  We study the inverse Langevin function $\mathscr{L}^{-1}(x)$ because of its
importance in modelling limited-stretch elasticity where the stress and strain
energy become infinite as a certain maximum strain is approached, modelled here
by $x\to1$. The only real singularities of the inverse Langevin function
$\mathscr{L}^{-1}(x)$ are two simple poles at $x=\pm1$ and we see how to remove
their effects either multiplicatively or additively. In addition, we find that
$\mathscr{L}^{-1}(x)$ has an infinity of complex singularities. Examination of
the Taylor series about the origin of $\mathscr{L}^{-1}(x)$ shows that the four
complex singularities nearest the origin are equidistant from the origin and
have the same strength; we develop a new algorithm for finding these four
complex singularities. Graphical illustration seems to point to these complex
singularities being of a square root nature. An exact analysis then proves
these are square root branch points.
