Geometry of Complex Numbers
By Hans Schwerdtfeger
* Publisher: Dover Publications
* Number Of Pages: 200
* Publication Date: 1980-02-01
* ISBN-10 / ASIN: 0486638308
* ISBN-13 / EAN: 9780486638300
* Binding: Paperback
Product Description:
Illuminating, widely praised book on analytic geometry of circles, the Moebius transformation, and 2-dimensional non-Euclidean geometries.
Summary: plane geometry and complex numbers
Rating: 5
I discovered this book some twenty years ago while trying to improve my knowledge of plane geometry; I used it especially to work on circle pencils: a part of geometry I had already encountered time and again; setting up circles through two-rowed hermitian matrices and linear transforms {z->(az+b)/(cz+d) }as done in the book is both very pretty and efficient. The appendix (numbered 3) describing the use and applications of the characteristic parallelogram really appealed to me. I was also quite impressed by the way the cross ratio of 4 complex numbers is dealt with in the book; to put icing on the cake, one can find within those 200 pages some knowledge of non euclidian plane geometry plan...and dynamical systems associated with linear transforms in the complex plane; very informative and quite refreshing.
Summary: a good beginning
Rating: 5
Schwerdtfeger's nice little book starts at the beginning with geometry of circles, Moebius transformations (a third of the book), and it covers some selected aspects of complex function theory, but the emphasis is on elementary geometry. Harmonic and analytic functions are only touched peripherically.
The central topics are (in this order): geometry of circles, Moebius transformations, geometry of the plane, complex numbers, transformation groups, a little hyperbolic geometry, and ending with a brief chapter on spherical and elliptic geometry.
The book was published first in 1962, but reprinted since by Dover. It is suitable as a supplement in a standard course in complex function theory, at the late undergraduate level, or perhaps at beginning graduate. While it contains attractive geometric concepts, it leaves out a systematic treatment of power series. Some readers might want to begin with that; using some of the other Dover titles on complex functions. We recommend the books by Volkovyskii et al, Flanigan, and Silverman. Review by Palle Jorgensen, August 5, 2006.
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