Applications of Complex Variables: Asymptotics and Integral Transforms (De Gruyter Textbook)

✍ Scribed by Foluso Ladeinde

• Publisher: De Gruyter
• Year: 2024
• Tongue: English
• Leaves: 606
• Edition: 1
• Category: Library

✦ Synopsis

The subject of applied complex variables is so fundamental that most of the other topics in advanced engineering mathematics (AEM) depend on it. The present book contains complete coverage of the subject, summarizing the more elementary aspects that you find in most AEM textbooks and delving into the more specialized topics that are less commonplace. The book represents a one-stop reference for complex variables in engineering. The applications of conformal mapping in this book are significantly more extensive than in other AEM textbooks. The treatments of complex integral transforms enable a much larger class of functions that can be transformed, resulting in an expanded use of complex-transform techniques in engineering analysis. The inclusion of the asymptotics of complex integrals enables the analysis of models with irregular singular points. The book, which has more than 300 illustrations, is generous with realistic example problems.

✦ Table of Contents

About the author
Preface
Acknowledgments
Contents
Abbreviations
Nomenclature
Part I: Introduction to complex variables
Introduction to Part I
1 Introductory concepts
1.1 Preliminary concepts
1.2 Polar form of complex variables
1.3 Roots of numbers
1.4 Functions of a complex variable
1.5 Trigonometric and hyperbolic functions
1.6 Inverse trigonometric and hyperbolic functions
1.7 Differentiation of a function of a complex variable
1.8 A note on the adequacy of the C-R conditions
1.9 Regular functions
1.10 Multivalued functions
1.11 Riemann surface
1.12 Two-dimensional integrals
1.13 Cauchy-Goursat theorem
1.14 Singularities and Cauchy’s theorems
1.15 Poisson’s integral formula
1.16 Miscellaneous examples
1.17 Problems
1.18 Suggested reading
2 Laurent series, residue theorem, and contour integration
2.1 Taylor series
2.2 Analytic continuation
2.3 Laurent series
2.4 Classification of isolated singularities
2.5 Residue theorem and contour integration
2.6 Contour integration
2.7 A few other theorems on improper integrals
2.8 Miscellaneous examples
2.9 Problems
2.10 Suggested reading
3 Conformal mapping and its applications
3.1 Preservation of angles and sense of direction
3.2 Invariance of Laplace equation under conformal mapping
3.3 Common transformations
3.4 Application of eccentric circle conformal mapping in engineering analysis
3.5 Transformation of boundary conditions on harmonic functions
3.6 Multiple transformations
3.7 Finding a harmonic function in the UHP
3.8 More examples of conformal transformation
3.9 The Schwarz-Christoffel transformation
3.10 Miscellaneous examples
3.11 Problems
3.12 Suggested reading
4 Application of complex variable theory and conformal mapping to perfect fluid flow
4.1 Notations
4.2 The complex potential for familiar flows
4.3 Blasius integral laws
4.4 Joukowski transformation and conformal mapping in airfoil design
4.5 Suggested reading
Part II: Complex integral transforms
Introduction to Part II
5 The techniques of complex Laplace transform
5.1 Theoretical background
5.2 Applications to partial differential equations (PDES)
5.3 Generalized unsteady diffusion problems
5.4 Miscellaneous examples
5.5 Problems
5.6 Suggested reading
6 Asymptotic behavior of complex integrals
6.1 Introduction
6.2 Asymptotic expansion
6.3 Asymptotics of integrals
6.4 Miscellaneous examples
6.5 Problems
6.6 Suggested reading
7 The techniques of complex Fourier transform
7.1 Generalization of Fourier transform
7.2 Function splitting
7.3 Convolution in Fourier transform
7.4 Multidimensional Fourier transform
7.5 The Wiener–Hopf techniques
7.6 Miscellaneous examples
7.7 Problems
7.8 Suggested reading
8 Modern applications of complex variables
8.1 Recent advances in prime function-based mappings
8.2 Ordinary differential equations in the complex plane
8.3 Advances in relevant Schwarz-Christoffel (SC) mappings
8.4 Numerical complex variable simulation
8.5 Suggested reading
Appendix A Table of Laplace transforms pairs in diffusion analysis
Appendix B Table of general properties of Laplace transforms
Appendix C Table of common Laplace transform pairs
Appendix D Table of special functions
Appendix E Bessel Functions
Appendix F Special functions
Appendix G Miscellaneous functions
Appendix H Table of transformations of regions
List of figures
Index