Visual Complex Analysis

Book Description

This radical approach to complex analysis replaces the standard calculational arguments with new geometric ones. With several hundred diagrams, and far fewer prerequisites than usual, this is the first visually intuitive introduction to complex analysis. Although designed for use by undergraduates in mathematics and science, the novelty of the approach will also interest professional mathematicians. PDF extracts of the book can be read at http://www.usfca.edu/vcal/.

Customer Reviews:

Beautiful.......2006-05-11

This is one of the best math books ever written. It is insightful, reader friendly,and has an excellent set of exercises. It also covers a broad range of topics including applications to physics.

Great Intuitive Coverage of Complex Variables.......2006-02-17

This is a great book for a Complex analysis course, especially when combined with a more traditional text like Ahlfors or Knopp's Theory of Functions. Needham's geometric approach builds 'feel' for the subject and manages to include a huge range of topics. Sometimes just trying to work out what you are seeing in a particular visualization can lead to new insights. All in all its an innovative and well written book.

Pretty but not a substitute for traditional text........2005-10-09

My slightly harsh rating is an antidote to all the gushing about
this book. It is a nice book with lots of pretty pictures and
genuine geometrical insights and is well worth reading as a
supplement to traditional complex analysis texts. The geometrical
topics are actually quite good. If you are a maths major then
this book will be of limited use because its coverage of the
traditional topics is simply too weak. The geometrical approach
quickly runs out of steam, in my opinion, once it gets into
complex integration. Homotopy does not even rate a mention in the
index. My pet dislike was the almost complete omission of the
calculus of residues. The author dimisses that topic as being
old-fashioned. True, the application to computing real integrals
is reduced since the advent of computers. But I think that a
maths major would need to be aware of Jordan's Lemma and other
techniques to estimate the asymptotic behaviour of integrals
along curves. I also found that the treatment of multi-valued
functions and branch cuts quite confusing, which is surprising
in a book which is supposed to have a strong geometrical focus.

Exceptionally pretty complex analysis book.......2005-04-09

This is a very exciting introduction to complex analysis. Its most striking feature is the many excellent illustrations; pictures are used to explain things whenever possible. Needham is always eager to explain, and also to show meaning. Thus Möbius transformations are not just charming quirks; instead Needham gives a self-contained introduction to non-Euclidean geometry to show them in action. And when one needs to understand analytic functions as flows, then Needham gives a self-contained introduction to the ideas of vector analysis. Because the book always spills over on other topics like this, I keep my copy within reach at all times, as a treasure mine of beautiful, visual explanations of topics even outside of complex analysis proper. This wide scope works very well most of the time but it should be said that there are probably too many minor side topics than are appropriate in a first textbook. Sometimes Needham seems to include results not because they fit in or add something important, but because he has thought of such a pretty proof. But we quickly forgive him, for the book is so extremely loveable, and it is still by far my first choice as a first textbook of complex analysis.

essential.......2005-01-13

It is possible to memorize definitions, master proofs and work endless exercises and still feel that you don't understand what's going on. This is especially true in complex analysis. This book emphasizes the visual/geometric aspect of analytic functions at the expense of some loss of rigor. These insights are priceless.

Needham employs a wider range of mathematical tools than other books aimed at the upper level undergraduate, e.g., Palka and Brown/Churchill. This would include simple group theory, linear algebra, vector calculus and obviously geometry. Often this works very well at the expense of some digression. At other times, the more traditional algebraic approach is better.

This book is unique and fills an important need.

Complex Hyperbolic Geometry (Oxford Mathematical Monographs)

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Complex Hyperbolic Geometry (Oxford Mathematical Monographs)
William M. Goldman
Manufacturer: Oxford University Press, USA
ProductGroup: Book
Binding: Hardcover

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ASIN: 019853793X

Book Description

Complex hyperbolic geometry is a particularly rich area of study, enhanced by the confluence of several areas of research including Riemannian geometry, complex analysis, symplectic and contact geometry, Lie group theory, and harmonic analysis. The boundary of complex hyperbolic geometry, known as spherical CR or Heisenberg geometry, is equally rich, and although there exist accounts of analysis in such spaces there is currently no account of their geometry. This book redresses the balance and provides an overview of the geometry of both the complex hyperbolic space and its boundary. Motivated by applications of the theory to geometric structures, moduli spaces and discrete groups, it is designed to provide an introduction to this fascinating and important area and invite further research and development.

