable. This text then examines the properties of analytical functions, which are all consequences of the differentiability of the function. Other chapters consider the converse of Taylor's Theorem, namely that convergent power series are analytical functions in their domain of convergence. This book discusses as well the Residue Theorem, which is of fundamental significance in complex analysis and is the core concept in the development of the techniques. The final chapter deals with the method of steepest descent, which is useful in determining the asymptotic behavior of integral representations of analytic functions. This book is a valuable resource for undergraduate students in engineering and mathematics.

Table of Contents


  • Preface

    Table of Symbols

    1. Analytic Functions

    1.1. Complex Numbers

    1.2. Properties of the Complex Plane

    1.3. Functions of a Complex Variable

    1.4. Sufficient Conditions for Analyticity

    1.5. Some Elementary Functions

    1.6. Continuation

    Notes

    2. Complex Integration

    2.1. Line Integrals

    2.2. The Cauchy-Goursat Theorem

    2.3. The Fundamental Theorem of Integration

    2.4. The Cauchy Integral Formula

    2.5. Liouville's Theorem and the Maximum Principle

    Notes

    3. Infinite Series

    3.1. Taylor Series

    3.2. Uniform Convergence of Series

    3.3. Laurent Series

    3.4. Isolated Singularities

    3.5. Analytic Continuation

    3.6. Riemann Surfaces

    Notes

    4. Contour Integration

    4.1. The Residue Theorem

    4.2. Evaluation of Improper Real Integrals

    4.3. Continuation

    4.4. Integration of Multivalued Functions

    4.5. Other Integration Techniques

    4.6. The Argument Principle

    Notes

    5. Conformal Mappings

    5.1. General Properties

    5.2. Linear Fractional Transformations

    5.3. Continuation

    5.4. The Schwarz-Christoffel Formula

    5.5. Physical Applications

    Notes

    6. Boundary-Value Problems

    6.1. Harmonic Functions

    6.2. Poisson's Integral Formula

    6.3. Applications

    Notes

    7. Fourier and Laplace Transformations

    7.1. Fourier Series

    7.2. Fourier Transforms

    7.3. Laplace Transforms

    7.4. Properties of Laplace Transforms

    Notes

    8. Asymptotic Expansions

    8.1. Definitions and Properties

    8.2. Method of Steepest Descent

    8.3. Continuation

    Notes

    Appendix

    A.1. Table of Conformal Mappings

    References

    Index

Product details

  • No. of pages: 232
  • Language: English
  • Copyright: © Academic Press 1972
  • Published: January 1, 1972
  • Imprint: Academic Press
  • eBook ISBN: 9781483260488

About the Author

William R. Derrick

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