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A Brief Introduction to Theta Functions

Dover Publications
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What a strange book! It is a good and interesting book, but nevertheless seems to come to me from an alien culture. And I don’t just mean 1961, the date of original publication.

Theta functions are some of the most interesting of the many special functions that mathematicians have studied. For one thing, they seem to pop up in lots of places. For another, and closer to the heart of what this book is all about, they satisfy the most amazing identities. The goal of this book is not so much to provide a careful treatment of theta functions as to lead the reader on a tour of a vast territory, stopping to point out the sights and to take particularly inviting detours.

There are very few formal definitions in the book, but the first chapter introduces to the functions we are going to study:

Consider the function of \(z\) defined by the infinite series

\[ f(z) = \sum_{n=-\infty}^{\infty} e^{-n^2t+2\pi i z}.\]

Here \(z\) is a complex variable, permitted to assume any value, while \(t\) is a complex parameter satisfying the condition \( Re(t)>0.\)

There follows the first of many “it is easy to see” statements, to the effect that the “series converges absolutely and uniformly in any bounded region of the plane,” and we’re off to the races.

This, together with some historical comments, is the content of the first of 70 short chapters. The book has 78 pages, so we can see that many “chapters” will be less than one page. Each chapter is followed by a paragraph or so of “Comments and References.”

The general approach is to highlight ingenious computations, while anything that requires a delicate estimate or a straightforward computation is left to the reader. So in chapter 2 we meet the function

\[ \theta_3(z,t)=1 + 2q\cos 2z + 2q^4\cos 4z + 2 q^9 \cos 6z + \dots \]

Is this a special case of the \( f(z)\) of chapter 1? The author tells us \( q=e^{\pi i t}\), but doesn’t even raise the issue: it’s up to the reader to decide.

And so it goes. In chapter 4 we meet the basic transformation formula for \(\theta_3\) and are told that it

has amazing ramifications in the fields of algebra, number theory, geometry, and other parts of mathematics. In fact, it is not easy to find another identity of comparable significance. We shall make some effort in the coming pages to justify this apparently extravagant statement.

The following short chapters discuss various parts of Fourier analysis including Poisson summation, which is used, in chapter 9, to prove the transformation formula. In chapter 10, it is used again, to sum a slowly convergent series. Then we meet modular functions and Eisenstein series, look at the one-dimensional heat equation, “deduce” (the author calls it a “formal derivation”) the functional equation of \(\theta_3\) from the theory of the heat equation. We are now on page 20.

The whole book is like that: quick glances at lots of deep ideas, beautiful formulas, some impressive computations and some dodgy ones, Laplace and Mellin transforms, the zeta function. The whole of chapter 38 is a list of “interesting expansions due to Watson which can be established as above by use of Liouville’s theorem.” It’s fun, fascinating, and weird.

Don’t give this book to a student if you’ve just spent a semester teaching her about the importance of checking convergence and conditions of validity. From that point of view, this book is one vast problem set. But if you happen to meet a student who seems to delight in formulas and identities, who enjoys the sheer breadth and variety of this kind of material, who seems to have a small portion of Ramanujan’s gift — well, then, this is the book for her.

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME.

Date Received: 
Thursday, December 26, 2013
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Richard Bellman
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Fernando Q. Gouvêa
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Thursday, December 26, 2013