Brillouin and Langevin functions

The Brillouin and Langevin functions are a pair of special functions that appear when studying an idealized paramagnetic material in statistical mechanics.
Brillouin function.
The Brillouin function is a special function defined by the following equation:
The function is usually applied (see below) in the context where formula_2" is a real variable and formula_3 is a positive integer or half-integer. In this case, the function varies from -1 to 1, approaching +1 as formula_4 and -1 as formula_5.
The function is best known for arising in the calculation of the magnetization of an ideal paramagnet. In particular, it describes the dependency of the magnetization formula_6 on the applied magnetic field formula_7 and the total angular momentum quantum number J of the microscopic magnetic moments of the material. The magnetization is given by:
where
Note that in the SI system of units formula_7 given in Tesla stands for the magnetic field, formula_18, where formula_19 is the auxiliary magnetic field given in A/m and formula_20 is the permeability of vacuum.
Takacs proposed the following approximation to the inverse of the Brillouin function:
where the constants formula_22 and formula_23 are defined to be
Langevin function.
In the classical limit, the moments can be continuously aligned in the field and formula_3 can assume all values (formula_27). The Brillouin function is then simplified into the Langevin function, named after Paul Langevin:
For small values of , the Langevin function can be approximated by a truncation of its Taylor series:
An alternative, better behaved approximation can be derived from the
Lambert's continued fraction expansion of :
For small enough , both approximations are numerically better than a direct evaluation of the actual analytical expression, since the latter suffers from loss of significance.
The inverse Langevin function is defined on the open interval (−1, 1). For small values of , it can be approximated by a truncation of its Taylor series
and by the Padé approximant
Since this function has no closed form, it is useful to have approximations valid for arbitrary values of . One popular approximation, valid on the whole range (−1, 1), has been published by A. Cohen:
This has a maximum relative error of 4.9% at the vicinity of . Greater accuracy can be achieved by using the formula given by R. Jedynak:
valid for . The maximal relative error for this approximation is 1.5% at the vicinity of x = 0.85. Even greater accuracy can be achieved by using the formula given by M. Kröger:
The maximal relative error for this approximation is less than 0.28%. More accurate approximation was reported by R. Petrosyan:
valid for . The maximal relative error for the above formula is less than 0.18%.
New approximation given by R. Jedynak, is the best reported approximant at complexity 11:
formula_37
valid for . Its maximum relative error is less than 0.076%.
Current state-of-the-art diagram of the approximants to the inverse Langevin function 
presents the figure below. It is valid for the rational/Padé approximants,
A recently published paper by R. Jedynak, provides a series of the optimal approximants to the inverse Langevin function. The table below reports the results with correct asymptotic behaviors.
Comparison of relative errors for the different optimal rational approximations, which were computed with constraints (Appendix 8 Table 1)
Also recently, an efficient near-machine precision approximant, based on spline interpolations, has been proposed by Benítez and Montáns, where Matlab code is also given to generate the spline-based approximant and to compare many of the previously proposed approximants in all the function domain.
High-temperature limit.
When formula_38 i.e. when formula_39 is small, the expression of the magnetization can be approximated by the Curie's law:
where formula_41 is a constant. One can note that formula_42 is the effective number of Bohr magnetons.
High-field limit.
When formula_43, the Brillouin function goes to 1. The magnetization saturates with the magnetic moments completely aligned with the applied field: