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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