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Special functions. A unified theory based on singularities. With a foreword by Alfred Seeger. (English) Zbl 1064.33006
Oxford Mathematical Monographs. Oxford: Oxford University Press (ISBN 0-19-850573-6/hbk). xvi, 293 p. £ 65.00 (2000).

From the text: The topic of special functions, normally presented as a mere collection of functions exhibiting particular properties, is treated from a fresh and unusual perspective in this book. The authors have based the special functions on the theory of second-order ordinary differential equations in the complex domain. Several physical applications are presented. Numerous tables and figures will help the reader find his way through the subject.

In this book, the authors present a theory of special functions as a field of knowledge lying at the intersection of mathematics, physics, and computer science. The book is accompanied by a software package (also available on the market) which the reader can use to learn the technical part of the subject with the help of a computer. While it was not intended to set great store by rigorous proofs, attention is paid to various links between equations and functions, thus providing good insight into the theory. Illustrations of the theory by examples, tables, schemes, and figures are widely spread in the text, enabling, hopefully, a better understanding of the material.

Special functions appear as solutions of differential equations, of difference equations, of integrals, in group representation theories, etc. Here, ordinary differential equations are dealt with – in particular, second-order equations.

Readers experienced in the field might wonder why they will not find a historical presentation of the equations, or standard notations and conventional derivations of formulas. The reason for this is that, in order to present the new theory, new structural elements had to be chosen as a basis. Primarily, these are the following:

(1) The notions of the $s$-rank of a singularity and of the $s$-multisymbol of a differential equation are efficient tools for the classification of the equations. They permit the division of linear second-order ordinary differential equations (ODE) into classes and into types. They also appear to be useful in studying the (nonlinear) Painlevé equations.

(2) The form of an equation, distinguishing the various elements of the set of equivalent equations, is another structural element that is important for practical needs. The standard forms: (i) canonical, (ii) normal, and (iii) self-adjoint- – which frequently appear in physical theories – are emphasized. Formulas for the standard forms are more compact than those for more general forms.

(3) The third—so to say, dynamic—structural element consists of confluence and reduction processes of equations. The confluence process controls the coalescence of two singularities of an equation accompanied by limiting processes in the parameter space. It permits the extension of properties of solutions of the basic equation to confluent equations. The reduction process of an equation is a mechanism for changing the type of an equation by specializing parameters. If two singularities coalesce without limiting processes in the parameter space of the equation, then we speak of a merging process.

Several quite recent researches of the authors are included in the book. These are: (i) new asymptotics (S. Slavyanov: Chapter 3), (ii) generalization of Jaffé expansions (W. Lay: Chapter 1, Chapter 3, Addendum), (iii) new integral relations (S. Slavyanov: Chapter 3), (iv) generalizations of Riemann’s scheme (W. Lay, S. Slavyanov: Chapter 1), (v) links between Heun equations and Painlevé equations (S. Slavyanov: Chapter 5), (vi) recurrent calculations of matrix elements (S. Slavyanov: Addendum), (vii) merging processes (W. Lay, S. Slavyanov: Chapter 3). Full references including the names of the co-authors of the relevent publications can be found in the bibliography.

Most of the book was written in Baden-Württemberg – the land where the great philosopher G. W. F. Hegel was born. Perhaps this was the reason why many structural elements appear as triads. Without writing their precise definitions (which can be found below), we exhibit the most important of them: hypergeometric equations, Heun equations, Painlevé equations; Fuchsian equation, confluent equations, reduced confluent equations; class of equations, type of equation, form of equation; differential equation, difference equation, integral equation; regular singularity, irregular ramified singularity, irregular unramified singularity; local solution, eigensolution, polynomial solution; local parameter, scaling parameter, accessory parameter; confluence process, reduction process, merging process.

The first chapter presents the basic general theory dealing with linear homogeneous second-order differential equations with polynomial coefficients and their solutions. The reader is assumed to know basic facts in differential equations and complex variable theory.

The next two chapters deal with equations and special functions belonging to the hypergeometric and Heun class respectively. In the fourth chapter, various examples of applications in physics are presented. The authors tried to choose these examples from different areas. The fifth chapter is an introductory text to Painlevé equations. No preliminary knowledge is required by the reader.

Several items which do not fit the chapters mentioned, but still appear to be useful from the point of view of the authors, are collected in the Appendices A–C.

##### MSC:
 33Cxx Hypergeometric functions 33-01 Textbooks (special functions) 34B30 Special ODE (Mathieu, Hill, Bessel, etc.)