This unusually lively textbook on complex variables introduces the theory of analytic functions, explores its diverse applications and shows the reader how to harness its powerful techniques. "Complex Analysis" offers new and interesting motivations for classical results and introduces related topics that do not appear in this form in other texts. Stressing motivation and technique, and containing a large number of problems and their solutions, this volume may be used as a text both in classrooms and for self-study. Topics covered include: The complex numbers; functions of a complex variable; analytic functions; line integrals and entire functions; properties of entire functions and of analytic functions; simply connected domains; isolated singularities; the residue theorem and applications; contour integral techniques; conformal mapping and the riemann mapping theorem; maximum-modulus theorems for unbounded domains; harmonic functions; forms of analytic functions; analytic continuation; the gamma and zeta functions; application to other areas of mathematics. For this second edition, the authors have revised some of the existing material and have provided new exercises and solutions.
Table of contents:
Preface; 1. The Complex Numbers; 2. Functions of the Complex Variablez; 3. Analytic Functions; 4. Line Integrals and Entire Functions; 5.Properties of Entire Functions; 6. Properties of Analytic Functions;7. Further Properties of Analytic Functions; 8. Simply ConnectedDomains; 9. Isolated Sigularities of an Analytic Function; 10. TheResidue Theorem; 11. Applications of The Residue Theorem to theEvaluation of Integrals Sums; 12. Further Contour Integral Techniques;13. Introduction to Conformal Mapping; 14. The Riemann MappingTheorem; 15. Maximum-Modulus Theorems for Unbounded Domains; 16.Harmonic Functions; 17. Different Forms of Analytic Functions; 18.Analytic Continuation; The Gamma and Zeta Functions; 19. Applicationsto Other Areas of Mathematics; Appendices; Answers; Bibliography;Index
Table of contents:
Preface; 1. The Complex Numbers; 2. Functions of the Complex Variablez; 3. Analytic Functions; 4. Line Integrals and Entire Functions; 5.Properties of Entire Functions; 6. Properties of Analytic Functions;7. Further Properties of Analytic Functions; 8. Simply ConnectedDomains; 9. Isolated Sigularities of an Analytic Function; 10. TheResidue Theorem; 11. Applications of The Residue Theorem to theEvaluation of Integrals Sums; 12. Further Contour Integral Techniques;13. Introduction to Conformal Mapping; 14. The Riemann MappingTheorem; 15. Maximum-Modulus Theorems for Unbounded Domains; 16.Harmonic Functions; 17. Different Forms of Analytic Functions; 18.Analytic Continuation; The Gamma and Zeta Functions; 19. Applicationsto Other Areas of Mathematics; Appendices; Answers; Bibliography;Index
From the reviews of the third edition:
"The book of the known mathematicians J. Bak and D. Newman is an excellent introduction into the theory of analytic functions of one complex variable. The book is written on an elementary level and so it supports students in the early stages of their mathematical studies. ... The book also contains many illustrations, examples and exercises, which give additional information and explanations." (Konstantin Malyutin, Zentralblatt MATH, Vol. 1205, 2011)
"The book of the known mathematicians J. Bak and D. Newman is an excellent introduction into the theory of analytic functions of one complex variable. The book is written on an elementary level and so it supports students in the early stages of their mathematical studies. ... The book also contains many illustrations, examples and exercises, which give additional information and explanations." (Konstantin Malyutin, Zentralblatt MATH, Vol. 1205, 2011)