The following inversion formula for the Laplace transform is given. Let be an analytic function in the complex plane cut along the negative real axis. Assume that and that the limiting value , exist for almost all . If (A) for , for , uniformly in any sector , ; (B) there exists such that for every ,
where does not depend on and for any . Then,
After presenting two illustrative examples of the inversion formula, the inversion formula is applied to the calculation of a class of exact eternal solutions of the Boltzmann equations, recently found by the authors [J. Stat. Phys. 106, 1019-1038 (2002; Zbl 1001.82090)].