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Symbolic and Numerical Computation on Bessel Functions of Complex Argument
and Large Magnitude

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Jun Zhang

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Department of Mathematics, The George Washington University, Washington, DC 20052

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

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ABSTRACT

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).