Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM) (ISBN 978-0-898716-34-4/pbk; 978-0-89871-782-2/ebook). xiv, 417 p. $ 99.00 (2007).

This book systematically extends the monography by {\it N. M. Temme} [Special functions. An introduction to the classical functions of mathematical physics. J. Wiley, New York (1996;

Zbl 0856.33001)]. It covers the most important functions the applied mathematician, physicist, and engineer is likely to need, most of them (Bessel, Legendre, Kummer functions and special cases, notably Airy functions, etc.) being of hypergeometric type. It leaves aside functions of a very limited range of application that can be found elsewhere. The book is intended to help the practicioner in both study and research in this area that has been affected by the advent of the computer more strongly than many other branches of applied mathematics. As the title indicates, the book emphasizes numerics from the standpoint of software, providing a guide to computational methods and explaining when to use them for computing values or for obtaining deeper insight into the nature of numerical processes and corresponding software. The major part (Part I, about 40% of the 400 pages of the book) concerns the most important methods of a general nature, based on series (convergent and divergent ones), Chebyshev expansions, recurrence relations and continued fractions, and numerical integration. Airy functions are selected as typical examples for illustrating these general methods, particularly for solving ordinary differential equations (ODEs). This includes a discussion of practical limitations due to computer capacity, as formulated for the student in very simple terms. It also motivates corresponding general concepts, such as recessive solution, Liouville-Green approximation, condition of a process, the use of asymptotic expansions in this context, the direction of solving a recurrence to obtain stability, a discussion of an “intermediate range”, etc. Series expansions are followed by a discussion of Chebyshev polynomials and interpolation, Chebyshev expansions, and the evaluation of Chebyshev sums by Clenshaw’s method. The next topic, quadrature methods, concerns the numerical integration of integral representations of special functions, notably by the trapezoidal rule, with discussions of how to avoid heavy oscillations of the integrand by employing complex path integration, in particular, saddle point methods. Part II, entitled Further tools and methods, begins with numerical aspects of continued fractions (Stieltjes and Jacobi fractions and their relation to Padé (approximants), their convergence and their numerical evaluation, showing also Legendre’s example of fractions for the incomplete gamma function as well as fractions for the hypergeometric function. Much space is devoted to the important task of computing real and complex zeros. This consists of local strategies notably, approximations, the global strategies of matrix methods (for orthogonal polynomials as well as for minimal solutions of three-term recurrence relations) and fixed-point methods (for instance, for Bessel functions), and asymptotics for several important classes of functions. Uniform asymptotic expansions are then presented for incomplete gamma functions and for Airy-type expansions of Bessel and some other functions. The last chapter of Part II concerns Padé approximations, sequence transformations, best rational approximations, numerics for ODEs, and other integration methods (Romberg, Fejér, Clenshaw-Curtis, and Gauss). Part III includes the inversion of functions needed in probability theory and statistics, Euler summation, elliptic integrals, and the numerical inversion of Laplace transforms. Finally, the over thirty pages of Part IV are assigned to algorithms for computing selected classes of functions. Corresponding routines can be downloaded under \url{http://functions.unican.es}. They are based on algorithms that have been published by the authors in ACM Transactions on Mathematical Software or in Computer Physics Communications. This monography is topically very rich and many-sided and reflects the great practical experience of the authors. Very remarkable is its balance between the various aspects of the subject matter, a detailed simple Introduction, providing an overview that will be particularly helpful to the student, and the extensive and up-to-date list of almost 250 references at the end.