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Conformal Mapping
L. Bieberbach

AMS Chelsea Publishing
1953; 234 pp; hardcover
Volume: 90
Reprint/Revision History:
first AMS reprinting 2000
ISBN-10: 0-8218-2105-9
ISBN-13: 978-0-8218-2105-3
List Price: US$32
Member Price: US$28.80
Order Code: CHEL/90.H
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Translated from the fourth German edition by F. Steinhardt, with an expanded Bibliography.


"The author presents, in a thorough and painstaking manner, the fundamentals of the principal topics which arise in the theory of ... conformal representation ... The style is lucid and clear and the material well arranged ... The pedagogical excellence of the book is particularly to be recommended. It is an excellent text."

-- Bulletin of the AMS

"The first book in English to give an elementary, readable account of the Riemann Mapping Theorem and the distortion theorems and uniformisation problem with which it is connected ... Presented in very attractive and readable form."

-- Mathematical Gazette

"Engineers will profitably use this book for its accurate exposition."

-- Applied Mechanics Reviews

"An excellent book. It contains sufficient [information] in the first four chapters to satisfy all the needs of students who wish to study only the usual transformations, and many interesting details are given that are rarely to be found elsewhere. The final chapter will appeal to those readers whose interests are in very general problems, and here also is collected material that is not readily accessible elsewhere."

-- Science Progress

Table of Contents

  • Foundations. Linear functions: 1.1 Analytic functions and conformal mapping; 1.2 Integral linear functions; 1.3 The function \(w=1\slash z\); 1.3a Appendix to 1.3: Stereographic projection; 1.4 Linear functions; 1.5 Linear functions (continued); 1.6 Groups of linear functions
  • Rational Functions: 2.7 \(w=z^n\); 2.8 Rational functions
  • General considerations: 3.9 The relation between the conformal mapping of the boundary and that of the interior of a region; 3.10 Schwarz' principle of reflection
  • Further study of mappings represented by given formulas: 4.11 Further study of the geometry of \(w=z^2\); 4.12 \(w=z+1\slash z\); 4.13 The exponential function and the trigonometric functions; 4.14 The elliptic integral of the first kind
  • Mappings of given regions: 5.15 The mapping of a given region onto the interior of a circle (illustrative examples); 5.16 Vitali's theorem on double series; 5.17 A limit theorem for simple mappings; 5.18 Proof of Riemann's mapping theorem; 5.19 On the actual construction of the conformal mapping of a given region onto a circular disc; 5.20 Potential-theoretic considerations; 5.21 The correspondence between the boundaries under conformal mapping; 5.22 Distortion theorems for simple mappings of the disc \(\vert z\vert< 1\); 5.23 Distortion theorems for simple mappings of \(\vert z\vert > 1\); 5.24 On the conformal mapping of non-simple, simply-connected regions onto a circular disc; 5.24a Remark on the mapping of non-simple, multiply-connected regions onto simple regions; 5.25 The problems of uniformization; 5.26 The mapping of multiply-connected plane regions onto canonical regions
  • Bibliography
  • Index
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