\documentclass[border=5]{standalone} 
\usepackage{tikz} 
\directlua{ function coth (i)%定义双曲正切   
return math.cosh(i) / math.sinh(i) endfunction 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] \draw [help lines] (-5,0) -- (10,0); \draw [help lines, -stealth] (0,-1) -- (0,1.5); \draw [densely dotted] (0,{ Brillouin(1, 100)} ) -- ++(10,0); \draw [red]   plot [domain=-5:10, samples=100] (\x, { Brillouin(1, \x)}); \draw [green] plot [domain=-5:10, samples=100] (\x, { Brillouin(5, \x)}); \draw [blue]  plot [domain=-5:10, samples=100] (\x, { Brillouin(50, \x)}); \end{tikzpicture} \end{document}