The inverse Laplace transform of some analytic functions with an application to the eternal solutions of the Boltzmann equation
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 \(\overline{F(p)}=F(\overline p)\) and that the limiting value \(F^{\pm}(t)=\lim_{\phi\to\pi-0} F\left(t e^{\pm i\phi}\right)\), \(F^+(t)=\overline{F^{-}(t)}\) exist for almost all \(t>0\). If (A) \(F(p)=o(1)\) for \(|p|\to \infty\), \(F(p)=o\left(|p|^{-1}\right)\) for \(|p|\to 0\), uniformly in any sector \(|\arg p|< \pi-\eta\), \(\pi>\eta>0\); (B) there exists \(\varepsilon >0\) such that for every \(\pi -\varepsilon<\phi \leq \pi\), \[ \frac{F\left(r e^{\pm i\phi}\right)}{1+r}\in L^1(\mathbf R_+),\qquad \left|F\left(r e^{\pm i\phi}\right)\right|\leq a(r), \] where \(a(r)\) does not depend on \(\phi\) and \(a(r) e^{-\delta r}\in L^1(\mathbf R_+)\) for any \(\delta >0\). Then, \[ \mathcal L^{-1}[F](x)=\frac{1}{\pi}\int_0^\infty dt e^{-xt} \Im F^{-}(t). \] 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)].
scientific article; zbMATH DE number 1878886
scientific article