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Zbl 1025.44002
Bobylev, A.V.; Cercignani, C.
The inverse Laplace transform of some analytic functions with an application to the eternal solutions of the Boltzmann equation.
(English)
[J] Appl. Math. Lett. 15, No.7, 807-813 (2002). ISSN 0893-9659

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 $\pi -\varepsilon<\phi \leq \pi$, $$\frac{F\left(r e^{\pm i\phi}\right)}{1+r}\in L^1(\bold R_+),\qquad \left|F\left(r e^{\pm i\phi}\right)\right|\leq a(r),$$ where $a(r)$ does not depend on $\phi$ and $a(r) e^{-\delta r}\in L^1(\bold R_+)$ for any $\delta >0$. Then, $$\Cal L^{-1}[F](x)=\frac{1}{\pi}\int_0^\infty dt e^{-xt} \Im F^{-}(t).$$ 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)].
[Osman Yürekli (Ithaca)]
MSC 2000:
*44A10 Laplace transform
44A20 Integral transforms of special functions
82C40 Kinetic theory of gases

Keywords: Laplace transform; Boltzmann equation; inversion formula

Citations: Zbl 1001.82090

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