An Introduction to Complex Function Theory

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1 Bruce P. Palka An Introduction to Complex Function Theory With 138 luustrations Springer

2 1 Contents Preface vü I The Complex Number System 1 1 The Algebra and Geometry of Complex Numbers The Field of Complex Numbers Conjugate, Modulus, and Argument 5 2 Exponentials and Logarithms of Complex Numbers Raising e to Complex Powers Logarithms of Complex Numbers Raising Complex Numbers to Complex Powers Functions of a Complex Variable Complex Functions Combining Functions Functions as Mappings 20 4 Exercises for Chapter I 25 II The Rudiments of Plane Topology 33 1 Basic Notation and Terminology Disks Interior Points, Open Sets Closed Sets Boundary, Closure, Interior Sequences Convergence of Complex Sequences Accumulation Points of Complex Sequences 37 2 Continuity and Limits of Functions Continuity Limits of Functions 43 3 Connected Sets 47

3 Contents 3.1 Disconnected Sets Connected Sets Domains Components of Open Sets 50 4 Compact Sets Bounded Sets and Sequences Cauchy Sequences Compact Sets Uniform Continuity 57 5 Exercises for Chapter II 58 Analytic Functions 62 1 Complex Derivatives Differentiability Differentiation Rules Analytic Functions 67 2 The Cauchy-Riemann Equations The Cauchy-Riemann System of Equations Consequences of the Cauchy-Riemann Relations Exponential and Trigonometrie Functions Entire Functions Trigonometrie Functions The Principal Arcsine and Arctangent Functions Branches of Inverse Functions Branches of Inverse Functions Branches of the p fh -root Function Branches of the Logarithm Function Branches of the A-power Function 92 5 Differentiability in the Real Sense Real Differentiability The Functions / z and /j 98 6 Exercises for Chapter III 101 Complex Integration Paths in the Complex Plane Paths Smooth and Piecewise Smooth Paths Parametrizing Line Segments Reverse Paths, Path Sums 115

4 xiii 1.5 Change of Parameter Integrals Along Paths Complex Line Integrals Properties of Contour Integrals Primitives Some Notation Rectifiable Paths Rectifiable Paths Integrals Along Rectifiable Paths Exercises for Chapter IV 136 Cauchy's Theorem and its Consequences The Local Cauchy Theorem Cauchy's Theorem For Rectangles Integrals and Primitives The Local Cauchy Theorem Winding Numbers and the Local Cauchy Integral Formula Winding Numbers Oriented Paths, Jordan Contours The Local Integral Formula Consequences of the Local Cauchy Integral Formula Analyticity of Derivatives Derivative Estimates The Maximum Principle More About Logarithm and Power Functions Branches of Logarithms of Functions Logarithms of Rational Functions Branches of Powers of Functions The Global Cauchy Theorems Iterated Line Integrals Cycles Cauchy's Theorem and Integral Formula Simply Connected Domains Simply Connected Domains Simple Connectivity, Primitives, and Logarithms Homotopy and Winding Numbers Homotopic Paths 197

5 xiv Contents 7.2 Contractible Paths Exercises for Chapter V 204 VI Harmonie Functions Harmonie Functions Harmonie Conjugates The Mean Value Property The Mean Value Property Functions Harmonie in Annuli The Dirichlet Problem for a Disk A Heat Flow Problem Poisson Integrals Exercises for Chapter VI 238 VII Sequences and Series of Analytic Functions Sequences of Functions Uniform Convergence Normal Convergence Infinite Series Complex Series Series of Functions Sequences and Series of Analytic Functions General Results Limit Superior of a Sequence Taylor Series Laurent Series Normal Families Normal Subfamilies of C(U) Equicontinuity The Arzelä-Ascoli and Montel Theorems Exercises for Chapter VII 286 VIII Isolated Singularities of Analytic Functions Zeros of Analytic Functions The Factor Theorem for Analytic Functions Multiplicity Discrete Sets, Discrete Mappings Isolated Singularities 309

6 Contents xv 2.1 Definition and Classification of Isolated Singularities Removable Singularities Poles Meromorphic Functions Essential Singularities Isolated Singularities at Infinity The Residue Theorem and its Consequences The Residue Theorem Evaluating Integrals with the Residue Theorem Consequences of the Residue Theorem Function Theory on the Extended Plane The Extended Complex Plane The Extended Plane and Stereographic Projection Functions in the Extended Setting Topology in the Extended Plane Meromorphic Functions and the Extended Plane Exercises for Chapter VIII 362 IX Conformal Mapping Conformal Mappings Curvilinear Angles Diffeomorphisms Conformal Mappings Some Standard Conformal Mappings Self-Mappings of the Plane and Unit Disk Conformal Mappings in the Extended Plane Möbius Transformations Elementary Möbius Transformations Möbius Transformations and Matrices Fixed Points Cross-ratios Circles in the Extended Plane Reflection and Symmetry Classification of Möbius Transformations Invariant Circles Riemann's Mapping Theorem Preparations The Mapping Theorem The Caratheodory-Osgood Theorem Topological Preliminaries Double Integrals 426

7 xvi Contents 4.3 Conformal Modulus Extending Conformal Mappings of the Unit Disk Jordan Domains Oriented Boundaries Conformal Mappings onto Polygons Polygons The Reflection Principle The Schwarz-Christoffel Formula Exercises for Chapter IX 466 X Constructing Analytic Functions The Theorem of Mittag-Leffler Series of Meromorphic Functions Constructing Meromorphic Functions The Weierstrass P-function The Theorem of Weierstrass Infinite Products Infinite Products of Functions Infinite Products and Analytic Functions The Gamma Function Analytic Continuation Extending Functions by Means of Taylor Series Analytic Continuation Analytic Continuation Along Paths Analytic Continuation and Homotopy Algebraic Function Elements Global Analytic Functions Exercises for Chapter X 535 Appendix A Background on Fields Fields The Field Axioms Subfields Isomorphic Fields Order in Fields Ordered Fields Complete Ordered Fields Implications for Real Sequences 546

8 l Contents xvii Appendix B Winding Numbers Revisited Technical Facts About Winding Numbers The Geometrie Interpretation Winding Numbers and Jordan Curves 550 Index 556

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