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\begin{align}
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!{\rm P}\pars{\xi}&=
\int_{-\infty}^{\infty}\dd x\,{1 \over \root{2\pi}\sigma}\,
\exp\pars{-\,{x^{2} \over 2\sigma^{2}}}
\int_{-\infty}^{\infty}\dd y\,{1 \over \root{2\pi}\sigma}\,
\exp\pars{-\,{y^{2} \over 2\sigma^{2}}}\
\overbrace{\delta\pars{\xi - xy}}^{\ds{\delta\pars{y - \xi/x} \over \verts{x}}}
\\[3mm]&={1 \over 2\pi\sigma^{2}}\int_{-\infty}^{\infty}
\exp\pars{-\,{1 \over 2\sigma^{2}}\bracks{x^{2} + {\xi^{2} \over x^{2}}}}\,
{\dd x \over \verts{x}}
\\[3mm]&={1 \over \pi\sigma^{2}}\int_{0}^{\infty}
\exp\pars{-\,{1 \over 2\sigma^{2}}\bracks{x^{2} + {\xi^{2} \over x^{2}}}}\,
{\dd x \over x}\tag{1}
\end{align}
$\ds{\delta\pars{x}}$ es la
La Función Delta De Dirac.
Con $\ds{x \equiv A\expo{\theta/2}\,,\quad A > 0\,,\quad\theta \in {\mathbb R}}$:
\begin{align}
{\rm P}\pars{\xi}&={1 \over \pi\sigma^{2}}
\int_{-\infty}^{\infty}
\exp\pars{-\,{1 \over 2\sigma^{2}}
\bracks{A^{2}\expo{\theta} + {\xi^{2} \over A^{2}}\expo{-\theta}}}\,
\pars{A\expo{\theta/2}\,\dd\theta/2 \over A\expo{\theta/2}}
\end{align}
Podemos optar $\ds{A}$ tal que
$\ds{A^{2} = {\xi^{2} \over A^{2}}\quad\imp\quad A = \verts{\xi}^{1/2}}$:
\begin{align}
{\rm P}\pars{\xi}&={1 \over 2\pi\sigma^{2}}
\int_{-\infty}^{\infty}
\exp\pars{-\,{\verts{\xi} \over \sigma^{2}}\cosh\pars{\theta}}\,\dd\theta
={1 \over \pi\sigma^{2}}
\int_{0}^{\infty}
\exp\pars{-\,{\verts{\xi} \over \sigma^{2}}\cosh\pars{\theta}}\,\dd\theta
\end{align}
$$\color{#00f}{\large%
{\rm P}\pars{\xi} = {1 \over \pi\sigma^{2}}\,
{\rm K}_{0}\pars{\verts{\xi} \over \phantom{2}\sigma^{2}}}
$$
donde $\ds{{\rm K}_{\nu}\pars{z}}$ es un
El Segundo Tipo De Función De Bessel.