Preface |
Introduction |
Historical Summary |
Mwbius Transformation / I: |
Conformal representation in general |
Invariance of the cross-ratio |
Pencils of circles |
Bundles of circles |
Inversion with respect to a circle |
Geometry of Mwbius Transformations |
Non-Euclidean Geometry / II: |
Inversion with respect to the circles of a bundle |
Representation of a circular area on itself |
Angle and distance |
The triangle theorem |
Non-Euclidean length of a curve |
Geodesic curvature:45-47 |
Non-Euclidean motions |
Parallel curves |
Elementary Transformations:49-51 / III: |
The exponential function |
Representation of a rectilinear strip on a circle: |
Representation of a circular crescent |
Representation of Riemann surfaces |
Representation of the exterior of an ellipse |
Representation of an arbitrary simply-connected domain on a bounded domain |
Schwarz's Lemma: / IV: |
Schwarz's Theorem: |
Theorem of uniqueness for the conformal representation of simply-connected domains: |
Liouville's Theorem |
Invariant enunciation of Schwarz's Lemma: |
Functions with positive real parts: |
Harnack's Theorem: |
Functions with bounded real parts |
Surfaces with algebraic and logarithmic branch-points |
Representation of simple domains |
Representation upon one another of domains containing circular areas: |
Problem |
Extensions of Schwarz's Lemma |
Julia's Theorem |
The Fundamental Theorems of Conformal Representation / V: |
Continuous convergence |
Limiting oscillation |
Normal families of bounded functions |
Existence of the solution in certain problems of the calculus of variations |
Normal families of regular analytic functions |
Application to conformal representation |
The main theorem of conformal representation: |
Normal families composed of functions which transform simple domains into circles |
The kernel of a sequence of domains: |
Examples |
Simultaneous conformal transformation of domains lying each within another |
Transformation of the Frontier / VI: |
An inequality due to Lindelwf |
Lemma 1, on representation of the frontier |
Lemma 2 |
Transformation of one Jordan domain into another |
Inversion with respect to an analytic curve |
The inversion principle |
Transformation of corners |
Conformal transformation on the frontier |
Transformation of Closed Surfaces / VII: |
Blending of domains |
Conformal transformation of a three-dimensional surface |
Conformal representation of a closed surface on a sphere |
The General Theorem of Uniformisation / VIII: |
Abstract surfaces |
The universal covering surface: 167 |
Domains and their boundaries: 168 |
The Theorem of van der Waerden: 169 |
Riemann surfaces |
The Uniformisation Theorem: 172 |
Conformal representation of a torus |
Bibliographical |
Notes |
Preface |
Introduction |
Historical Summary |
Mwbius Transformation / I: |
Conformal representation in general |
Invariance of the cross-ratio |