Symbolic and numerical computation on Bessel functions of complex argument and large magnitude

The Lanczos tau-method, with perturbations proportional to Faber polynomials, is employed to approximate the Bessel functions of the first kind J(v)(z) and the second kind Y-v(z), the Hankel functions of the first kind H-v((1))(z) and the second kind Hv((2))(z) of integer order v for specific outer regions of the complex plane, i.e. \z\ greater than or equal to 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 J(0)(z) and Y-0(z) for \z\ greater than or equal to 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\ less than or equal to 8. Some concluding remarks and suggestions on future research are given.