Geometry of Complex Numbers (Dover Books on Mathematics) by Hans Schwerdtfeger (PDF)

Description

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 ReviewSince 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.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐A wonderful gem. From circles represented as 2×2 Hermitian matrices, through classification of Moebius transformations, to non-Euclidean geometry. What a ride!The development is careful and thorough.I think a prospective reader would benefit from some prior exposure to pencils of circles, which I first came into contact with in Johnson’s wonderful “Advanced Euclidean Geometry” (another Dover book). Schwerdtfeger will show you what pencils of circles can do for you!A bit of projective geometry would also be helpful, although the demands are minimal. The “completed z-plane” is after all the complex projective line, and Moebius transformations are just projective transformations of that line. A prior exposure to cross-ratio, and its special case, harmonic ratio, would be part of this. Schwerdtfeger doesn’t pre-suppose any of this, but I think prior knowledge would help the reader focus more on what’s really new and exciting here.At some point the author expects the reader to be comfortable with viewing real 3-space as the affine part of real projective 3-space, although terminology like this isn’t used. (We get from a discussion of the complex plane to real 3-space via stereographic projection of the plane onto the unit sphere, which is a key move for the overall presentation.)All of the necessaries are in Pedoe’s “Geometry” (another Dover book).It helped me keep track of things by having been exposed to the representation of circles and lines in the real plane as points in real projective 3-space, in Pedoe’s “Geometry”. In this representation, orthogonality of circles becomes conjugacy of points (with respect to a certain quadric), and inversion in a circle becomes a certain harmonic homology. Schwerdtfeger uses a dual representation, in which circles and lines become *planes* in 3-space. Although this is isomorphic to Pedoe’s representation, it plays well with the focus on the unit sphere.The exercises (or “examples” as Schwerdtfeger calls them) are interesting!

⭐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.

⭐I would have to wait months before I can write a real review about the book because I would have to finish reading it. However for now I can say that the book was shipped fast and in very good condition for a price that everyone can afford.

⭐Good!

⭐The book is retired from a library

⭐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.

⭐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.

⭐Ho acquistato il libro per studiare all’università. Spedizione molto veloce e libro nuovo. ottima stampa e rilegatura. lo consiglio a tutti gli studenti.J’ai découvert ce livre en 1985… en préparant l’agrégation … de mathématiques. Je l’ai utilisé pour présenter une leçon sur les faisceaux de cercles qui eut un franc succès auprès du professeur Avez lors de son exposition.La présentation algébrique des faisceaux de cercles (à l’aide des matrices hermitiennes de taille 2 et des transformations homographiques complexes {z->(az+b)/(cz+d) }présentée dans ce livre est à la fois élégante et efficace. L’appendice sur l’emploi du parallèlogramme caractéristique m’avait beaucoup plu. Le traitement du birapport de quatre nombres complexes est tout à fait réussi; en prime, le livre traite des géométries non-euclidiennes du plan…et des systèmes dynamiques associés aux homographies.Old book, old theory, bad printing, bad quality. Content is good…but it is 1962 book printed in 2019 with no changes!!!

⭐Good value for the money.

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