A Course in Complex Analysis : From Basic Results to Advanced Topics

This carefully written textbook is an introduction to the beautiful concepts and results of complex analysis. It is intended for international bachelor and master programmes in Germany and throughout Europe; in the Anglo-American system of university education the content corresponds to a beginning graduate course. The book presents the fundamental results and methods of complex analysis and applies them to a study of elementary and non-elementary functions (elliptic functions, Gamma- and Zeta function including a proof of the prime number theorem ...) and – a new feature in this context! – to exhibiting basic facts in the theory of several complex variables. Part of the book is a translation of the authors’ German text “Einführung in die komplexe Analysis”; some material was added from the by now almost “classical” text “Funktionentheorie” written by the authors, and a few paragraphs were newly written for special use in a master’s programme. Content Analysis in the complex plane - The fundamental theorems of complex analysis - Functions on the plane and on the sphere - Integral formulas, residues and applications - Non-elementary functions - Meromorphic functions of several variables - Holomorphic maps: Geometric aspects Readership Advanced undergraduates (bachelor students) and beginning graduate students (master's programme) Lecturers in mathematics About the authors Professor Dr. Ingo Lieb, Department of Mathematics, University of Bonn Professor Dr. Wolfgang Fischer, Department of Mathematics, University of Bremen Annotation This carefully written textbook is an introduction to the beautiful concepts and results of complex analysis. It is intended for international bachelor and master programmes in Germany and throughout Europe; in the Anglo-American system of university education the content corresponds to a beginning graduate course. The book presents the fundamental results and methods of complex analysis and applies them to a study of elementary and non-elementary functions (elliptic functions, Gamma- and Zeta function including a proof of the prime number theorem) and a new feature in this context! to exhibiting basic facts in the theory of several complex variables. Part of the book is a translation of the authors German text Einführung in die komplexe Analysis; some material was added from the by now almost classical text Funktionentheorie written by the authors, and a few paragraphs were newly written for special use in a masters programme. ContentAnalysis in the complex plane - The fundamental theorems of complex analysis - Functions on the plane and on the sphere - Integral formulas, residues and applications - Non-elementary functions - Meromorphic functions of several variables - Holomorphic maps: Geometric aspectsReadershipAdvanced undergraduates (bachelor students) and beginning graduate students (master's programme) Lecturers in mathematicsAbout the authorsProfessor Dr. Ingo Lieb, Department of Mathematics, University of BonnProfessor Dr. Wolfgang Fischer, Department of Mathematics, University of Bremen Front Matter....Pages i-viii Analysis in the complex plane....Pages 1-35 The fundamental theorems of complex analysis....Pages 36-70 Functions on the plane and on the sphere....Pages 71-107 Integral formulas, residues, and applications....Pages 108-147 Non-elementary functions....Pages 148-192 Meromorphic functions of several variables....Pages 193-213 Holomorphic maps: Geometric aspects....Pages 214-257 Back Matter....Pages 258-272