Numerically stable conditions on rational and essential singularities

TL;DR

This paper explores relaxed conditions on Taylor series coefficients to better understand singularities, extending classical results by allowing coefficients to be small rather than zero, thus enhancing the analysis of function behavior near singularities.

Contribution

It introduces a novel approach that relaxes classical singularity and convergence conditions by permitting small coefficients, broadening the scope of singularity analysis.

Findings

Extended classical results on singularities with small coefficients

Connected coefficients of Taylor series to singularity types

Provided new conditions for convergence and overconvergence

Abstract

This paper demonstrates some connections between the coefficients of a Taylor series $f(z)=\ds\sum_{n=0}^\infty a_n z^n$ and singularities of the function. There are many known results of this type, for example, counting the number of poles on the circle of convergence, and doing convergence or overconvergence for $f$ on any arc of holomorphy. A new approach proposed here is that these kinds of results are extended by relaxing the classical conditions for singularities and convergence theorems. This is done by allowing the coefficients to be sufficiently small instead of being zero.