Title Complex Variables and Analytic Functions
Subtitle An Illustrated Introduction
Author Bengt Fornberg, Cécile Piret
ISBN 9781611975970
List price USD 84.00
Price outside India Available on Request
Original price
Binding Paperback
No of pages 372
Book size 178 x 254 mm
Publishing year 2020
Original publisher SIAM - Society for Industrial and Applied Mathematics (Eurospan Group)
Published in India by .
Exclusive distributors Viva Books Private Limited
Sales territory India, Sri Lanka, Bangladesh, Pakistan, Nepal, .
Status New Arrival
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Description:

At almost all academic institutions worldwide, complex variables and analytic functions are utilized in courses on applied mathematics, physics, engineering, and other related subjects. For most students, formulas alone do not provide a sufficient introduction to this widely taught material, yet illustrations of functions are sparse in current books on the topic. This is the first primary introductory textbook on complex variables and analytic functions to make extensive use of functional illustrations.

 

Aiming to reach undergraduate students entering into the world of complex variables and analytic functions, this book:

  • utilizes graphics to visually build on familiar cases and illustrate how these same functions extend beyond the real axis;
  • covers several important topics that are omitted in nearly all recent texts, including techniques for analytic continuation and discussions of elliptic functions and of Wiener–Hopf methods; and
  • presents current advances in research, highlighting the subject’s active and fascinating frontier.


Contents:

Preface

Chapter 1. Complex Numbers • How to think about different types of numbers • Definition of complex numbers • The complex number plane as a tool for planar geometry • Stereographic projection • Supplementary materials • Exercises

Chapter 2. Functions of a Complex Variable • Derivative • Some elementary functions generalized to complex argument by means of their Taylor expansion • Additional observations on Taylor expansions of analytic functions • Singularities • Multivalued functions—Branch cuts and Riemann sheets • Sequences of analytic functions • Functions defined by integrals • Supplementary materials • Exercises

Chapter 3. Analytic Continuation • Introductory examples • Some methods for analytic continuation • Exercises

Chapter 4. Introduction to Complex Integration • Integration when a primitive function F(z) is available • Contour integration • Laurent series • Supplementary materials • Exercises

Chapter 5. Residue Calculus • Residue calculus • Infinite sums • Analytic continuation with use of contour integration • Weierstrass products and Mittag–Leffler expansions • Supplementary materials • Exercises

Chapter 6. Gamma, Zeta, and Related Functions • The gamma function • The zeta function • The Lambert W-function • Supplementary materials • Exercises

Chapter 7. Elliptic Functions • Some introductory remarks on simply periodic functions • Some basic properties of doubly periodic functions • The Weierstrass P-function • The Jacobi elliptic functions • Supplementary materials • Exercises

Chapter 8. Conformal Mappings • Relations between conformal mappings and analytic functions • Mappings provided by bilinear functions • Riemann’s mapping theorem • Mappings of polygonal regions • Some applications of conformal mappings • Revisiting the Jacobi elliptic function sn(z, k) • Supplementary materials • Exercises

Chapter 9. Transforms • Fourier transform • Laplace transform • Mellin transform • Hilbert transform • z-transform • Three additional transforms related to rotations • Supplementary materials • Exercises

Chapter 10. Wiener–Hopf and Riemann–Hilbert Methods • The Wiener–Hopf method • A brief primer on Riemann–Hilbert methods • Supplementary materials • Exercises

Chapter 11. Special Functions Defined by ODEs • Airy’s equation • Bessel functions • Hypergeometric functions • Converting linear ODEs to integrals • The Painlevé equations • Exercises

Chapter 12. Steepest Descent for Approximating Integrals • Asymptotic vs convergent expansions • Euler–Maclaurin formula • Laplace integrals • Steepest descent • Supplementary materials • Exercises

Bibliography

Index


About the Authors:

Bengt Fornberg has been a professor of applied mathematics at the University of Colorado, Boulder since 1995. His primary research interests focus on computational methods for solving PDEs and numerical methodologies related to analytic functions. He has held positions at the European Organization for Nuclear Research (CERN), California Institute of Technology, and Exxon Corporate Research. In 2014, he was elected a SIAM Fellow.

Cécile Piret is a faculty member in the department of mathematical sciences at Michigan Technological University. A former postdoctoral fellow at Catholic University of Louvain and the National Center for Atmospheric Research (NCAR), her research interests revolve around the development of high order methods for numerically solving differential equations.


Target Audience:

The primary audience for this textbook is undergraduate students taking an introductory course on complex variables and analytic functions. It is also geared towards graduate students taking a second semester course on these topics, engineers and physicists who use complex variables in their work, and students and researchers at any level who want a reference book on the subject.

 

 
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