*(English)*Zbl 0857.33012

This book studies elliptic functions as meromorphic functions of a complex variable $x$ with two periods $u$ and $v$ whose ratio $\tau =v/u$ has positive imaginary part. This study, which begins in Chapter 3, is preceded by Chapter 0 (ten pages summarizing the required background in analysis), Chapter 1 (30 pages) developing the circular functions via Eisenstein series of the form ${E}_{k}\left(x\right)={\sum}_{n=-\infty}^{\infty}{(x+n)}^{-k}$ for integer $k\ge 1$, and Chapter 2 (25 pages) developing the Gamma function and related functions via series of the form ${H}_{k}\left(x\right)={\sum}_{n=0}^{\infty}{(x+n)}^{-k}$ for integer $k\ge 2$. This approach emphasizes the analogy between the circular and elliptic functions and reveals familiar objects from a new point of view. For example, in Chapter 1 the circular functions are shown to have power series expansions, to satisfy addition formulas and to be related to infinite products and geometry.

In Chapter 3 the basic elliptic functions are introduced by sums of the form ${F}_{k}\left(x\right)={\sum}_{n,e}{E}_{k}(x+n\tau )$ taken over all integers $n$ in $\mathbb{Z}$, where the subscript $e$ denotes the Eistenstein summation convention of grouping together the terms in $\pm n$. Addition formulas and nonlinear relations between the functions ${F}_{k}$ are derived from corresponding results obtained earlier for circular functions. Chapter 4 treats theta functions and extends the infinite product relations introduced in Chapter 1. Chapter 5 develops Jacobi elliptic functions as quotients of theta functions.

Chapter 6 deals with elliptic integrals and inversion of elliptic functions, using the method of Chapter 1 for inverting the sine. Chapter 7 is devoted to functions depending only on the lattice constant $\tau $. This leas naturally to modular groups, fundamental regions, and the modular function $\lambda \left(\tau \right)$. There is also a brief mention of Dedekind’s function $\eta \left(\tau \right)$.

Finally, Chapter 8 treats applications to four topics: the Korteweg and de Vries wave equation, Picard’s theorem on entire functions, quadratic residues and theta-multiplier systems, and elliptic curves. Each chapter concludes with exercises. There is a bibliography and an index, but no index of symbols.

In his preface, the author states: “Most of our results appear as formulae and this has allowed us to replace the ‘theorem-proof’ style of exposition with a freer form in which each result grows naturally out of the ones before.” This freer form works well in a lecture, but it does not help the reader who wishes to browse through the book and obtain a bird’s-eye view of the landscape, or the reader who wants to use the book as a reference.

Because most of the results do appear as formulas, the lack of an index of symbols compounds this problem. For example, a reader interested in the Weierstrass function $\wp \left(x\right)$ will find it defined on p. 80 as ${F}_{2}\left(x\right)-{\delta}_{2}$, but if one has not read the material six pages earlier there may be some difficulty trying to decipher the meaning of ${\delta}_{2}$ since this is not standard notation. The author has done an admirable job in briefly outlining at the beginning of each chapter the main ideas of that chapter. The book contains a wealth of ideas, and its value would be greatly enhanced, both as a text and as a reference, if a summary of important formulas was appended at the end of each chapter.