Taylor expansion of the inverse function with application to the Langevin function

Abstract

A Taylor power series is a powerful mathematical tool, which can be used to express an inverse function especially if it is given in an implicit form. This is for example the case for the inverse Langevin function, which is an indispensable ingredient of full-network rubber models. In the present paper, we propose a simple recurrence procedure for calculating Taylor series coefficients of the inverse function. This procedure is based on the Taylor series expansion of the original function and results in a simple recurrence formula. This formula is further applied to the inverse Langevin function. The convergence radius of the resulting series is evaluated. Within this convergence radius the obtained approximation of the inverse Langevin function demonstrates better agreement with the exact solution in comparison to different Padé approximants.