Complex Analysis: The Geometric Viewpoint (Carus Mathematical Monographs, Series Number 23) 2nd Edition by Steven G. Krantz (PDF)

Description

In this second edition of a Carus Monograph Classic, Steven G. Krantz, a leading worker in complex analysis and a winner of the Chauvenet Prize for outstanding mathematical exposition, develops material on classical non-Euclidean geometry. He shows how it can be developed in a natural way from the invariant geometry of the complex disk. He also introduces the Bergmann kernel and metric and provides profound applications, some of which have never appeared in print before. In general, the new edition represents a considerable polishing and re-thinking of the original successful volume. A minimum of geometric formalism is used to gain a maximum of geometric and analytic insight. The climax of the book is an introduction to several complex variables from the geometric viewpoint. Poincaré’s theorem, that the ball and bidisc are biholomorphically inequivalent, is discussed and proved.

User’s Reviews

Editorial Reviews: Review ‘A first-rate book, which can be used either as a text or a reference.’ Choice’In five very nicely written chapters this book gives an introduction to the approach to function theory via Riemannian geometry. Very little function-theoretic background is needed and no knowledge whatsoever of differential geometry is assumed.’ Mathematical Reviews Book Description Advanced textbook on central topic of pure mathematics. About the Author Steven G. Krantz, currently Professor and Chairman of Mathematics at Washington University in St. Louis, earned hid PhD at Princeton University and has taught at UCLA, Princeton University and Pennsylvania State University. He is the recipient of the UCLA Alumni Foundation Distinguished Teaching Award, the MAA’s Chauvenet Prize, the MAA Beckenbach Book Award, and the Outstanding Academic Book Award of the Current Review of Academic Libraries. He has written numerous books including: “Function Theory of Several Complex Variables,” “Real Analysis and Foundations, The Geometry of Domains in Space” (with Harold R. Parks),” Function Theory of One Complex Variable” (with Robert E. Greene), “The Implicit Function Theorem “(with Harold Parks and “A Panorama of Harmonic Analysis” and “Mathematical Apocrypha” (for the MAA). He is also the author of over one-hundred research articles. Read more

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

⭐If you are a math nerd who can follow books likeTheoremProofCorollaryPropositionProofCorollaryPropositionLemmaProofCorollaryTheoremProofCorollary…[Some Explanatory Text]…TheoremProofCorollaryPropositionProofCorollaryPropositionLemmaProof…[Figure]…Then this is the book for you. If you know Complex analysisalready, you may enjoy this one. IT IS NOT INTENDED FOR SOMEONEWHO WANT’S TO LEARN C.A. FROM SCRATCH.I was personally looking for something more of the like of”Visual Complex Analysis” by Tristan Needham. Lots of Fun Indeed.

⭐As one of the reviewers said this is not for the beginner in complex analysis but if you have been exposed to a standard course on complex analysis and have some knowledge of differential geometry, simple knowledge of tangent spaces should be more than sufficient, then this book is a delight. The introduction of a metric structure recasts old proofs in a new light and shifts the focus from counting zeroes and poles to understanding analytic maps as geometric objects that reveal a lot of their properties through their interaction with the metric structure. Highly recommended.

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