Customer Reviews

Geometry of Complex Numbers (Dover Books on Mathematics)

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This inexpensive book covers material not easily found elsewhere but key in understanding complex functions. The problem with complex functions is they are hard to visualize because the input is a plane and the output is another plane. The book covers Circles, Moebius transforms, and Non-Euclidean Geometry. The level is senior undergraduate, 1st year graduate. The book is easy to understand with good exercises. I really like this book.

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 ...and dynamical systems associated with linear transforms in the complex plane; very informative and quite refreshing.

I was interested in projecting a network onto hyperbolic space using the upper half plane projection. This book contained the equations relating to that, particularly the moebius transformation z' = (az+b) / (cz + d), and also stuff on stereographic mapping which I found useful.

I have not taken the trouble to understand much of the more in-depth parts of the book, but it is so clear and step-by-step that even though I am not a math student, I'm fairly confident that I could. The whole thing was fairly mind-opening.

Interestingly, after reading this and developing my own intuitions (eg: that flat translation, rotation and scaling are special cases of parabolic, elliptical and hyperbolic transformations with a fixed point at infinity), a re-reading discovered these conclusions in the book. So you can take the exposition and run with it. What I'd really like is to be able to get the n'th root of a transformation (to animate them). I suspect that that's in there too.

The book does not cover real-world applications (aerodynamics, electrodynamics), but that's cool. It's purely about the math.

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.