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plane geometry and complex numbers
a good beginning
This book contained the stuff I wanted to know
Should be a "must read" for math students

Geometry of Complex Numbers
Hans Schwerdtfeger
Manufacturer: Dover Publications
ProductGroup: Book
Binding: Paperback

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ASIN: 0486638308

Book Description

Illuminating, widely praised book on analytic geometry of circles, the Moebius transformation, and 2-dimensional non-Euclidean geometries.

Customer Reviews:

plane geometry and complex numbers.......2006-10-07

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.

a good beginning.......2006-08-06

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.

This book contained the stuff I wanted to know.......2003-05-10

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.

Should be a "must read" for math students.......2002-03-10

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.

Complex Functions: An Algebraic and Geometric Viewpoint

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A postmodern second course in complex analysis
invitation to elliptic functions and Riemann surfaces

Complex Functions: An Algebraic and Geometric Viewpoint
Gareth A. Jones , and David Singerman
Manufacturer: Cambridge University Press
ProductGroup: Book
Binding: Paperback

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ASIN: 052131366X

Book Description

Elliptic functions and Riemann surfaces played an important role in nineteenth-century mathematics. At the present time there is a great revival of interest in these topics not only for their own sake but also because of their applications to so many areas of mathematical research from group theory and number theory to topology and differential equations. In this book the authors give elementary accounts of many aspects of classical complex function theory including Möbius transformations, elliptic functions, Riemann surfaces, Fuchsian groups and modular functions. A distinctive feature of their presentation is the way in which they have incorporated into the text many interesting topics from other branches of mathematics. This book is based on lectures given to advanced undergraduates and is well-suited as a textbook for a second course in complex function theory. Professionals will also find it valuable as a straightforward introduction to a subject which is finding widespread application throughout mathematics.

Customer Reviews:

A postmodern second course in complex analysis.......2005-12-22

First there are two short chapters on the Riemann sphere and Möbius transformations, probably partly familiar to many readers. Chapter 3 on elliptic functions opens with the following sentence, which illustrates the type of unconvincing pseudo-motivation that occur throughout the book: "Having considered the sphere and its meromorphic functions in the first part of this book, we now turn our attention to another compact surface, the torus, and its meromorphic functions." From here the theory of elliptic functions unwinds along a path that is largely hidden from us, although towards the end of the chapter we are rewarded for fumbling ahead with a discussion of elliptic curves and even a vague allusion to number theory. Now we are warmed up for a general treatment of Riemann surfaces (chapter 4). People often lament that such a simple and beautiful idea requires so much technical machinery to be treated rigourously. Jones & Singerman certainly don't prove them wrong. Chapter 5 is called "PSL(2,R) and its discrete subgroups". This is a heavily geometrical topic, especially since PSL(2,R) is half the isometry group of the half plane model of hyperbolic geometry. Indeed, instead of praising this as a pleasantly geometrical part of function theory, it is perhaps even more satisfying to treat it geometrically altogether (cf. Stillwell). Anyway, the prime example of those "discrete subgroups" of chapter 5 is the modular group, which gets the final chapter 6 all to itself. The whole book has a thoroughly modern feel to it. Admittedly, there is an impressive amount of mathematics for such a modestly sized book, but personally I think there are still many virtues of a more classical and less mysterious approach as well, such as Hurwitz & Courant or Siegel (neither of which is in the bibliography).

invitation to elliptic functions and Riemann surfaces.......2004-12-23

this book goes from the Riemann sphere up to the modular function and the great Picard Theorem via geometry and group theory; it is an excellent book to begin with from an elementary knowledge of complex functions since it is rather self-contained; I see it as a fine blending of different area of mathematics and as such it should help its reader towards more understanding (through serious work...) of those. As an example, I opened the book at random an found the rule for adding points of a cubic (page 119). As a matter of facts, when I first met hyperbolas as groups via geometric addition of their points, I was rather dumbfounded. It's a pity that the hardback edition cannot be found anymore...

