= Brillouin function
{c}
{title2=$B_J(x)$}
The <Brillouin function> gives the normalized <magnetization> of independent angular-momentum-$J$ moments. It is odd, increasing, has slope $(J+1)/(3J)$ at zero and tends to $\pm1$ at large positive or negative argument. Explicitly,
$$
B_J(x)=\frac{2J+1}{2J}\coth\!\left(\frac{2J+1}{2J}x\right)-\frac1{2J}\coth\!\left(\frac{x}{2J}\right).
$$
It follows by differentiating the logarithm of $\sum_{m=-J}^J\exp(mx/J)$, the one-ion <partition function>. Its limiting value gives <magnetic saturation>, and its initial slope gives the <Curie law>.
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