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 Fatou's Lemma 2.2 Jordan Contour 2.2.1 HSlder 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 2.6.6 Schwarz Reflection Principle 2.6.7 Conformal Equivalence 2.6.8 Riemann Sphere 2.6.9 Bieberbach Conjecture 2.6.10 Mercator's Projection 2.7 Taylor Series Approximations 2.7.1 Interior of the Unit Circle 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 Cassini's Ovals and Lenmiscate 3.1.8 Cardioid and Lima~ons 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 ���� Part 2��Numerical Conformal Mapping Part 3: Applications