Method of self-similar factor approximants

TL;DR

The paper extends the self-similar factor approximants method to odd orders, demonstrating its effectiveness in approximating transcendental functions and extrapolating their behavior at infinity using limited series data.

Contribution

It introduces a new way to construct odd-order approximants from power series, enhancing the method's accuracy and applicability to complex functions and models.

Findings

Accurately approximates transcendental functions with few series terms

Enables extrapolation of function behavior at infinity from small-argument series

Shows numerical convergence across multiple examples

Abstract

The method of self-similar factor approximants is completed by defining the approximants of odd orders, constructed from the power series with the largest term of an odd power. It is shown that the method provides good approximations for transcendental functions. In some cases, just a few terms in a power series make it possible to reconstruct a transcendental function exactly. Numerical convergence of the factor approximants is checked for several examples. A special attention is paid to the possibility of extrapolating the behavior of functions, with arguments tending to infinity, from the related asymptotic series at small arguments. Applications of the method are thoroughly illustrated by the examples of several functions, nonlinear differential equations, and anharmonic models.