Handbook of conformal mappings and applications Cover Half Title Title Page Copyright Page Table of Contents Preface Notations Definitions and Acronyms Part 1: Theory and Conformal Maps 1: Introduction 1.1 Historical Background 1.2 Modern Developments 1.3 In Retrospect 2: Conformal Mapping 2.1 Definitions 2.1.1 Analytic Functions 2.1.2 Integration 2.1.3 Fatous Lemma 2.2 Jordan Contour 2.2.1 Hölder Condition 2.3 Metric Spaces 2.4 Basic Theorems 2.4.1 Singularities 2.4.2 Residues 2.4.3 Boundary Values for Cauchy Integral 2.4.4 Argument Principle 2.4.5 Plemelj Formulas 2.5 Harmonic Functions 2.5.1 Harmonic Conjugate 2.5.2 Capacity 2.6 Univalent Functions 2.6.1 Conformality and Uniqueness 2.6.2 Conformal and Isogonal Mappings 2.6.3 Conformal Mapping of an Area Element 2.6.4 Analytic Continuation 2.6.5 Chain Property Map 2.1. Map 2.2. Map 2.3. Map 2.4. Map 2.5. Map 2.6. Map 2.7. Map 2.8. Map 2.9. 2.6.6 Schwarz Reflection Principle 2.6.7 Conformal Equivalence 2.6.8 Riemann Sphere 2.6.9 Bieberbach Conjecture 2.6.10 Mercators Projection 2.7 Taylor Series Approximations 2.7.1 Interior of the Unit Circle Map 2.10. 3: Linear and Bilinear Transformations 3.1 Definitions of Certain Curves 3.1.1 Line 3.1.2 Circle 3.1.3 Ellipse 3.1.4 Hyperbola 3.1.5 Rectangular Hyperbola 3.1.6 Parabola 3.1.7 Cassinis Ovals and Lemniscate 3.1.8 Cardioid and Limaçons 3.2 Bilinear Transformations 3.2.1 Fixed Points 3.2.2 Linear Transformation 3.2.3 Composition of Bilinear Transformations Map 3.1. Involutory Transformation Map 3.2. Three Points onto Three Points Map 3.3. Sequence of Bilinear Transformations 3.3 Cross Ratio 3.3.1 Symmetric Points 3.3.2 Symmetry Principle 3.3.3 Special Cases Map 3.4.