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.