In high school we learn about functions and then we work with them in calculus courses. All of those functions are "elementary," and we learn their properties, usually pictured in a graph. It happens often in integrating, however, that we start needing to find anti-derivatives of functions with no known elementary answers. We also run into things like the Gamma function, a natural generalization of the standard factorial notation. Many of these special functions appear in integral forms. Others appear in other problems with no elementary answer; such as solving differential equations. These problems, and therefore the functions that are needed to solve them, are important both in mathematics and in applied mathematics and physics.
There are many methods for doing numerical computions with functions, depending on the nature of function. In the case of special functions, they are a little bit different from the elementary ones, and sometimes harder in view of how these functions are defined.
The book under review focuses on numerical methods for special functions. It includes four parts, which the on-line preface of the book explains in detail. Briefly, the book studies how to compute the values of special functions and their zeros (real and complex), and gives some algorithms for this.
There is no doubt that considering all of special functions for numerical methods in a book would take a (much too) big volume, so the authors focus on some of the more important special functions: the Airy function, gamma and beta functions, hypergeometric functions, and the Bessel functions. The methods are in fact general, so one can think of these as examples for describing their methods.
The book is very well-written. It includes enough tables and diagrams to help the reader understand examples, and it collects some very useful tools for computing special functions. Furthermore, developing a numerical method usually requires an abstract background in analysis, and since the book under review tries to describe the methods in a fundamental way, the book is useful as a collection of examples of applications of the main concepts from undergraduate courses in fundamental analysis.
This is a research monograph rather than a text, but it will be useful for advanced courses in numerical analysis; note, however, that it doesn’t contain problems or exercises. It will also be useful to those who need to implement such computations. I recommend the book strongly for researchers in numerical analysis and the theory of special functions.
Mehdi Hassani is a co-tutelle Ph.D. student in Mathematics in the Institute for Advanced Studies in Basic Science in Zanjan, Iran, and the Université de Bordeaux I, under supervision of the professors M.M. Shahshahani and J-M. Deshouillers.