Complex Analysis

This unusually lively textbook introduces the theory of analytic functions, explores its diverse applications and shows the reader how to harness its powerful techniques. The book offers new and interesting motivations for classical results and introduces related topics that do not appear in this fo [...]
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This unusually lively textbook introduces the theory of analytic functions, explores its diverse applications and shows the reader how to harness its powerful techniques. The book offers new and interesting motivations for classical results and introduces related topics that do not appear in this form in other texts. For the second edition, the authors have revised some of the existing material and have provided new exercises and solutions. The third edition contains a new section on Schwarz-Christoffel mappings, an improved treatment of the Schwarz Reflection Principle, and an expanded section on applications to number theory. TOC:Preface.- The Complex Numbers.- Functions of the Complex Variable z.- Analytic Functions.- Line Integrals and Entire Functions.- Properties of Entire Functions.- Properties of Analytic Functions.- Further Properties of Analytic Functions; 8. Simply Connected Domains.- Isolated Sigularities of an Analytic Function.- The Residue Theorem.- Applications of The Residue Theorem to the Evaluation of Integrals Sums.- Further Contour Integral Techniques.- Introduction to Conformal Mapping.- The Riemann Mapping Theorem.- Maximum-Modulus Theorems for Unbounded Domains.- Harmonic Functions.- Different Forms of Analytic Functions.- Analytic Continuation; The Gamma and Zeta Functions.- Applications to Other Areas of Mathematics.- Appendices.- Answers.- Bibliography.- Index.-