Numerical Calculation of Bessel, Hankel and Airy Functions


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.



Keywords and Phrases

Airy Function; Asymptotic Expansion; Convergence Acceleration; High Order; Mathematical Structure; Nonlinear Sequences; Numerical Calculation; Numerical Evaluations; Numerical Example; Problematic Issues; Recurrence Relations; Reference Values; Resummation; Saddlepoint Method; Special Functions; Taylor Expansions; Uniform Asymptotic Expansion; Algorithms; Asymptotic Analysis; Computer Simulation; Expansion; Function Evaluation; Mathematical Transformations; Number Theory; Numerical Methods; Hankel Functions

International Standard Serial Number (ISSN)


Document Type

Article - Journal

Document Version


File Type





© 2012 Elsevier, All rights reserved.

Publication Date

01 Mar 2012