Function theory of one complex variable. (Q2770575)

This is the second edition of an an interesting text book for first-year graduate students in complex analysis. All material usually treated in such a course is covered in this book. However, it is based on principles that differ somewhat from those underlying most introductory graduate texts on this subject. NEWLINENEWLINENEWLINEFirst of all, the authors develop the idea that an introductory book on the present subject should emphasize how complex analysis is a natural outgrowth of multivariable calculus. On the other hand, they made a systematic attempt to separate analytical ideas, belonging to complex analysis in the strictest sense, from topological considerations. Topological questions arise naturally, but the authors collected all of the difficult topological issues in Chapter 11 (the exposition of topological matters in Chapter 11 has been revisited by the authors in this second edition of the book) and leave the way open for more direct problems. In addition, in the final part of the book, the authors include a number of special topics, such as Hilbert spaces of holomorphic functions, the Bergman kernel, special functions, that bring the reader close to subjects of current research. A large number of exercises are also included in the book. NEWLINENEWLINENEWLINEThe book is arranged as follows: In Chapter 1 are given fundamental concepts. In Chapter 2 are presented several results about complex line integrals, while Chapter 3 provides applications of the Cauchy integral. In Chapter 4 are discussed meromorphic functions and residues. Chapter 5 provides results concerning the zeros of a holomorphic function. In Chapter 6 are discussed several problems concerning holomorphic functions as geometric mappings, while Chapter 7 provides certain results about harmonic functions. In Chapter 8 are studied infinite series and products, and in Chapter 9 are presented applications of infinite sums and products. Chapter 10 yields the analytic continuation. Chapter 11 contains an exposition of topological matters. Chapter 12 is devoted to the rational approximation theory, while Chapter 13 is concerned with special classes of holomorphic functions. In Chapter 14 are studied Hilbert spaces of holomorphic functions, the Bergman kernel, and biholomorphic mappings. Chapter 15 provides results about special functions such as the Gamma and Beta functions, the Riemann Zeta function. In Chapter 16 is studied the Prime Number theorem. Also the book contains Appendix A: Real Analysis and Appendix B: The statement and proof of Goursat's theorem. NEWLINENEWLINENEWLINEThe book is very well written and rigously treated. Also its subject is very interesting. It is very useful to graduate students who are interested in Complex Analysis, as well as to all researchers specialized in this subject.