Customer Reviews

Complex Functions: An Algebraic and Geometric Viewpoint

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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...

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).

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.