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 Cover......Page 1 Title......Page 3 Copyright......Page 4 Preface......Page 5 Contents......Page 7 0. Notations and basic concepts......Page 9 1. Holomorphic functions......Page 11 2. Real and complex differentiability......Page 15 Exercises......Page 20 3. Uniform convergence and power series......Page 21 4. Elementary functions......Page 26 5. Integration......Page 32 Exercises......Page 39 6. Several complex variables......Page 40 Exercises......Page 43 1. Primitive functions......Page 44 2. The Cauchy integral theorem......Page 48 Exercises......Page 51 3. The Cauchy integral formula......Page 52 4. Power series expansions of holomorphic functions......Page 56 Exercises......Page 62 5. Convergence theorems, maximum modulus principle, and open mapping theorem......Page 63 Exercises......Page 68 6. Isolated singularities and meromorphic functions......Page 69 7. Holomorphic functions of several variables......Page 74 Exercises......Page 78 1. The Riemann sphere......Page 79 2. Polynomials and rational functions......Page 83 3. Entire functions......Page 88 4. Möbius transformations......Page 90 5. Logarithms, powers, and roots......Page 95 6. Partial fraction decompositions......Page 103 7. Product Expansions......Page 110 Exercises......Page 115 1. The general Cauchy integral theorem......Page 116 2. The inhomogeneous Cauchy integral formula......Page 125 3. Laurent decomposition and Laurent expansion......Page 127 4. Residues......Page 131 Exercises......Page 134 5. Residue calculus......Page 135 Exercises......Page 144 6. Counting zeros......Page 145 7. The Weierstrass preparation theorem......Page 149 Exercises......Page 155 1. The Γ-function......Page 156 Exercises......Page 163 2. The ζ-function and the Prime Number Theorem......Page 164 Exercises......Page 174 3. The functional equation of the ζ-function......Page 175 4. Elliptic functions......Page 182 Exercises......Page 194 5. Elliptic functions and plane cubics......Page 196 Exercises......Page 200 1. Zero sets of holomorphic functions......Page 201 2. Meromorphic functions......Page 203 3. The inhomogeneous Cauchy-Riemann equation in dimension 1......Page 205 4. The Cauchy-Riemann equations with compact support......Page 207 5. The Cauchy-Riemann equations in a polydisk......Page 209 6. Principal parts: the first Cousin problem......Page 212 Exercises......Page 214 7. Divisors: the second Cousin problem......Page 215 8. Meromorphic functions revisited......Page 219 Exercises......Page 221 1. Holomorphic automorphisms......Page 222 2. The hyperbolic metric......Page 226 3. Hyperbolic geometry......Page 232 Historical remark......Page 242 Exercises......Page 243 4. The Riemann mapping theorem......Page 244 5. Harmonic functions......Page 249 6. Schwarz’s reflection principle......Page 255 Exercises......Page 258 7. The modular map λ......Page 259 Exercises......Page 262 8. Theorems of Picard and Montel......Page 263 Exercises......Page 265 Hints and solutions of selected exercises......Page 266 Bibliography......Page 271 Index of Symbols......Page 273 C......Page 274 E......Page 275 I......Page 276 M......Page 277 P......Page 278 S......Page 279 Z......Page 280 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