In [20], Siewert and Burniston obtained an exact analytical solution of the equation:

$$\begin{array}{}{\displaystyle x\mathrm{coth}x=\alpha {x}^{2}+1}\end{array}$$(11)

It can be written in terms of the Langevin function

$$\begin{array}{}{\displaystyle L\left(x\right)=\mathrm{coth}x-\frac{1}{x}}\end{array}$$(12)

as:

$$\begin{array}{}{\displaystyle L\left(x\right)=\alpha x}\end{array}$$(13)

So, in [20], Siewert and Burniston obtained explicitly the function *x*(*α*), or, equivalently, the inverse of the function *L*(*x*) /*x*. Mezö and Baritz had obtained the inverse of the Langevin function as a generalized Lambert function with one upper and one lower parameter, *W*(*t*; *s*; *a*) . Actually, if *L*(*x*) = *a*, the function *L*^{−1}(*a*) is:

$$\begin{array}{}{\displaystyle {L}^{-1}\left(a\right)=-2W\left(\frac{2}{a+1};\frac{2}{a-1};\frac{a-1}{a+1}\right)}\end{array}$$(14)

according to [29, [30, [9].

It is easy to see that, if 0 < *x* < ∞, then:

$$\begin{array}{}{\displaystyle 0<L\left(x\right)<1}\end{array}$$(15)

and:

$$\begin{array}{}{\displaystyle 0<\frac{L\left(x\right)}{x}<\frac{1}{3}}\end{array}$$(16)

Figure 1 The plot of *L*(*x*) (solid, black), tan*h*(*x*) (purple), *L*(*x*)/x (red) and of the constant function *y*(*x*) = 1 (dotted).

It is a simple exercise to check that the solution of (13), so the inverse of the function *L*(*x*) /*x* is a *W*(*t*_{1},*t*_{2}; *s*_{1}, *s*_{2}; *a*) generalized Lambert function, namely:

$$\begin{array}{}{\displaystyle x\left(\alpha \right)=\frac{1}{2}W(\frac{1+\sqrt{1-4\alpha}}{\alpha},\frac{1-\sqrt{1-4\alpha}}{\alpha};}\\ {\displaystyle \phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}-\frac{1+\sqrt{1-4\alpha}}{\alpha},-\frac{1-\sqrt{1-4\alpha}}{\alpha};1)}\end{array}$$(17)

So, the inverse of the Langevin function *L*(*x*) is a generalized Lambert function with one upper and one lower parameter, and the inverse of *L*(*x*) /*x* is a generalized Lambert function with two upper and two lower parameters. However, as there is no explicit formula for the dependence of *W*(*t*_{1}, *t*_{2}; *s*_{1}, *s*_{2}; *a*) of its parameter, eq. (17) seems to be of limited practical value.

Let us recall now the main applications of the Langevin function in physics. It has been firstly introduced in the context of classical theory of paramagnetism, where it gives the magnetization *M* as a function of the external magnetic field *H* and temperature *T*:

$$\begin{array}{}{\displaystyle M=n\mu L\left(\frac{\mu H}{{k}_{B}T}\right)}\end{array}$$(18)

(see for instance [31], eq. (9.2)). Actually, this is the equation of state for a classical paramagnet. The same formula is valid for superparamagnetic nanoparticles (particles which are small enough to contain one single magnetic domain), at high enough values of temperature *T* [32], [33].

For physicists, it is important that the Langevin function is a particular case of a function which has a key role in the mean field theory of magnetism, the Brillouin function *B*_{S}, defined as:

$$\begin{array}{}{\displaystyle {B}_{S}\left(x\right)=\frac{2S+1}{2S}\mathrm{coth}\left(\frac{2S+1}{2S}x\right)-\frac{1}{2S}\mathrm{coth}\left(\frac{1}{2S}x\right)}\end{array}$$(19)

Indeed,

$$\begin{array}{}{\displaystyle {B}_{\mathrm{\infty}}\left(x\right)=L\left(x\right)}\end{array}$$(20)

Another important particular case corresponds to the half-integer spin, *S* = 1/2, when

$$\begin{array}{}{\displaystyle {B}_{1/2}\left(x\right)=\mathrm{tanh}x}\end{array}$$(21)

The Langevin function and its inverse are relevant not only for magnetism, but also for other domains of physics with important practical applications, as polymers (polymer deformation and flow) [34], [36], [37] or rubber elasticity [35]. An idealized model for rubber-like chain is given in Kubo’s treatise of statistical mechanics [38], Problem 17, Ch. 2, p. 135, and is illustrative for understanding how the Langevin function is used in this domain. Another application of the Langevin function pertains to solar energy conversion (daily clearness index) [40]. Actually, Keady noticed that the average of the daily clearness index can be expressed in terms of a Langevin function [39]. Researchers in these fields proposed a large number of useful analytical approximations for *L*(*x*) and
*L*^{−1}(*x*). We shall give an example.

