*(English)*Zbl 0856.33001

“The book has been written with students of mathematics, physics and engineering in mind, and also researchers in these areas who need special functions in their work and for whom the results are too scattered in the general literature”, writes the author in the preface. Indeed, the book gives a concise and very well written introduction to the classical canon of special functions. Despite the many books on the subject that can be found in mathematical libraries, most of them quite classical, the new book by N. Temme should not be missing in any of them. Contrary to most books that cover the full spectrum, Temme’s new one is much more than a compilation of formulas. It is very well suited as a text book for a course in special functions, but of course has also its value as a reference book. For students it may be important that the presentation is selfcontained on a certain level. Thus, for instance, mathematical tools like elements of asymptotic analysis or differential equations in the complex plane are explained in short chapters. Each class of functions is typically introduced in detail by one or two subclasses, the remaining material then follows in a much shorter and a more summarizing form. Brief historical notes are given throughout the text, and each chapter ends with a section containing several exercises, many of them partially worked through with hints and directions for a proof, and this considerably augments the amount of material that the book is effectively able to cover.

In Chapter 1 the author introduces and discusses Bernoulli, Euler and Stirling numbers. Chapter 2 is devoted to the statement of some standard theorems in analysis that are useful tools in advanced calculus and asymptotic analysis and includes Watson’s lemma and a discussion of the saddle point method. Chapter 3 examines the properties of the Gamma function and closely related functions such as the Beta function, the reciprocal Gamma function and Riemann’s Zeta function. Chapter 4 is concerned with the theory of regular and singular points of linear second-order differential equations in the complex plane. In Chapter 5, the main properties of the Gauss hypergeometric function are given. Chapter 6 outlines the theory of general orthogonal polynomials and then discusses and compares the properties of the different classical orthogonal polynomial families. Chapter 7 is concerned with confluent hypergeometric functions, while Chapter 8 and 9 develop the properties of Legendre functions and Bessel functions, respectively. Chapter 10 introduces special coordinate systems and discusses boundary value problems in the framework of the separation of the wave equation. Chapter 11 concerns special statistical distribution functions with particular emphasis on various kinds of asymptotic expansions. Chapter 12 introduces elliptic integrals and elliptic functions, and Chapter 13 contains a discussion, with several examples, of numerical aspects of special functions.

In summary: This is a very valuable book. It covers a lot of the standard material on special functions in an efficient and very readable way, and it provides a sound springboard for more specialized knowledge in the subject. About 50 percent of the 199 references appeared in the last 20 years.

##### MSC:

33-02 | Research monographs (special functions) |