Geometry of Complex Numbers
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Beschreibung:
"This book should be in every library, and every expert in classical function theory should be familiar with this material. The author has performed a distinct service by making this material so conveniently accessible in a single book." — Mathematical Review
Since its initial publication in 1962, Professor Schwerdtfeger's illuminating book has been widely praised for generating a deeper understanding of the geometrical theory of analytic functions as well as of the connections between different branches of geometry. Its focus lies in the intersection of geometry, analysis, and algebra, with the exposition generally taking place on a moderately advanced level. Much emphasis, however, has been given to the careful exposition of details and to the development of an adequate algebraic technique.
In three broad chapters, the author clearly and elegantly approaches his subject. The first chapter, Analytic Geometry of Circles, treats such topics as representation of circles by Hermitian matrices, inversion, stereographic projection, and the cross ratio. The second chapter considers in depth the Moebius transformation: its elementary properties, real one-dimensional projectivities, similarity and classification of various kinds, anti-homographies, iteration, and geometrical characterization. The final chapter, Two-Dimensional Non-Euclidean Geometries, discusses subgroups of Moebius transformations, the geometry of a transformation group, hyperbolic geometry, and spherical and elliptic geometry. For this Dover edition, Professor Schwerdtfeger has added four new appendices and a supplementary bibliography.
Advanced undergraduates who possess a working knowledge of the algebra of complex numbers and of the elements of analytical geometry and linear algebra will greatly profit from reading this book. It will also prove a stimulating and thought-provoking book to mathematics professors and teachers.
CHAPTER I. ANALYTIC GEOMETRY OF CIRCLES
1. Representation of Circles by Hermitian Matrices
a. One circle
b. Two circles
c. Pencils of circles
Examples
2. The Inversion
a. Definition
b. Simple properties of the inversion
Examples
3. Stereographic Projection
a. Definition
b. Simple properties of the stereographic projection
c. Stereographic projection and polarity
Examples
4. Pencils and Bundles of Circles
a. Pencils of circles
b. Bundles of circles
Examples
5. The Cross Ratio
a. The simple ratio
b. The double ratio or cross ratio
c. The cross ratio in circle geometry
Examples
CHAPTER II. THE MOEBIUS TRANSFORMATION
6. Definition: Elementary Properties
a. Definition and notation
b. The group of all Moebius transformations
c. Simple types of Moebius transformations
d. Mapping properties of the Moebius transformations
e. Transformation of a circle
f. Involutions
Examples
7. Real One-dimensional Projectivities
a. Perpectivities
b. Projectivities
c. Line-circle perspectivity
Examples
8. Similarity and Classification of Moebius Transformations
a. Introduction of a new variable
b. Normal forms of Moebius transformations
c. "Hyperbolic, elliptic, loxodromic transformations"
d. The subgroup of the real Moebius transformations
e. The characteristic parallelogram
Examples
9. Classification of Anti-homographies
a. Anti-homographies
b. Anti-involutions
c. Normal forms of non-involutory anti-homographies
d. Normal forms of circle matrices and anti-involutions
e. Moebius transformations and anti-homographies as products of inversions
f. The groups of a pencil
Examples
10. Iteration of a Moebius Transformation
a. General remarks on iteration
b. Iteration of a Moebius transformation
c. Periodic sequences of Moebius transformations
d. Moebius transformations with periodic iteration
e. Continuous iteration
f. Continuous iteration of a Moebius transformation
Examples
11. Geometrical Characterization of the Moebius Transformation
a. The fundamental theorem
b. Complex projective transformations
c. Representation in space
Examples
CHAPTER III. TWO-DIMENSIONAL NON-EUCLIDEAN GEOMETRIES
12. Subgroups of Moebius Transformations
a. The group U of the unit circle
b. The group R of rotational Moebius transformations
c. Normal forms of bundles of circles
d. The bundle groups
e. Transitivity of the bundle groups
Examples
13. The Geometry of a Transformation Group
a. Euclidean geometry
b. G-geometry
c. Distance function
d. G-circles
Examples
14. Hyperbolic Geometry
a. Hyperbolic straight lines and distance
b. The triangle inequality
c. Hyperbolic circles and cycles
d. Hyperbolic trigonometry
e. Applications
Examples
15. Spherical and Elliptic Geometry
a. Spherical straight lines and distance
b. Additivity and triangle inequality
c. Spherical circles
d. Elliptic geometry
e. Spherical trigonometry
Examples
APPENDICES
1. Uniqueness of the cross ratio
2. A theorem of H. Haruki
3. Applications of the characteristic parallelogram
4. Complex Numbers in Geometry by I. M. Yaglom
BIBLIOGRAPHY
SUPPLEMENTARY BIBLIOGRAPHY
INDEX