Numerical Methods for Special Functions

Excerpt

Probably, the most extended (pseudo) definition of the set of functions known as “special functions” refers to those mathematical functions which are widely used in scientific and technical applications, and of which many useful properties are known. These functions are typically used in two related contexts:

1. as a way of obtaining simple closed formulas and other analytical properties of solutions of problems from pure and applied mathematics, statistics, physics, and engineering;

2. as a way of understanding the nature of the solutions of these problems, and for obtaining numerical results from the representations of the functions.

Our book is intended to provide assistance when a researcher or a student needs to get the numbers from analytical formulas containing special functions. This book should be useful for those who need to compute a function by their own means, or for those who want to know more about the numerical methods behind the available algorithms. Our main purpose is to provide a guide of available methods for computations and when to use them. Also, because of the large variety of numerical methods that are available for computing special functions, we expect that a broader “numerical audience” will be interested in many of the topics discussed (particularly in the first part of the book). Several levels of reading are possible in this book and most of the chapters start with basic principles. Examples are given to illustrate the use of the methods, pseudoalgorithms are given to describe technical details, and published algorithms for computing a selection of functions are described as practical illustrations for the basic methods of this book.

The presentation of the topics is organized in four parts: Basic Methods, Further Tools and Methods, Related Topics and Examples, and Software. The first part (Basic Methods) describes a set of methods which, in our experience, are the most popular and important ones for computing special functions. This includes convergent and divergent series, Chebyshev expansions, linear recurrence relations, and quadrature methods. These basic chapters are mostly self-contained and start from first principles. We expect that many of the contents are appropriate for advanced numerical analysis courses (parts of the chapters are in fact based on classroom notes); however, because the main focus is on special functions, detailed examples of application are also provided.

The second part of the book (Further Tools and Methods) contains a set of important methods for computing special functions which, however, are probably not so well known as the basic methods (at least for readers who are not very familiar with special functions).