On the stability to perturbations of Stieltjes continued fractions with complex elements

Abstract

This paper investigates the conditions for stability to perturbations of Stieltjes continued fractions with complex elements. An analytical approach is proposed that allows for the evaluation of the influence of perturbations of the fraction's coefficients and its complex variable on the value of its approximant. Recurrence relations for the relative errors of the approximants are established and presented in the form of a continued fraction. Sufficient conditions for stability to perturbations are obtained, as well as explicit estimates for the relative errors of the approximants. Sets of stability to perturbations are constructed, in particular, semi-circular and circular sets in the complex plane.