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A Course in Complex Analysis : From Basic Results to Advanced Topics / by Wolfgang Fischer, Ingo Lieb
Contributor(s): Resource type: Ressourcentyp: Buch (Online)Book (Online)Language: English Series: SpringerLink BücherPublisher: Wiesbaden : Vieweg+Teubner Verlag / Springer Fachmedien Wiesbaden GmbH, Wiesbaden, 2012Description: Online-Ressource (VIII, 272p. 21 illus, digital)ISBN:- 9783834886613
- 515/.9
- 515
- QA299.6-433
Contents:
Summary: 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.PPN: PPN: 1651087652Package identifier: Produktsigel: ZDB-2-SEB | ZDB-2-SXMS | ZDB-2-SMA
""Preface""; ""Contents""; ""Chapter I. Analysis in the complex plane""; ""0. Notations and basic concepts""; ""1. Holomorphic functions""; ""Exercises""; ""2. Real and complex differentiability""; ""Exercises""; ""3. Uniform convergence and power series""; ""Exercises""; ""4. Elementary functions""; ""Exercises""; ""5. Integration""; ""Exercises""; ""6. Several complex variables""; ""Exercises""; ""Chapter II. The fundamental theorems of complex analysis""; ""1. Primitive functions""; ""Exercises""; ""2. The Cauchy integral theorem""; ""Exercises""; ""3. The Cauchy integral formula""
""Exercises""""4. Power series expansions of holomorphic functions""; ""Exercises""; ""5. Convergence theorems, maximum modulus principle, and open mapping theorem""; ""Exercises""; ""6. Isolated singularities and meromorphic functions""; ""Exercises""; ""7. Holomorphic functions of several variables""; ""Exercises""; ""Chapter III. Functions on the plane and on the sphere""; ""1. The Riemann sphere""; ""Exercises""; ""2. Polynomials and rational functions""; ""Historical note""; ""Exercises""; ""3. Entire functions""; ""Exercises""; ""4. Möbius transformations""; ""Exercises""
""5. Logarithms, powers, and roots""""Exercises""; ""6. Partial fraction decompositions""; ""Exercises""; ""7. Product Expansions""; ""Exercises""; ""Chapter IV. Integral formulas, residues, and applications""; ""1. The general Cauchy integral theorem""; ""Exercises""; ""2. The inhomogeneous Cauchy integral formula""; ""3. Laurent decomposition and Laurent expansion""; ""Exercises""; ""4. Residues""; ""Exercises""; ""5. Residue calculus""; ""Exercises""; ""6. Counting zeros""; ""Exercises""; ""7. The Weierstrass preparation theorem""; ""Exercises""; ""Chapter V. Non-elementary functions""
""1. The г-function""""Exercises""; ""2. The ζ-function and the Prime Number Theorem""; ""Historical Note""; ""Exercises""; ""3. The functional equation of the ζ-function""; ""Exercises""; ""4. Elliptic functions""; ""Exercises""; ""5. Elliptic functions and plane cubics""; ""Exercises""; ""Chapter VI. Meromorphic functions of several variables""; ""1. Zero sets of holomorphic functions""; ""Exercises""; ""2. Meromorphic functions""; ""Exercises""; ""3. The inhomogeneous Cauchy-Riemann equation in dimension 1""; ""Exercises""; ""4. The Cauchy-Riemann equations with compact support""
""Exercises""""5. The Cauchy-Riemann equations in a polydisk""; ""6. Principal parts: the first Cousin problem""; ""Exercises""; ""7. Divisors: the second Cousin problem""; ""8. Meromorphic functions revisited""; ""Exercises""; ""Chapter VII. Holomorphic maps: Geometric aspects""; ""1. Holomorphic automorphisms""; ""Exercises""; ""2. The hyperbolic metric""; ""Exercises""; ""3. Hyperbolic geometry""; ""Historical remark""; ""Exercises""; ""4. The Riemann mapping theorem""; ""Exercises""; ""5. Harmonic functions""; ""Exercises""; ""6. Schwarz�s reflection principle""; ""Exercises""
""7. The modular map λ""
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