This paper is to complete and improve the work reported in earlier papers Tau-method approximations for the Bessel function Y0(z) and Tau-method approximations for the Bessel function Y1(z) . These methods use the Lanczos tau-method (in Coleman's version) to approximate the Bessel functions Y0(z) and Y1(z). We introduce symbolic representations of the scaled Faber polynomials on any fan-shaped section of the complex plane. These Faber polynomials are used as the perturbation terms in the tau-method. Numerical comparison among the power series, the Chebyshev series and the tau-method are conducted to show the accuracy improvement achieved by this new version of the tau-method. Some concluding remarks and suggestions on future research are given.
Key words and phrases: Tau-method approximation, Bessel function.