Plot of the Brillouin Function

The Brillouin function[1][2] is a special function, used for example in statistical mechanics.

It's plotting using standard TikZ, pgfplots is not required. The calculation is done using Lua, so LuaLaTeX has to be used for compiling.

This example was written by Mark Wibrow answering a question on TeX.SE, with modifications by Stefan Kottwitz (formula, scaling, styles for axis and plot).

Edit and compile if you like:

% Brillouin Function
% Author: Mark Wibrow
\documentclass[tikz,border=10pt]{standalone}
\usepackage{amsmath}
\usetikzlibrary{arrows.meta}
\directlua{
function coth (i) 
  return math.cosh(i) / math.sinh(i)
end

function brillouin (J, x) 
  if x == 0 then
    return 0
  else
   return (2*J+1)/(2*J)*coth((2*J+1)/(2*J)*x) - 
        1/(2*J)*coth(1/(2*J)*x)
  end
end
}
\pgfmathdeclarefunction{Brillouin}{2}{%
  \edef\pgfmathresult{%
     \directlua{tex.print("" .. brillouin(#1,#2))}%
   }%  
}
\begin{document}
\begin{tikzpicture}[
    x                = 2cm/10,
    scale            = 3,
    axis/.style      = {help lines, -{Stealth[length = 1.5ex]}},
    brillouin/.style = {domain = -5:10, samples = 100}
  ]
  \draw [axis] (-5,0) -- (10,0);
  \draw [axis] (0,-1) -- (0,1.5);
  \draw [densely dotted] (0,{ Brillouin(1, 100)} ) -- ++(10,0);
  \draw [red]   plot [brillouin] (\x, { Brillouin(1,  \x)});
  \draw [green] plot [brillouin] (\x, { Brillouin(5,  \x)});
  \draw [blue]  plot [brillouin] (\x, { Brillouin(50, \x)});
  \node [align = center, anchor = west] at (1,1.3) {%
    $\begin{alignedat}{2}
      B_J(x) &= \tfrac{2J + 1}{2J}
                &&\coth \left ( \tfrac{2J + 1}{2J} x \right ) \\
             &\quad - \tfrac{1}{2J}
                &&\coth \left ( \tfrac{1}{2J} x \right )
     \end{alignedat}$};
\end{tikzpicture}
\end{document}