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Complex analysis. 4th ed. (English) Zbl 0933.30001
Graduate Texts in Mathematics. 103. New York, NY: Springer. xiv, 485 p. (1999).
The first and second editions of this book have been reviewed earlier: see Zbl 0366.30001 and 819.30001. The plan of the book has been retained. Part I (290 pages) contains the basic theory up to the calculus of residues, and basic results about harmonic functions. Part II (43 pages), entitled Geometric Function Theory, contains the Riemann mapping theorem, the Schwarz reflection principle, and a proof of Picard’s theorem via the module function. Part III (116 pages) presents several topics for the more advanced reader: Hadamard’s three circles theorem and applications, Weierstraß’s product theorem, elliptic functions, and the proof of the prime number theorem following Newman and Korevaar. In an Appendix (24 pages) useful material on difference equations, analytic differential equations, fixed points of linear transformations, and Cauchy’s formula for \(C^\infty\)-functions is presented.
Compared with the third edition, some material on harmonic functions and vector fields has been added, and the wealth of examples and exercises has still been increased. These exercises, plus their solutions, have been collected in a separate book; see the following review.
We noticed that proof reading is not the strength of the author. Thus, as already present in the third edition, there are two sections VII, §4; VIII, §4; VIII, §5. The statement of the reflection principle is still not complete, and Fig. 12 on p. 222 is not correct. We also note that in the Bibliography important recent texts, even Springer Graduate Texts, are missing, and that there is a complete absence of historical remarks. In spite of this, it is a highly recommendable book for a two semester course on complex analysis.
Reviewer: D.Gaier (Gießen)

MSC:
30-02 Research exposition (monographs, survey articles) pertaining to functions of a complex variable
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