Conformal Representation
Based on lectures by a noted mathematician, this text offers an essential background in conformal representation. Subjects include the Möbius transformation, non-Euclidean geometry, elementary transformations, Schwarz's Lemma, transformation of the frontier and closed surfaces, and the general theorem of uniformization. Clearly detailed proofs accompany this lucid introduction.
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Comprehensive introduction discusses the Möbius transformation, non-Euclidean geometry, elementary transformations, Schwarz's Lemma, transformation of the frontier and closed surfaces, and the general theorem of uniformization. Detailed proofs.
Preface; Introduction. Historical Summary I. Möbius Transformation §5. Conformal representation in general §§6-9. Möbius Transformation §§10-12. Invariance of the cross-ratio §§13-15. Pencils of circles §§16-22. Bundles of circles §§23-25. Inversion with respect to a circle §§26-30. Geometry of Möbius Transformations II. Non-Euclidean Geometry §§31-34. Inversion with respect to the circles of a bundle §35. Representation of a circular area on itself §§36, 37. Non-Euclidean Geometry §§38-41. Angle and distance §42. The triangle theorem §43. Non-Euclidean length of a curve §44. Geodesic curvature §45-47. Non-Euclidean motions §48. Parallel curves III. Elementary Transformations §49-51. The exponential function §§52, 53. Representation of a rectilinear strip on a circle §54. Representation of a circular crescent §§55-59. Representation of Riemann surfaces §§60, 61. Representation of the exterior of an ellipse §§62-66. Representation of an arbitrary simply-connected domain on a bounded domain IV. Schwarz's Lemma §67. Schwarz's Theorem §68. Theorem of uniqueness for the conformal representation of simply-connected domains §69. Liouville's Theorem §§70-73. Invariant enunciation of Schwarz's Lemma §74. Functions with positive real parts §75. Harnack's Theorem §76. Functions with bounded real parts §§77-79. Surfaces with algebraic and logarithmic branch-points §§80-82. Representation of simple domains §§83-85. Representation upon one another of domains containing circular areas §86. Problem §§87, 88. Extensions of Schwarz's Lemma §§89-93. Julia's Theorem V. The Fundamental Theorems of Conformal Representation §94. Continuous convergence §§95, 96. Limiting oscillation §§97-99. Normal families of bounded functions §100. Existence of the solution in certain problems of the calculus of variations §§101-103. Normal families of regular analytic functions §104. Application to conformal representation §§105-118. The main theorem of conformal representation §119. Normal families composed of functions which transform simple domains into circles §§120-123. The kernel of a sequence of domains §124. Examples §§125-130. Simultaneous conformal transformation of domains lying each within another VI. Transformation of the Frontier §§131-133. An inequality due to Lindelöf §§134, 135. Lemma 1, on representation of the frontier §136. Lemma 2 §§137, 138. Transformation of one Jordan domain into another §§139, 140. Inversion with respect to an analytic curve §§141-145. The inversion principle §§146-151. Transformation of corners §§152, 153. Conformal transformation on the frontier VII. Transformation of Closed Surfaces §§154, 155. Blending of domains §§156. Conformal transformation of a three-dimensional surface §§157-161. Conformal representation of a closed surface on a sphere VIII. The General Theorem of Uniformisation §§162, 163, 164. Abstract surfaces §§165, 166. The universal covering surface §167. Domains and their boundaries §168. The Theorem of van der Waerden §169. Riemann surfaces §§170, 171. The Uniformisation Theorem §172. Conformal representation of a torus Bibliographical Notes
