Generalization of Taylor series

Abstract

Scope and Method of Study: This study is composed of a short familiarization with ordinary Taylor series followed by a detailed development of a generalized Taylor's expansion about two points instead of the usual one. It is followed by the development of an interpolation correction formula arising as a result of the generalized expansion. The latter part of this study is devoted to the development of various approximation formulas for well known functions such as ex, arc tan x, and ln{l+x}. The results of these approximations are compared with results obtained from ordinary Taylor series and the well known Hasting's approximations for these same functions. Last of all, an approximation formula for modified Bessel functions of the second kind is developed and the results verified and tabulated by means of the IBM 650 computer.

Findings and Conclusions: The generalized Taylor's expansion with which we worked was found to converge much faster than the ordinary Taylor series for all functions expanded. The approximation formulas for the functions mentioned above, compared favorably with the Hasting's approximations over a limited range depending upon the points about which the function was expanded. The number of calculations necessary to evaluate a particular function was usually, however, more, in the approximation derived by the generalized expansion. The approximation formula derived for the previously mentioned Bessel functions proved to be accurate in general to seven significant figures over the ranges investigated.

The author would like to state in conclusion that only a few of the possible applications of this generalization have been investigated here. Besides the many other functions which might be approximated by this method, almost any application of ordinary Taylors series would bear investigation with respect to this generalization.