On a New Two-Point Taylor Expansion With Applications

Abstract

A new two-point Taylor series expansion is proposed. The expansion is slightly different than the classical definition. The coefficients are calculated as recursive relations in a general form. The two-point Taylor expansion is applied to several functions which are odd, even, neither odd nor even. Functions having finite interval of convergence or infinite interval of convergence are investigated. The conditions for convergence are derived and the results are compared with the results of single-point Taylor expansions as well as two-point Taylor expansions reported in the literature. It is found that for a finite radius of convergence, two-point Taylor expansions can have a single convergence interval as well as two separate convergence intervals. Generally speaking, two-point Taylor expansions better represent the real function when the series is truncated. The new two-point expansion and the classical two-point expansion produced identical results for all the problems treated. Based on the results of this analysis, the asymmetric two-point Taylor expansion presented here does not have an advantage compared to the classical symmetric expansion. An application of the series to solution of a variable coefficient differential equation is also treated.