Conformal Representation

Title: Conformal Representation......Page 1
Second Edition, Reprinted 1969......Page 2
PREFACE TO SECOND EDITION......Page 3
PREFACE TO THE FIRST EDITION......Page 4
CONTENTS......Page 5
HISTORICAL SUMMARY......Page 9
Conformal representation in general.......Page 11
Mobius Transformation......Page 12
Invariance of the cross-ratio.......Page 13
Pencils of circles......Page 15
Bundles of circles.......Page 16
Inversion with respect to a circle......Page 19
Geometry of Moius Transformations......Page 21
Inversion with respect to the circles of a bundle......Page 24
Representation of a circular area on itself.......Page 25
Non-Euclidean Geometry (8) (9)......Page 26
Angle and distance......Page 27
The triangle theorem......Page 29
Geodesic curvature......Page 30
Non-Euclidean motions......Page 31
Parallel curves......Page 33
The exponential function......Page 34
Representation of a circular crescent......Page 36
Representation of Rlemann surfaces.......Page 37
Representation of the exterior of an ellipse......Page 39
Representation of an arbitrary simply-connected domain on a bounded domain......Page 40
Schwarz\'s Theorem......Page 47
Liouville\'s Theorem......Page 48
Invariant enunciation of Schwarz\'s Lemma......Page 49
Functions with positive real parts.......Page 51
Harnack\'s Theorem.......Page 52
Surta.ces with algebraic and logarithmic branch-points.......Page 53
Representation of simple domains......Page 54
Representation upon one anotherof domains containing circular areas.......Page 58
Extensions of Schwarz\'s Lemma......Page 60
Julia\'s Theorem......Page 61
Limiting oscillation......Page 66
Normal fkmilies of bounded fUnctioni (17) (18......Page 69
Existence of the solution in certain problems of the calculus of variations.......Page 70
Normal families of regular analytic flinctions......Page 71
The main theorem of conformal representatIon (21).......Page 74
Normal tkmilies composed of ftznction. which transform simple domains into circles......Page 81
The kernel of a sequence of domains......Page 82
Simultaneous conformal transformation of domains lying each within another......Page 85
An inequality due to Lindel鰂.......Page 89
Lemma 1, on representation of the frontier.......Page 90
Transformation of one Jordan domain into another (23)......Page 93
Inversion with respect to an anal:vtfc curve......Page 95
The inversion principle......Page 96
Transformation of corners......Page 99
Conformal traniforma.tiou on the frontier......Page 104
BlendIng of domains......Page 106
Conformal transformation of a three-dimensional surface (28).......Page 107
Conformal representation of a closed surface on a sphere......Page 108
Abstract surfaces......Page 111
The universal covering surface......Page 112
Domains and their boundaries......Page 113
The Theorem of van der Waerden (30)......Page 114
Riemann surfaces......Page 116
The Uniformisation Theorem......Page 118
Conformal representation of a torus......Page 119
II. NOTES AND PAPERS......Page 121