Complex analysis. An introduction to the theory of analytic functions of one complex variable. 3rd ed. (English) Zbl 0395.30001

International Series in Pure and Applied Mathematics. Düsseldorf etc.: McGraw-Hill Book Company. xiv, 331 p. DM 44.10; $ 19.75 (1979).
From the preface: For the first edition (1953) see the review of A. Dinghas in Zbl 0052.07002. The third edition contains no radical innovations. The introductory chapters are virtually unchanged. In few places throughout the book, it was desirable to clarify certain points that experience has shown to have been a source of possible misunderstanding or difficulties. Misprints and minor errors have been corrected. The main differences between the second edition (1966; Zbl 0154.31904) and the third edition can be summarized as follows:
1. Notations and terminology have been modernized, but the style has not changed in any significant way.
2. In Chapter 2 a brief section on the change of length and area under conformal mapping has been added. To some degree this infringes on the otherwise self-contained exposition, for it forces the reader to fall back on calculus for the definition and manipulation of double integrals. The disadvantage is minor.
3. In Chapter 4 there is a new and simpler proof of the general form of Cauchy’s theorem. It is due to A. F. Beardon, who permitted to reproduce it. It complements but does not replace the old proof, which has been retained and improved.
4. A short section on the Riemann zeta function has been included. The functional equation illustrates the use of residues in a less trivial situation than the mere computation of definite integrals.
5. Large parts of Chapter 8 have been completely rewritten. The main purpose was to introduce the reader to the terminology of germs and sheaves while emphasizing all the classical concepts. It goes without saying that nothing beyond the basic notions of sheaf theory would have been compatible with the elementary nature of the book.
6. The author has successfully resisted the temptation to include Riemann surfaces as one-dimensional complex manifolds.
The book would lose much of its usefulness if it went beyond its purpose of being no more than an introduction to the basic methods and results of complex function theory in the plane.


30-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functions of a complex variable