Basic Methods for Computing Special Functions

Abstract

This paper gives an overview of methods for the numerical evaluation of special functions, that is, the functions that arise in many problems from mathematical physics, engineering, probability theory, and other applied sciences. We consider in detail a selection of basic methods which are frequently used in the numerical evaluation of special functions: converging and asymptotic series, including Chebyshev expansions, linear recurrence relations, and numerical quadrature. Several other methods are available and some of these will be discussed in less detail. We give examples of recent software for special functions where these methods are used. We mention a list of new publications on computational aspects of special functions available on our website.2000 Mathematics Subject Classification: 65D20, 41A60, 33C05, 33C10, 33C15.

Numerical calculation of Bessel, Hankel and Airy functions

“…The method relies on the use of the uniform asymptotic expansions given in Eqs. (12), (18), (19) and (20). These asymptotic expansions involve Ai and Bi functions, and their derivatives.…”

“…However, formidable numerical cancellations still prevent us from using the uniform asymptotic expansions directly when ν ≈ z. In this case, the region near ν = z is bridged by the application of a convergence accelerator (Weniger transformation), applied in this case to the infinite asymptotic series defining the expansion for large ν in the uniform asymptotic formulas (12), (18), (19) and (20). The transformation is applied after the order ν is shifted away [see Eq.…”

“…Expansions about the saddle points give rise to the above mentioned uniform asymptotic expansions (12), (18), (19) and (20). In general, the evaluation of Hankel, and Bessel functions can be written in terms of two saddle points which dominate the paths of integration.…”

“…The uniform asymptotic formulas (12), (18), (19) and (20) are expansions for large ν for argument z = ν y of the Bessel functions. They can be written in a form that is amenable to the application of the convergence acceleration transformation (30), as follows.…”

“…Indeed, Bessel, Hankel and Airy functions belong to the most important special functions used in theoretical physics; the much revived interest in these is also manifest in a recent monograph on Airy functions [16]. The renewed interest in the theory of special functions is also manifest in a number of other recent books and review articles [17][18][19][20].…”