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. 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.

I The Complex Number System.- 1 The Algebra and Geometry of Complex Numbers.- 1.1 The Field of Complex Numbers.- 1.2 Conjugate, Modulus, and Argument.- 2 Exponentials and Logarithms of Complex Numbers.- 2.1 Raising e to Complex Powers.- 2.2 Logarithms of Complex Numbers.- 2.3 Raising Complex Numbers to Complex Powers.- 3 Functions of a Complex Variable.- 3.1 Complex Functions.- 3.2 Combining Functions.- 3.3 Functions as Mappings.- 4 Exercises for Chapter I.- II The Rudiments of Plane Topology.- 1 Basic Notation and Terminology.- 1.1 Disks.- 1.2 Interior Points, Open Sets.- 1.3 Closed Sets.- 1.4 Boundary, Closure, Interior.- 1.5 Sequences.- 1.6 Convergence of Complex Sequences.- 1.7 Accumulation Points of Complex Sequences.- 2 Continuity and Limits of Functions.- 2.1 Continuity.- 2.2 Limits of Functions.- 3 Connected Sets.- 3.1 Disconnected Sets.- 3.2 Connected Sets.- 3.3 Domains.- 3.4 Components of Open Sets.- 4 Compact Sets.- 4.1 Bounded Sets and Sequences.- 4.2 Cauchy Sequences.- 4.3 Compact Sets.- 4.4 Uniform Continuity.- 5 Exercises for Chapter II.- III Analytic Functions.- 1 Complex Derivatives.- 1.1 Differentiability.- 1.2 Differentiation Rules.- 1.3 Analytic Functions.- 2 The Cauchy-Riemann Equations.- 2.1 The Cauchy-Riemann System of Equations.- 2.2 Consequences of the Cauchy-Riemann Relations.- 3 Exponential and Trigonometric Functions.- 3.1 Entire Functions.- 3.2 Trigonometric Functions.- 3.3 The Principal Arcsine and Arctangent Functions.- 4 Branches of Inverse Functions.- 4.1 Branches of Inverse Functions.- 4.2 Branches of the pth-root Function.- 4.3 Branches of the Logarithm Function.- 4.4 Branches of the ?-power Function.- 5 Differentiability in the Real Sense.- 5.1 Real Differentiability.- 5.2 The Functions fz and fz.- 6 Exercises for Chapter III.- IV Complex Integration.- 1 Paths in the Complex Plane.- 1.1 Paths.- 1.2 Smooth and Piece wise Smooth Paths.- 1.3 Parametrizing Line Segments.- 1.4 Reverse Paths, Path Sums.- 1.5 Change of Parameter.- 2 Integrals Along Paths.- 2.1 Complex Line Integrals.- 2.2 Properties of Contour Integrals.- 2.3 Primitives.- 2.4 Some Notation.- 3 Rectiflable Paths.- 3.1 Rectifiable Paths.- 3.2 Integrals Along Rectifiable Paths.- 4 Exercises for Chapter IV.- V Cauchy’s Theorem and its Consequences.- 1 The Local Cauchy Theorem.- 1.1 Cauchy’s Theorem For Rectangles.- 1.2 Integrals and Primitives.- 1.3 The Local Cauchy Theorem.- 2 Winding Numbers and the Local Cauchy Integral Formula.- 2.1 Winding Numbers.- 2.2 Oriented Paths, Jordan Contours.- 2.3 The Local Integral Formula.- 3 Consequences of the Local Cauchy Integral Formula.- 3.1 Analyticity of Derivatives.- 3.2 Derivative Estimates.- 3.3 The Maximum Principle.- 4 More About Logarithm and Power Functions.- 4.1 Branches of Logarithms of Functions.- 4.2 Logarithms of Rational Functions.- 4.3 Branches of Powers of Functions.- 5 The Global Cauchy Theorems.- 5.1 Iterated Line Integrals.- 5.2 Cycles.- 5.3 Cauchy’s Theorem and Integral Formula.- 6 Simply Connected Domains.- 6.1 Simply Connected Domains.- 6.2 Simple Connectivity, Primitives, and Logarithms.- 7 Homotopy and Winding Numbers.- 7.1 Homotopic Paths.- 7.2 Contractible Paths.- 8 Exercises for Chapter V.