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Complex Functions: An Algebraic and Geometric Viewpoint, by Gareth A. Jones, David Singerman
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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.
- Sales Rank: #296487 in Books
- Published on: 1987-03-27
- Original language: English
- Number of items: 1
- Dimensions: 8.98" h x .79" w x 5.98" l, 1.11 pounds
- Binding: Paperback
- 360 pages
Review
"...a very nice treatment which emphasizes the unity of mathematics...Several years ago the reviewer wanted to teach an undergraduate course that gave an introduction to hyperbolic geometry, Möbius transformations and discrete groups. There was no suitable undergraduate text. This book fills that void," Mathematical Reviews
"Well motivated with good selection of problems at the end of each chapter." American Mathematical Monthly
"...clear and well written...Its message, admirably conveyed, is that mathematics is not a collection of neat parcels, the contents of one being blind to the contents of another." Times Higher Education Supplement
"...clearly presented...a delightful book..." American Scientist
About the Author
Gareth A. Jones is Senior Lecturer in Development Geography, and Dennis Rodgers is Lecturer in Urban Development, both at the London School of Economics and Political Science.
Most helpful customer reviews
11 of 11 people found the following review helpful.
invitation to elliptic functions and Riemann surfaces
By Gilles Benson
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...
2 of 3 people found the following review helpful.
Great mix
By Mark
This book was very useful to me during an undergrad course on modular functions. A good blend of number theory, algebraic geometry, and complex analysis at an approachable level. Lots of interesting results presented very clearly.
4 of 7 people found the following review helpful.
A postmodern second course in complex analysis
By Viktor Blasjo
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).
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