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 J_v(z) and the second kind F_v(z), the Hankel functions of the first kind H⁽¹⁾_v(z) and the second kind H⁽²⁾_v(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 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.