# Complex Analysis : An Introduction to The Theory of Analytic Functions of One Complex Variable

###### Lars Ahlfors
Publisher:
McGraw-Hill
Publication Date:
1979
Number of Pages:
345
Format:
Hardcover
Edition:
3
Price:
286.67
ISBN:
9780070006577
Category:
Textbook
BLL Rating:

The Basic Library List Committee considers this book essential for undergraduate mathematics libraries.

[Reviewed by
Allen Stenger
, on
07/16/2015
]

This classic text is revered by many mathematicians, although it seems less so by students. It is concise, clear, and thorough, and is still fresh today, thirty-five years after its last revision. Its main weakness today is its ridiculous cover price: nearly $300. In present-day terms the book works best as a graduate text: it’s not especially difficult, nor does it go very deep, but there’s not much handholding and there are hardly any worked examples. Most of the exercise sections start with two or three relatively easy exercises, covering the kinds of results that would otherwise be worked examples. The remainder of the exercises are quite challenging and prove a lot of standard results. The prerequisites are modest, being mostly calculus and an introduction to real analysis. The book develops the properties of complex numbers, the line integral, and the needed point-set topology. The book is slanted toward the geometric side, with a lot of material on conformal mapping, the Riemann mapping theorem, Dirichlet’s problem (the existence of a harmonic function having given boundary values), the monodromy theorem, and considerations of the kinds of regions that the Cauchy integral theorem holds for. It has relatively less coverage of power series, contour integrals, and infinite products. The coverage of special functions is concise but reasonably complete. This is a pure-math book aimed at math majors and generally omits any applications, even to math. For example, it has a very nice section on the Riemann zeta function, but mentions only in passing that it is useful in analytic number theory, and nowhere uses the term “prime number theorem”. The book concludes with two chapters on more specialized topics. One chapter is on elliptic (doubly-periodic) functions in general, and the Weierstrass $\wp$-function in particular. The other is on global analytic functions, that is, a way of formalizing multi-valued functions; the approach here is through sheaves. (Despite the geometric emphasis, the book makes only modest use of Riemann surfaces.) I recommend against using this book as the text for a course, because of its high price. Wonderful though it is, you are only going to cover a small portion of the text in any course, and there are much cheaper though less comprehensive texts that do just as well for an introductory text. Needham’s Visual Complex Analysis is well-regarded by many people and also emphasizes the geometric perspective, although it is very different from any other text on the market. There are lots and lots of introductory complex analysis texts that lean toward the power series and integral side. Of these, I like Bak & Newman’s Complex Analysis and Fisher’s Complex Variables (the latter a bargain at under$20). Another well-regarded modern book, that I have not seen, is Boas’s Invitation to 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.