We study the inverse Langevin function 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→1. The only real singularities of the inverse Langevin function
L−1(x) are two simple poles at x=±1 and we see how to remove
their effects either multiplicatively or additively. In addition, we find that
L−1(x) has an infinity of complex singularities. Examination of
the Taylor series about the origin of 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.Comment: 25 pages, 10 figures, 4 tables, 50 equations, 28 reference