If you’ve been out of graduate school for a while and haven’t been involved with complex function theory in your professional life, then you may be hard-pressed to recall Rouché’s Theorem, Schwarz’s Lemma, or Harnack’s Principle. Do you remember the difference between Picard’s Great Theorem and Picard’s Little Theorem?
Although aimed at graduate students faced with a qualifying exam in complex analysis, this slender volume is a quick, reasonably complete guide for any student, scientist, or mathematician to this “beautiful and compelling” area of mathematics, encompassing its algebraic, analytical, and topological aspects.
As expected in such a compact treatment, there has to be a form of triage with respect to proofs: Short proofs are given, medium-length proofs are outlined, and long proofs are sketched. There are plenty of figures, examples, summarizing tables, references, and good expository commentary. Right after the basic algebraic and topological aspects of complex numbers are reviewed, Krantz defines the fundamental concept of a holomorphic (or analytic) function — first in terms of the Cauchy-Riemann equations (where the continuity of the partial derivatives is assumed), then in terms of the vanishing of the partial derivatives with respect to the complex conjugate of z, next in terms of the complex derivative in a neighborhood, and eventually in terms of power series.
A gentle introduction to the nature of complex line integrals leads to a nice exposition of Cauchy’s integral theorems. Discussion of other important topics involving mappings, harmonic functions, infinite products, analytic continuation, and Riemann surfaces follow.
The book doesn’t have exercise sets, although there are in-text exercises. There is a thorough Glossary (“accumulation point” to “zeta function”) and a Bibliography.
Jacques Hadamard has said that “the shortest path between two truths in the real domain passes through the complex domain.” The book under review will shorten the journey through the complex domain while missing few sights along the way. It is a suitable companion to A Handbook of Real Variables: With Applications to Differential Equations and Fourier Analysis by the same author and a worthy addition to the Dolciani Mathematical Expositions series.
Henry Ricardo (firstname.lastname@example.org) is Professor of Mathematics at Medgar Evers College of The City University of New York and serves as Secretary (until May, 2009) and Governor of the Metropolitan NY Section of the MAA. He is the author of A Modern Introduction to Differential Equations, which has just been translated into Spanish; and he is currently writing a linear algebra text.