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Elliptic functions. (English) Zbl 1105.14001
London Mathematical Society Student Texts 67. Cambridge: Cambridge University Press (ISBN 0-521-78563-4/pbk; 0-521-78078-0/hbk; 0-511-23974-2/ebook). xiii, 387 p. £ 24.95, $42.99/pbk; £ 50.00,$ 90.00/hbk; \$ 34.00/e-book (2006).

The book under review provides an introduction to the analytic theory of elliptic functions for advanced undergraduates or beginning graduate students. This fascinating topic, one of the greatest achievements of mathematics in the 19th century, has been one of the driving forces in the development of various branches of contemporary mathematics, including complex analysis, algebraic geometry, topology, mathematical physics, and others. During the past 175 years, keeping up with the continual progress in the theory of elliptic functions, their generalizations (abelian functions, theta functions), and their diverse applications in complex analysis, arithmetic and geometry, a great variety of excellent textbooks on the subject, both classical and modern, has been published. Therefore one might ask the legitimate question of what particular new aspects another introductory text on elliptic functions could possibly offer. As for the book under review, there are indeed some interesting new features distinguishing it from most of the many other texts on elliptic functions, which can be characterized as follows.

First of all, the text consists of two parts of different authorship, viewpoint, and flavour, which nevertheless form a coherent entirety. More precisely, the first six chapters of the book were written by the late W. F. Eberlein as a largely historical essay on the theory of elliptic functions à la C. G. J. Jacobi. In view of the very fact that the theory of elliptic functions was developed almost simultaneously, but independently, by N. H. Abel (1827/1828) and by C. G. J. Jacobi (1829) out of the inversion problem for elliptic integrals, the author tried to answer the just as hypothetical as gripping question: “What would the unfolding of the theory of elliptic functions have been like if N. H. Abel had lived long enough to fully develop his ideas, rather than it went, after Abel’s untimely death in 1829, under the sole influence of Jacobi’s approach?”

The author’s thesis is that Abel’s approach to elliptic functions was much more natural, rigorous, modern and promising than Jacobi’s (admittedly very elegant) methods, and thus the first six chapters focus on applying Abel’s methods, supplemented by the rudiments of complex function theory, to a treatment of Jacobi’s elliptic functions via special differential equations and the allied problem of inverting integrals. This highly interesting and enlightening, largely historically flavoured project of purely classical nature is carried out in six steps (chapters):

1. The differential equation of the simple pendulum and the Jacobian elliptic functions; 2. Jacobian elliptic functions of a complex variable; 3. General properties of elliptic functions; 4. Jacobi’s theta functions; 5. Further properties of Jacobian elliptic functions; 6. Transformation theory of elliptic functions and the elliptic modular function.

Throughout this part of the book, the author has put strong emphasis on motivating theorems and proofs by foregoing concrete examples, striking analogies, and “plausible inferences” on all possible occasions, which may be seen – apart from the above-mentioned special aim of this part – as another feature of the present book.

The second part of the book, containing the following chapters 7–12 , was written by J. V. Armitage. These chapters complement the foregoing historical essay by a more conventional approach to the theory of elliptic functions and elliptic integrals in the spirit of K. Weierstrass’s fundamental ideas’, together with various illustrating applications of elliptic functions in classical geometry, algebra, arithmetic, and theoretical physics. Seeking to preserve both the essentials and the vein of the first part of the book, the author treats the following six topics (chapters):

7. The Weierstrass elliptic functions (including modular forms, modular functions, the addition theorem for the Weierstrass $\wp$-function, the field of elliptic functions, the connection with the Jacobi elliptic functions, and the corresponding inversion problem); 8. Elliptic integrals (including normal forms, and basic reduction theory); 9. Applications of elliptic functions in geometry (Fagnano’s theorem, the area of the surface of an ellipsoid, properties of space curves, Poncelet’s poristic polygons, spherical trigonometry, the “Nine Circles Theorem”, and the group law on a complex elliptic curve); 10. An application of elliptic functions in algebra: solution of the general quintic equation; 11. An arithmetic application of elliptic functions: the representation of a positive integer as a sum of three squares (via binary forms, unimodular substitutions, and theta identities); 12. Applications in mechanics, statistics and other topics (including Euler’s dynamical equations, the computation of planetary orbits in general relativity, the spherical pendulum, Green’s function for a rectangle, correlation in statistics and elliptic functions, elliptic integrals in numerical analysis, rational maps with empty Fatou set, and the description of the heat-flow on a circle by using the Riemann zeta function).

In an appendix, several standard formulae in complex function theory are explained for the convenience of the reader.

This unique collection of classical applications of elliptic functions in different areas of mathematics, selected and compiled from various classical standard texts, represents another feature of the present book on elliptic functions: a panoramic view to this fascinating classical subject in all its beauty, ubiquity, and down-to-earth nature. In contrast to many recent, mostly more advanced and comprehensive texts on elliptic functions, this book pays special attention to both the origin and the genesis of the topic; and as such it may be seen as a highly valuable enrichment of the existing standard literature in the field. Moreover, the above-mentioned pedagogical strategy, together with the lucid and detailed exposition, makes this text a particularly student-friendly primer for beginners in the theory of elliptic functions.

##### MSC:
 14-02 Research monographs (algebraic geometry) 33-02 Research monographs (special functions) 14K25 Theta-functions 33E05 Elliptic functions and integrals 11E25 Sums of squares, etc 14-01 Textbooks (algebraic geometry) 33-01 Textbooks (special functions) 14K20 Analytic theory; abelian integrals and differentials 01A50 Mathematics in the 18th century 00A05 General mathematics