TUM Library

The subject of conformal mappings is a major part of geometric function theory that gained prominence after the publication of the Riemann mapping theorem -- for every simply connected domain of the extended complex plane there is a univalent and meromorphic function that maps such a domain conformally onto the unit disk. The Handbook of Conformal Mappings and Applications is a compendium of at least all known conformal maps to date, with diagrams and description, and all possible applications in different scientific disciplines, such as: fluid flows, heat transfer, acoustics, electromagnetic fields as static fields in electricity and magnetism, various mathematical models and methods, including solutions of certain integral equations.

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 Fatou's 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 Conjugate2.5.2 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 Mercator's 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 Curves3.1.1 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 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.; Map 3.5.; Map 3.6.; Map 3.7.; Map 3.8.; Map 3.9.

Map 3.10.Map 3.11.; Map 3.12.; Map 3.13.; Map 3.14.; Map 3.15.; Map 3.16.; Map 3.17.; Map 3.18.; Map 3.19.; Map 3.20.; Map 3.21.; Map 3.22.; Map 3.23.; Map 3.24.; Map 3.25.; Map 3.26.; Map 3.27.; Map 3.28.; Map 3.29.; Map 3.30.; Map 3.31.; Map 3.32.; Map 3.33.; Map 3.34.; Map 3.35.; Map 3.36.; Map 3.37.; Map 3.38.; Map 3.39.; Map 3.40.; Map 3.41.; Map 3.42.; Map 3.43.; Map 3.44.; Map 3.45.; Map 3.46.; Map 3.47.; Map 3.48.; Map 3.49.; Map 3.50(a)-(d). Cassini's ovals; Map 3.51. Cardioid and Limaçon; Map 3.52. Cardioid and Generalized Cardioids; 3.4 Straight Lines and Circles

Map 3.53. Lines parallel to the axesMap 3.54. Other lines and circles; Map 3.55. Circle onto another circle; Map 3.56(a)-(e). Three points onto three points; Map 3.57. Straight line onto straight line; Map 3.58. Angle onto itself, with arms interchanged; Map 3.59. Straight line onto circle; Map 3.60. Circle onto straight line; Map 3.61. Circle and line in contact onto two parallel lines; Map 3.62. Two circles in contact onto two parallel lines (inner contact); Map 3.63. Two circles in outer contact onto two parallel lines (outer contact)

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