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

The Lanczos τ-method, with perturbations proportional to Faber polynomials, is employed to approximate the Bessel functions of the first kind Jv(z) and the second kind Yv(z), the Hankel functions of the first kind Hv(1)(z) and the second kind Hv(2)(z) of integer order v 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 τ-method approximation are introduced. Both symbolic and numerical computation are discussed. In addition, numerical experiments are employed to test the proposed τ-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 τ-method approximations on the inner disk ¦z¦ ⩽ 8. Some concluding remarks and suggestions on future research are given.