Langevin function
= Langevin function
{c}
{title2=$\mathscr L(\xi)=\coth\xi-1/\xi$}
The <Langevin function> is the mean projection of a freely orientable three-dimensional unit vector weighted by $e^{\xi\cos\theta}$. Integrating over solid angle gives $Z_1=4\pi\sinh\xi/\xi$, and differentiation gives $\langle\cos\theta\rangle=\partial_\xi\log Z_1=\coth\xi-1/\xi$. It behaves as $\xi/3$ at small argument and $1-1/\xi$ at large positive argument.