You are here

Complex Variables

Stephen D. Fisher
Dover Publications
Publication Date: 
Number of Pages: 
BLL Rating: 

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Allen Stenger
, on

This is an unusual text in complex variables, in that it has only a moderate amount on the properties of analytic functions and a lot on mathematical techniques based on analytic functions. These techniques include differential equations, Green’s functions, and several kinds of transforms (Fourier, Laplace, and Z). The viewpoint is from mathematics and not from physics or engineering, but the book contains much that is valuable to physics and engineering students. The present volume is a 1999 slightly-corrected reprint of the 1990 second edition from Wadsworth & Brooks.

This is a very good introductory text for most students. The emphasis is on breadth rather than depth. It carefully builds on the student’s presumed knowledge of calculus and trigonometry, and starts with the theory of the complex numbers themselves, followed by complex sequences and convergence, and then contour integrals. The approach is analytic (through integrals and some power series) rather than geometric, although there is a good brief description of conformal maps, including the Riemann mapping theorem.

There are copious exercises, of reasonable difficulty. The author also takes the opportunity to use the exercises to introduce many traditional topics that did not fit in the main narrative, so the book actually covers much more ground that it appears from the table of contents.

A good alternative, that is pitched at the same level but is slanted more toward function theory than applications, is Bak & Newman’s Complex Analysis.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis.




The complex plane
  1.1 The complex numbers and the complex plane
    1.1.1 A formal view of the complex numbers
  1.2 Some geometry
  1.3 Subsets of the plane
  1.4 Functions and limits
  1.5 The exponential, logarithm, and trigonometric functions
  1.6 Line integrals and Green's theorem
2. Basic properties of analytic functions
  2.1 Analytic and harmonic functions; the Cauchy-Riemann equations
    2.1.1 Flows, fields, and analytic functions
  2.2 Power series
  2.3 Cauchy's theorem and Cauchy's formula
    2.3.1 The Cauchy-Goursat theorem
  2.4 Consequences of Cauchy's formula
  2.5 Isolated singularities
  2.6 The residue theorem and its application to the evaluation of definite integrals
3. Analytic functions as mappings
  3.1 The zeros of an analytic function
    3.1.1 The stability of solutions of a system of linear differential equations
  3.2 Maximum modulus and mean value
  3.3 Linear fractional transformations
  3.4 Conformal mapping
    3.4.1 Conformal mapping and flows
  3.5 The Riemann mapping theorem and Schwarz-Christoffel transformations
4. Analytic and harmonic functions in applications
  4.1 Harmonic functions
  4.2 Harmonic functions as solutions to physical problems
  4.3 Integral representations of harmonic functions
  4.4 Boundary-value problems
  4.5 Impulse functions and the Green's function of a domain
5. Transform methods
  5.1 The Fourier transform: basic properties
  5.2 Formulas Relating u and û
  5.3 The Laplace transform
  5.4 Applications of the Laplace transform to differential equations
  5.5 The Z-Transform
    5.5.1 The stability of a discrete linear system
Appendix 1. The stability of a discrete linear system
Appendix 2. A Table of Conformal Mappings
Appendix 3. A Table of Laplace Transforms
  Solutions to Odd-Numbered Exercises