Numerical calculation of Bessel, Hankel and Airy functions

Abstract

The numerical evaluation of an individual Bessel or Hankel function of large order and large argument is a notoriously problematic issue in physics. Recurrence relations are inefficient when an individual function of high order and argument is to be evaluated. The coefficients in the well-known uniform asymptotic expansions have a complex mathematical structure which involves Airy functions. For Bessel and Hankel functions, we present an adapted algorithm which relies on a combination of three methods: (i) numerical evaluation of Debye polynomials, (ii) calculation of Airy functions with special emphasis on their Stokes lines, and (iii) resummation of the entire uniform asymptotic expansion of the Bessel and Hankel functions by nonlinear sequence transformations.In general, for an evaluation of a special function, we advocate the use of nonlinear sequence transformations in order to bridge the gap between the asymptotic expansion for large argument and the Taylor expansion for small argument ("principle of asymptotic overlap"). This general principle needs to be strongly adapted to the current case, taking into account the complex phase of the argument. Combining the indicated techniques, we observe that it possible to extend the range of applicability of existing algorithms. Numerical examples and reference values are given.

Numerical Calculation of Bessel Functions

“…I expect that this general approach can be used for arbitrary complex values of the argument z and order ν, although I have not investigated those ideas. There are also some difficult problems in remote corners of Bessel function parameters [5]; and whether or not the present method might be better in all cases is an open question.…”

Asymptotic formulae for the Lommel and Bessel functions and their derivatives

“…This differs in our paper from [ 6 ], where the authors are looking for the values of J ν ( ct ) for large values of ν and ct . In particular, Jentschura & Lötstedt [ 6 ] present, via apparently heroic numerical efforts, the following value J ν ( ct )=0.002614463954691926 for ν =5000000.2 and ct =5000000.1. In this formula, the values of the argument ct and order ν of the Bessel function are the largest ones for which we know the value of the Bessel function from the scientific literature.…”

Convergence analysis of the summation of the factorially divergent Euler series by Padé approximants and the delta transformation

“…It is now common to call S (n) k (β, s n , ω n ) and its variants the Weniger transformation (see for example [13,16,17,51,71,90,121] The transformation S (n) k (β, s n , ω n ) was first used for the evaluation of auxiliary functions in molecular electronic structure calculations [146]. Later, predominantly the delta variant (5.6), which will be discussed later, was used with considerable success for the evaluation of special functions and related objects [79,80,83,126,127,128,129,133,137,141,145], the summation of divergent perturbation expansions [41,44,45,46,81,82,127,128,129,131,132,134,135,138,147,148], and the prediction of unknown perturbation series coefficients [8,81,82,135]. More recently, the delta transformation had also been employed in optics in the study of nonparaxial free-space propagation of optical wavefields [23,24,52,90,…”