On a New Two-Point Taylor Expansion with Applications

A new two-point Taylor series expansion is proposed. The expansion is slightly different from the classical definition. The coefficients are calculated as recursive relations in a general form. The two-point Taylor expansion is applied to several functions that are odd, even, or neither odd nor even. Functions having a finite interval of convergence or an 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 the solution of a variable coefficient differential equation is also treated.