This book provides a rigorous yet elementary introduction to the theory of analytic functions of a single complex variable. Starting from basic definitions, the text slowly and carefully develops the ideas of complex analysis to such a degree that Cauchy's theorem, the Riemann mapping theorem, and the theorem of Mittag-Leffler can be treated without side-stepping any issues of rigor. Each chapter concludes with a wide selection of exercises.
This book provides a rigorous yet elementary introduction to the theory of analytic functions of a single complex variable. Starting from basic definitions, the text slowly and carefully develops the ideas of complex analysis to such a degree that Cauchy's theorem, the Riemann mapping theorem, and the theorem of Mittag-Leffler can be treated without side-stepping any issues of rigor. Each chapter concludes with a wide selection of exercises.
Zusammenfassung
This book provides a rigorous yet elementary introduction to the theory of analytic functions of a single complex variable. Starting from basic definitions, the text slowly and carefully develops the ideas of complex analysis to such a degree that Cauchy's theorem, the Riemann mapping theorem, and the theorem of Mittag-Leffler can be treated without side-stepping any issues of rigor. Each chapter concludes with a wide selection of exercises.
Inhaltsverzeichnis
I The Complex Number System.- 1 The Algebra and Geometry of Complex Numbers.- 2 Exponentials and Logarithms of Complex Numbers.- 3 Functions of a Complex Variable.- 4 Exercises for Chapter I.- II The Rudiments of Plane Topology.- 1 Basic Notation and Terminology.- 2 Continuity and Limits of Functions.- 3 Connected Sets.- 4 Compact Sets.- 5 Exercises for Chapter II.- III Analytic Functions.- 1 Complex Derivatives.- 2 The Cauchy-Riemann Equations.- 3 Exponential and Trigonometric Functions.- 4 Branches of Inverse Functions.- 5 Differentiability in the Real Sense.- 6 Exercises for Chapter III.- IV Complex Integration.- 1 Paths in the Complex Plane.- 2 Integrals Along Paths.- 3 Rectiflable Paths.- 4 Exercises for Chapter IV.- V Cauchy's Theorem and its Consequences.- 1 The Local Cauchy Theorem.- 2 Winding Numbers and the Local Cauchy Integral Formula.- 3 Consequences of the Local Cauchy Integral Formula.- 4 More About Logarithm and Power Functions.- 5 The Global Cauchy Theorems.- 6 SimplyConnected Domains.- 7 Homotopy and Winding Numbers.- 8 Exercises for Chapter V.- VI Harmonic Functions.- 1 Harmonic Functions.- 2 The Mean Value Property.- 3 The Dirichlet Problem for a Disk.- 4 Exercises for Chapter VI.- VII Sequences and Series of Analytic Functions.- 1 Sequences of Functions.- 2 Infinite Series.- 3 Sequences and Series of Analytic Functions.- 4 Normal Families.- 5 Exercises for Chapter VII.- VIII Isolated Singularities of Analytic Functions.- 1 Zeros of Analytic Functions.- 2 Isolated Singularities.- 3 The Residue Theorem and its Consequences.- 4 Function Theory on the Extended Plane.- 5 Exercises for Chapter VIII.- IX Conformal Mapping.- 1 Conformal Mappings.- 2 Möbius Transformations.- 3 Riemann's Mapping Theorem.- 4 The Caratheodory-Osgood Theorem.- 5 Conformal Mappings onto Polygons.- 6 Exercises for Chapter IX.- X Constructing Analytic Functions.- 1 The Theorem of Mittag-Leffler.- 2 The Theorem of Weierstrass.- 3 Analytic Continuation.- 4 Exercises for ChapterX.- Appendix A Background on Fields.- 1 Fields.- 1.1 The Field Axioms.- 1.2 Subfields.- 1.3 Isomorphic Fields.- 2 Order in Fields.- 2.1 Ordered Fields.- 2.2 Complete Ordered Fields.- 2.3 Implications for Real Sequences.- Appendix B Winding Numbers Revisited.- 1 Technical Facts About Winding Numbers.- 1.1 The Geometric Interpretation.- 1.2 Winding Numbers and Jordan Curves.