A Primer of Real Analytic Functions, Second Edition

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A Primer of Real Analytic Functions, Second Edition
Steven G. Krantz , and Harold R. Parks
Manufacturer: Birkhäuser Boston
ProductGroup: Book
Binding: Hardcover

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The Implicit Function Theorem: History, Theory, and Applications

ASIN: 0817642641

Book Description

The subject of real analytic functions is one of the oldest in modern mathematics and is the wellspring of the theory of analysis, both real and complex. To date, there is no comprehensive book on the subject, yet the tools of the theory are widely used by mathematicians today. Key topics in the theory of real analytic functions that are covered in this text and are rather difficult to pry out of the literature include: the real analytic implicit function theorem, resolution of singularities, the FBI transform, semi-analytic sets, Faà di Bruno's formula and its applications, zero sets of real analytic functions, Lojaciewicz's theorem, Puiseaux's theorem. New to this second edition are such topics as: * A more revised and comprehensive treatment of the Faà di Bruno formula * An alternative treatment of the implicit function theorem * Topologies on the space of real analytic functions * The Weierstrass Preparation Theorem This well organized and clearly written advanced textbook introduces students to real analytic functions of one or more real variables in a systematic fashion. The first part focuses on elementary properties and classical topics and the second part is devoted to more difficult topics. Many historical remarks, examples, references and an excellent index should encourage student and researcher alike to further study this valuable and exciting theory.

From Holomorphic Functions to Complex Manifolds

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From Holomorphic Functions to Complex Manifolds
Klaus Fritzsche , and Hans Grauert
Manufacturer: Springer
ProductGroup: Book
Binding: Hardcover

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ASIN: 0387953957

Book Description

This book is an introduction to the theory of complex manifolds. The authors¿ intent is to familiarize the reader with the most important branches and methods in complex analysis of several variables and to do this as simply as possible. Therefore, the abstract concepts involving sheaves, coherence, and higher-dimensional cohomology have been completely avoided. Only elementary methods such as power series, holomorphic vector bundles, and one-dimensional cocycles are used. Nevertheless, deep results can be proved. The book can be used as a first introduction to several complex variables as well as a reference for the expert.

Geometric Function Theory: Explorations in Complex Analysis (Cornerstones)

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Geometric Function Theory: Explorations in Complex Analysis (Cornerstones)
Steven G. Krantz
Manufacturer: Birkhäuser Boston
ProductGroup: Book
Binding: Hardcover

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ASIN: 0817643397

Book Description

Complex variables is a precise, elegant, and captivating subject. Presented from the point of view of modern work in the field, this new book addresses advanced topics in complex analysis that verge on current areas of research, including invariant geometry, the Bergman metric, the automorphism groups of domains, harmonic measure, boundary regularity of conformal maps, the Poisson kernel, the Hilbert transform, the boundary behavior of harmonic and holomorphic functions, the inhomogeneous Cauchy–Riemann equations, and the corona problem.

The author adroitly weaves these varied topics to reveal a number of delightful interactions. Perhaps more importantly, the topics are presented with an understanding and explanation of their interrelations with other important parts of mathematics: harmonic analysis, differential geometry, partial differential equations, potential theory, abstract algebra, and invariant theory. Although the book examines complex analysis from many different points of view, it uses geometric analysis as its unifying theme.

This methodically designed book contains a rich collection of exercises, examples, and illustrations within each individual chapter, concluding with an extensive bibliography of monographs, research papers, and a thorough index. Seeking to capture the imagination of advanced undergraduate and graduate students with a basic background in complex analysis—and also to spark the interest of seasoned workers in the field—the book imparts a solid education both in complex analysis and in how modern mathematics works.

Introduction to Riemann Surfaces (CHEL/313)

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Introduction to Riemann Surfaces (CHEL/313)
George Springer
Manufacturer: Chelsea Publishing Company
ProductGroup: Book
Binding: Hardcover

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ASIN: 0828403139

Book Description

This well-known book is a self-contained treatment of the classical theory of abstract Riemann surfaces. The first five chapters cover the requisite function theory and topology for Riemann surfaces. The second five chapters cover differentials and uniformization. For compact Riemann surfaces, there are clear treatments of divisors, Weierstrass points, the Riemann-Roch theorem and other important topics.

Product Description

Functions of a complex variable are used to solve applications in various branches of mathematics, science, and engineering. Functions of a Complex Variable: Theory and Technique is a book in a special category of influential classics because it is based on the authors extensive experience in modeling complicated situations and providing analytic solutions. The book makes available to readers a comprehensive range of these analytical techniques based upon complex variable theory. Proficiency in these techniques requires practice. The authors provide many exercises, incorporating them into the body of the text. By completing a substantial number of these exercises, the reader will more fully benefit from this book. Based on graduate courses in applied mathematics, Functions of a Complex Variable: Theory and Technique is intended for applied mathematicians, scientists, engineers, and senior or graduate-level students who have advanced knowledge in calculus and are interested in such subjects as complex variable theory, function theory, mathematical methods, advanced engineering mathematics, and mathematical physics.

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