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An Introduction to Complex Function Theory
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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.
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Table of Contents

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 Arzela-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 Moebius 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.

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