Transformations to extend the range of application of power series solutions of differential equations of motion

Abstract

Let the solution of a differential equation, expanded in powers of the independent variablet, have radius of convergenceT. let τ, wheret=t(τ), be a new independent variable, and let the corresponding power series in τ have radius of convergenceS. Thent(S) will not in general be equal toT. Ift(S)>T, then the series in powers of τ may have advantages over those in powers oft. Mathematical consequences of this distinction have been appreciated since the time of Poincaré. In this note the practical applications of some transformations are investigated.