Symbolic Computation on Complex Polynomial Solution of Differential Equations

A symbolic computation scheme, based on the Lanczos τ-method, is proposed for obtaining exact polynomial solutions to some perturbed differential equations with suitable boundary conditions. The automated τ-method uses symbolic Faber polynomials as the perturbation terms for arbitrary circular sections of the complex plane and has advantages of avoiding rounding error and easy manipulation over the numerical counterpart. The method is illustrated by applying it to the modified Bessel function of the first kindI0(z) and the quality of the approximation is discussed.