A Course in Complex Analysis: From Basic Results to Advanced Topics
Springer Science & Business Media, Oct 21, 2011 - Mathematics - 280 pages
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.
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Chapter II The fundamental theorems of complex analysis
Chapter III Functions on the plane and on the sphere
Chapter IV Integral formulas residues and applications
Chapter V Nonelementary functions
Chapter VI Meromorphic functions of several variables
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arbitrary assume automorphism biholomorphic mappings bijection Cauchy integral formula Cauchy integral theorem Cauchy-Riemann choose closed path coefficients compact complex analysis complex differentiable complex numbers complex variables constant continuous function converges locally uniformly decomposition defined Definition denote derivatives disk divisor domain G elliptic functions entire function equation euclidean exists f is holomorphic function f functions on G geometry h-line harmonic function hence holomorphic function holomorphic on G hyperbolic identity theorem implies isolated singularities lattice Laurent lemma Let f Let G logarithm map f maximum modulus principle meromorphic function Möbius circle Möbius transformations multiplicity neighbourhood obtain open set path of integration point z0 pole of f polydisk polynomial Proposition prove radius rational function real differentiable removable singularity residue Riemann right hand side satisfies sequence Show simple poles simply connected subset triangle unique Weierstrass yields zeros