Department of Mathematics

The George Washington University

Washington, DC 20052, USA

The Lanczos tau-method, with perturbations proportional to Faber polynomials, is employed to approximate the Bessel functions of the first kind Jn(z) and the second kind Yn(z), the Hankel functions of the first kind Hn1(z) and the second kind Hn2(z) of integer order n for specific outer regions of the complex plane, i.e. |z| >= R for some R. The scaled symbolic representation of the Faber polynomials and the appropriate automated tau-method approximation are introduced. Both symbolic and numerical computation are discussed. In addition, numerical experiments are employed to test the proposed tau-method. Computed accuracy for J0(z) and Y0(z) for |z| >= 8 are presented. The results are compared with those obtained from the truncated Chebyshev series approximations and with those of the tau-method approximations on the inner disk |z| >= 8. Some concluding remarks and suggestions on future research are given.

This is a continuation of earlier papers

Tau-method approximations for the Bessel function Y0(z).

Tau-method approximations for the Bessel function Y1(z).

A note on the Tau-method approximations for the Bessel
functions Y0(z) and Y1(z).

Key words and phrases: Tau-method approximation, Bessel functions, automated tau-method, symbolic approximation.

This paper has been published in Journal of Computational and Applied Mathematics, Vol. 75, 99-118 (1996).