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The Langevin function $\mathcal{L}(x) = \mathrm{coth}(x) - \frac{1}{x}$ is named after the French physicist Paul Langevin (1872-1946). The Wikipedia article on the Brillouin and Langevin functions does not cite a publication by Langevin and other sources I consulted merely indicate that it originates in Langevin's work on paramagnetism.

In which publication by Langevin did this function make its first appearance?

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Best I could determine after searching for a couple of hours, this function was first introduced in the following publication, a scan of which is made available by the Bibliothèque nationale de France:

P. Langevin, "Magnétisme et théorie des électrons," Annales de chimie et de physique, 8:5 (1905), 70-127.

On page 117 one finds:

Le moment magnétique total des $\mathrm{N}$ molécules est évidemment dirigé parallèlement au champ et est égal à la somme des projections sur cette direction des moments composants. Pour l'unité de volume supposée contenir $\mathrm{N}$ molécules, ce moment résultant représente l'intensité d'aimantation $\mathrm{I}$ :

$$\mathrm{I}=\int{\mathrm{M}\cos \alpha}{\ dn} = \int_{-1}^{+1}{2\pi\mathrm{MK\ }x\ e^{ax}\ {dx}}.$$ Or $$\int_{-1}^{+1}{x\ }{e^{ax}}{\ dx} = 2 \left(\frac{\mathrm{Ch\ } {a}}{a}-\frac{\mathrm{Sh\ }a}{a^{2}}\right),$$ d'où $$\mathrm{I}=\mathrm{MN}\left(\frac{\mathrm{Ch\ }a}{\mathrm{Sh\ }{a}}-\frac{1}{a}\right).$$

Here $\mathrm{Ch}$ represents the hyperbolic cosine and $\mathrm{Sh}$ the hyperbolic sine, thus $\mathrm{I}=\mathrm {MN}\left(\mathrm{coth}(a) - \frac{1}{a}\right) = \mathrm{MN}\mathcal{L}(a).$

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