**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.