Elements of the theory of elliptic functions. Transl. from the Russian by H.H. McFaden; translation ed. by Ben Silver. (English) Zbl 0694.33001

Akhiezer, N. I.

Translations of Mathematical Monographs, 79. Providence, RI: American Mathematical Society (AMS). vii, 237 p. $ 93.00 (1990).

This book provides a well-written and concise introduction to the theory of elliptic functions, integrals and their applications for a reader with a basic knowledge of complex function theory. It is thus both more demanding and more sophisticated than the recent book of D. F. Lawden. This is also true of the applications treated, which include problems of conformal mapping, problems of approximation (à la Tchebycheff) and problems relating to differential equations. The chapter on approximation even contains a small hint of the author’s ideas which later led to the spectacular connections between Riemann surfaces and nonlinear partial differential equations of soliton type. Too bad this lovely piece of expository writing has been priced totally out of reach of those students who could most benefit from it!

Reviewer: H.H.Martens

MSC:

33E05	Elliptic functions and integrals
33-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to special functions

Keywords:

elliptic integrals; theta functions; elliptic functions