# On the properties of Laplace transform originating from one-sided L\\'evy stable laws

Source: [On the properties of Laplace transform originating from one-sided L\\'evy stable laws](<https://www.sota2.com/research/paper/68fa32c71e209131ce88a72a>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-24T19:40:17.310Z

Authors: K. A. Penson, K. G\\'orska

Published: 2014-06-15T08:40:58.000Z

## Abstract

We consider the conventional Laplace transform of $f(x)$, denoted by $\\mathcal{L}\[f(x); p\]~\\equiv~F(p)=\\int\_{0}^{\\infty} e^{-p x} f(x) dx$ with ${\\rm \\mathfrak{Re}}(p) \> 0$. For $0 \< \\alpha \< 1$ we furnish the closed form expressions for the inverse Laplace transforms $\\mathcal{L}^{-1}\[F(p^{\\alpha}); x\]$ and $\\mathcal{L}^{-1}\[p^{\\alpha-1}F(p^{\\alpha}); x\]$. In both cases they involve definite integration with kernels which are appropriately rescaled one-sided L\\'{e}vy stable probability distribution functions $g\_{\\alpha}(x)$, $0 \< \\alpha \< 1$, $x \> 0$. Since $g\_{\\alpha}(x)$ are exactly and explicitly known for rational $\\alpha$, \\textit{i.e.} for $\\alpha = l/k$ with $l, k=1, 2, \\ldots$, $l \< k$, our results extend the known and tabulated case of $\\alpha = 1/2$ to any rational $0 \< \\alpha \< 1$. We examine the integral kernels of this procedure as well as the resulting two kinds of L\\'{e}vy integral transformations.

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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/1406.3802>)

[PDF](<https://arxiv.org/pdf/1406.3802.pdf>)