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Heun’s differential equations. (English) Zbl 0847.34006
Oxford Science Publications. Oxford: Oxford University Press. xxiii, 354 p. (1995).
[The articles of this volume will not be indexed individually.] In 1889 {\it Karl Heun} published the paper `Zur Theorie der Riemannschen Funktionen zweiter Ordnung mit vier Verzweigungspunkten’ [Math. Ann. 33, 161-179 (1889)]. The paper was devoted to what has come to be called Heun’s equation. For a long time the equation and its solution were considered to be something of a curiosity, even though it contains many special functions of mathematical physics, including the hypergeometric function. In recent years an increasing number of physical problems have given rise to the need to study the properties of Heun’s equation and its solution. In 1989 a “centenial workshop” was held at Schloss Ringberg in Bavaria at which the participants strongly supported suggestions that an up-to-date account of Heun’s equation; its confluent forms as well as its applications should be brought to a wider audience. The result is the book under review. The book is composed of 5 parts. In part A, there is an overall account of Heun’s equation which begins by introducing appropriate notation and then explores various series solutions and their convergence properties. Of particular note are the series solutions in terms of hypergeometric functions. Integral equations and relations are introduced and discussed. Parts B, C, D and E cover all the confluent forms of Heun’s equation and give an up-to-date account of their properties. The book concludes with a comprehensive bibliography. There is a wealth of important results and open problems and the book is a welcome addition to the literature on these important special functions and their applications.

34-02Research monographs (ordinary differential equations)
34M99Differential equations in the complex domain
33C05Classical hypergeometric functions, ${}_2F_1$
33C15Confluent hypergeometric functions, Whittaker functions, ${}_1F_1$