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Introductory Complex and Analysis Applications
1st Edition - January 1, 1972
Author: William R. Derrick
eBook ISBN:
9 7 8 - 1 - 4 8 3 2 - 6 0 4 8 - 8
Introductory Complex and Analysis Applications provides an introduction to the functions of a complex variable, emphasizing applications. This book covers a variety of topics,… Read more
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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.
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
- No. of pages: 232
- Language: English
- Published: January 1, 1972
- Imprint: Academic Press
- eBook ISBN: 9781483260488