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The inverse Laplace transform of some analytic functions with an application to the eternal solutions of the Boltzmann equation. (English) Zbl 1025.44002

The following inversion formula for the Laplace transform is given. Let F(p) be an analytic function in the complex plane cut along the negative real axis. Assume that F(p) ¯=F(p ¯) and that the limiting value F ± (t)=lim ϕπ-0 Fte ±iϕ , F + (t)=F - (t) ¯ exist for almost all t>0. If (A) F(p)=o(1) for |p|, F(p)=o|p| -1 for |p|0, uniformly in any sector |argp|<π-η, π>η>0; (B) there exists ε>0 such that for every π-ε<ϕπ,

Fre ±iϕ 1+rL 1 (𝐑 + ),Fre ±iϕ a(r),

where a(r) does not depend on ϕ and a(r)e -δr L 1 (𝐑 + ) for any δ>0. Then,

-1 [F](x)=1 π 0 dte -xt F - (t)·

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)].

MSC:
44A10Laplace transform
44A20Integral transforms of special functions
82C40Kinetic theory of gases (time-dependent statistical mechanics)