The inverse laplace transform of some analytic functions with an application to the eternal solutions of the Boltzmann equation

When the Laplace transform F(p) of a function f(x) has no poles but is singular only on the real negative semiaxis because of a cut required to make it single-valued, the inverse transform f(x) can easily be computed by means of the integral of a real-valued function. This result is applied to the calculation of a class of exact eternal solutions of the Boltzmann equation, recently found by the authors. The new approach makes it easier to prove that these solutions are positive, as well as to study their asymptotics.