Complex Functions: An Algebraic and Geometric Viewpoint [Hardcover]

Gareth A. Jones (Author), David Singerman (Author)

Book Description

ISBN-10: 0521308933 | ISBN-13: 978-0521308939 | Publication Date: March 27, 1987 | Edition: 1

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.

Editorial Reviews

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

Book Description

An elementary account of many aspects of classical complex function theory, including Mobius transformations, elliptic functions, Riemann surfaces, Fuchsian groups and modular functions. The book is based on lectures given to advanced undergraduate students and is well suited as a textbook for a second course in complex function theory.

Product Details

• Hardcover: 356 pages
• Publisher: Cambridge University Press; 1 edition (March 27, 1987)
• Language: English
• ISBN-10: 0521308933
• ISBN-13: 978-0521308939
• Product Dimensions: 9 x 6 x 0.9 inches
• Shipping Weight: 1.4 pounds
• Average Customer Review: 4.3 out of 5 stars  See all reviews (3 customer reviews)
• Amazon Best Sellers Rank: #7,278,084 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews

9 of 9 people found the following review helpful:

4.0 out of 5 stars invitation to elliptic functions and Riemann surfaces, December 22, 2004

By 

Gilles Benson (Beauvais, France) - See all my reviews
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This review is from: Complex Functions: An Algebraic and Geometric Viewpoint (Paperback)

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...

3 of 4 people found the following review helpful:

4.0 out of 5 stars A postmodern second course in complex analysis, December 21, 2005

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Viktor Blasjo - See all my reviews
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This review is from: Complex Functions: An Algebraic and Geometric Viewpoint (Paperback)

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).

5.0 out of 5 stars Great mix, December 16, 2011

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Mark - See all my reviews

This review is from: Complex Functions: An Algebraic and Geometric Viewpoint (Paperback)

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.