The inverse Laplace transform of some analytic functions with an application to the eternal solutions of the Boltzmann equation
type: Item
instance of: scholarly article
title: The inverse Laplace transform of some analytic functions with an application to the eternal solutions of the Boltzmann equation
zbMATH Open document ID: 1025.44002
DOI: 10.1016/S0893-9659(02)00046-0
author: Carlo Cercignani
author: Alexander Bobylev
published in: Applied Mathematics Letters
publication date: 2003-03-10T00:00:00Z
review text: 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)].
reviewed by: Osman Yürekli
Mathematics Subject Classification ID: 44A10
Mathematics Subject Classification ID: 44A20
Mathematics Subject Classification ID: 82C40
zbMATH DE Number: 1878886
zbMATH Keywords: Laplace transform
zbMATH Keywords: Boltzmann equation
zbMATH Keywords: inversion formula
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1878886