- VI Harmonic Functions.- 1 Harmonic Functions.- 1.1 Harmonic Conjugates.- 2 The Mean Value Property.- 2.1 The Mean Value Property.- 2.2 Functions Harmonic in Annuli.- 3 The Dirichlet Problem for a Disk.- 3.1 A Heat Flow Problem.- 3.2 Poisson Integrals.- 4 Exercises for Chapter VI.- VII Sequences and Series of Analytic Functions.- 1 Sequences of Functions.- 1.1 Uniform Convergence.- 1.2 Normal Convergence.- 2 Infinite Series.- 2.1 Complex Series.- 2.2 Series of Functions.- 3 Sequences and Series of Analytic Functions.- 3.1 General Results.- 3.2 Limit Superior of a Sequence.- 3.3 Taylor Series.- 3.4 Laurent Series.- 4 Normal Families.- 4.1 Normal Subfamilies of C(U).- 4.2 Equicontinuity.- 4.3 The Arzelà-Ascoli and Montel Theorems.- 5 Exercises for Chapter VII.- VIII Isolated Singularities of Analytic Functions.- 1 Zeros of Analytic Functions.- 1.1 The Factor Theorem for Analytic Functions.- 1.2 Multiplicity.- 1.3 Discrete Sets, Discrete Mappings.- 2 Isolated Singularities.- 2.1 Definition and Classification of Isolated Singularities.- 2.2 Removable Singularities.- 2.3 Poles.- 2.4 Meromorphic Functions.- 2.5 Essential Singularities.- 2.6 Isolated Singularities at Infinity.- 3 The Residue Theorem and its Consequences.- 3.1 The Residue Theorem.- 3.2 Evaluating Integrals with the Residue Theorem.- 3.3 Consequences of the Residue Theorem.- 4 Function Theory on the Extended Plane.- 4.1 The Extended Complex Plane.- 4.2 The Extended Plane and Stereographic Projection.- 4.3 Functions in the Extended Setting.- 4.4 Topology in the Extended Plane.- 4.5 Meromorphic Functions and the Extended Plane.- 5 Exercises for Chapter VIII.- IX Conformal Mapping.- 1 Conformal Mappings.- 1.1 Curvilinear Angles.- 1.2 Diffeomorphisms.- 1.3 Conformal Mappings.- 1.4 Some Standard Conformal Mappings.- 1.5 Self-Mappings of the Plane and Unit Disk.- 1.6 Conformal Mappings in the Extended Plane.- 2 Möbius Transformations.- 2.1 Elementary Mobius Transformations.- 2.2 Mobius Transformations and Matrices.- 2.3 Fixed Points.- 2.4 Cross-ratios.- 2.5 Circles in the Extended Plane.- 2.6 Reflection and Symmetry.- 2.7 Classification of Mobius Transformations.- 2.8 Invariant Circles.- 3 Riemann’s Mapping Theorem.- 3.1 Preparations.- 3.2 The Mapping Theorem.- 4 The Caratheodory-Osgood Theorem.- 4.1 Topological Preliminaries.- 4.2 Double Integrals.- 4.3 Conformal Modulus.- 4.4 Extending Conformal Mappings of the Unit Disk.- 4.5 Jordan Domains.- 4.6 Oriented Boundaries.- 5 Conformal Mappings onto Polygons.- 5.1 Polygons.- 5.2 The Reflection Principle.- 5.3 The Schwarz-Christoffel Formula.- 6 Exercises for Chapter IX.- X Constructing Analytic Functions.- 1 The Theorem of Mittag-Leffler.- 1.1 Series of Meromorphic Functions.- 1.2 Constructing Meromorphic Functions.- 1.3 The Weierstrass -function.- 2 The Theorem of Weierstrass.- 2.1 Infinite Products.- 2.2 Infinite Products of Functions.- 2.3 Infinite Products and Analytic Functions.- 2.4 The Gamma Function.- 3 Analytic Continuation.- 3.1 Extending Functions by Means of Taylor Series.- 3.2 Analytic Continuation.- 3.3 Analytic Continuation Along Paths.- 3.4 Analytic Continuation and Homotopy.- 3.5 Algebraic Function Elements.- 3.6 Global Analytic Functions.- 4 Exercises for Chapter X.- 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.