Inverse Langevin and Brillouin functions: mathematical properties and physical applications

TL;DR

This paper reviews the mathematical properties of inverse Langevin and Brillouin functions, explores their applications across physics fields, and introduces new exact and approximate inverse functions using generalized Lambert functions.

Contribution

It provides a comprehensive review and introduces new exact and approximate inverse functions for Langevin and Brillouin functions using generalized Lambert functions.

Findings

Exact forms of inverse functions obtained in some cases

New approximate inverse functions proposed

Applications across multiple physics domains

Abstract

This paper gives a coherent and comprehensive review of the results concerning the inverse Langevin L(x) and Brillouin functions B_J (x) and of the inverse of L(x)/x and B_J (x)/x. As these functions are used in several fields of physics, without evident interconnections - magnetism (ferromagnetism, superparamagnetism, nanomagnetism, hysteretic physics), rubber elasticity, rheology, solar energy conversion - the new results are not always efficiently transferred from a domain to another. The increasing accuracy of experimental investigations claims an increasing accuracy in the knowledge of these functions, so it is important to compare the accuracy of various approximants and even to obtain, in some cases, the exact form of the inverses of L(x), B_J (x), L(x)/x and B_J (x)/x. This exact form can be obtained, in some cases, at least in principle, using the recently developed theory of generalized Lambert functions; in some particular - and also relevant - cases, explicit expressions for these new special functions are obtained. The paper contains also some new results, concerning both exact and approximate forms of the aforementioned inverse functions.