Taking the inverse Langevin function in both sides of (13), we get:

$$\begin{array}{}{\displaystyle {L}^{-1}\left(L\left(x\right)\right)={L}^{-1}\left(\alpha x\right)=x}\end{array}$$(22)

which also determines the function *x*(*α*), as eq. (13) or (11) do. Let us use, for *L*^{−1}(*x*), the very simple and precise approximation proposed by Kröger, see eq. (10) of [35]:

$$\begin{array}{}{\displaystyle {L}^{-1}\left(x\right)=\frac{3x}{\left(1-{x}^{2}\right)\left(1+0.5{x}^{2}\right)}}\end{array}$$(23)

In this case, the transcendental equation

$$\begin{array}{}{\displaystyle {L}^{-1}\left(\alpha x\right)=x}\end{array}$$(24)

has an approximate, but precise and simple algebraic equation, whose physically convenient root is:

$$\begin{array}{}{\displaystyle x\left(\alpha \right)=\frac{1}{\alpha}\sqrt{\frac{\sqrt{3\left(3-8\alpha \right)}-1}{2}}}\end{array}$$(25)

The identity

$$\begin{array}{}{\displaystyle f\left(\alpha \right)=\frac{x\left(\alpha \right)\mathrm{coth}x\left(\alpha \right)}{\alpha x{\left(\alpha \right)}^{2}+1}=1}\end{array}$$(26)

where *x*(*α*) is replaced with the approximate solution
(25), is fulfilled with a relative error less than 0.003, as we can see in the plot of Figure 2.

Figure 2 The plot of l.h.s. of the approximate identities (26) - blue, and (30) - brown, and of the constant functions 1 - red, and 0.996 - green.

So, we have the approximate relation:

$$\begin{array}{}{\displaystyle W(\frac{1+\sqrt{1-4\alpha}}{\alpha},\frac{1-\sqrt{1-4\alpha}}{\alpha};}\\ {\displaystyle \phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}-\frac{1+\sqrt{1-4\alpha}}{\alpha},-\frac{1-\sqrt{1-4\alpha}}{\alpha};1)}\\ {\displaystyle \phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\simeq \frac{1}{\alpha}\sqrt{2\left(\sqrt{3\left(3-8\alpha \right)}-1\right)}}\end{array}$$(27)

Also, with (14) and (23):

$$\begin{array}{}{\displaystyle W\left(\frac{2}{a+1};\frac{2}{a-1};\frac{a-1}{a+1}\right)\simeq -\frac{3}{2}\frac{a}{\left(1-{a}^{2}\right)\left(1+0.5{a}^{2}\right)}}\end{array}$$(28)

In conclusion, using a precise analytical approximation for the inverse Langevin function (23), we obtained precise approximate analytical expressions for particular cases of generalized Lambert functions, (27) and (28).

Kruger’s formula (23) is not the only very precise approximation of the inverse Langevin function; Keady [39] proposed the following one:

$$\begin{array}{}{\displaystyle {L}^{-1}\left(x\right)\simeq \frac{6}{\pi}\mathrm{tan}\left(\frac{\pi}{2}x\right)\frac{1+0.417\phantom{\rule{thinmathspace}{0ex}}70{\mathrm{tan}}^{2}\left(\frac{\pi}{2}x\right)}{1+0.507\phantom{\rule{thinmathspace}{0ex}}86{\mathrm{tan}}^{2}\left(\frac{\pi}{2}x\right)}}\end{array}$$(29)

which is even more precise, as we can see in Figure 2, but not useful for obtaining the inverse of *L*(*x*) /*x*. Actually, the error of Keady’s approximation is essentially the error of the identity

$$\begin{array}{}{\displaystyle g\left(x\right)=\frac{1}{x}L\left({L}^{-1}\left(x\right)\right)=1}\end{array}$$(30)

with *L*^{−1}(*x*) given by (27). The difference between the form of *f*(*x*) and *g*(*x*) is due to the difference between the arguments of the Langevin functions in (22) and (27).

The usefulness of Kroeger’s formula for putting the equation of state (18) in a more convenient form becomes more clear if we follow Arrot’s approach [41].

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