An Introduction to Complex Function Theory

This book provides a rigorous yet elementary introduction to the theory of analytic functions of a single complex variable. While presupposing in its readership a degree of mathematical maturity, it insists on no formal prerequisites beyond a sound knowledge of calculus. Starting from basic definitions, the text slowly and carefully develops the ideas of complex analysis to the point where such landmarks of the subject as Cauchy's theorem, the Riemann mapping theorem, and the theorem of Mittag-Leffler can be treated without sidestepping any issues of rigor. The emphasis throughout is a geometric one, most pronounced in the extensive chapter dealing with conformal mapping, which amounts essentially to a "short course" in that important area of complex function theory. Each chapter concludes with a wide selection of exercises, ranging from straightforward computations to problems of a more conceptual and thought-provoking nature.

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.