# Opt2

Find the associated power spectrum given the covariance function ${\displaystyle B_{x}(\tau )=\exp \left[-a^{2}\left({\frac {\tau }{\tau _{0}}}\right)^{2}\right],\tau _{0}>0,a>0\,}$

The power spectrum is defined as
${\displaystyle S_{x}(\omega )={\frac {1}{\pi }}\int _{0}^{\infty }B_{x}(\tau )\cos(\omega \tau )\,d\tau \,}$

${\displaystyle S_{x}(\omega )={\frac {1}{\pi }}\int _{0}^{\infty }\exp \left[-a^{2}\left({\frac {\tau }{\tau _{0}}}\right)^{2}\right]\cos(\omega \tau )\,d\tau \,}$

A useful integral relation is:

${\displaystyle \int _{0}^{\infty }x^{\mu -1}e^{-m^{2}x^{2}}\cos(bx)\,dx={\frac {\Gamma (\mu /2)}{2m^{\mu }}}\cdot {}_{1}F_{1}\left({\frac {\mu }{2}};{\frac {1}{2}};{\frac {-b^{2}}{4m^{2}}}\right),\{\mu >0,m>0,b>0\}\,}$

Here ${\displaystyle {}_{1}F_{1}\,}$ is the hypergeometric function defined as:

${\displaystyle {}_{1}F_{1}(p;c;z)=\sum _{n=0}^{\infty }{\frac {(p)_{n}}{(c)_{n}}}{\frac {z^{n}}{n!}},|z|<\infty \,}$

In the present problem, ${\displaystyle \mu =1,m=a/\tau _{0},b=\omega \,}$

${\displaystyle S_{x}(\omega )={\frac {1}{\pi }}{\frac {\Gamma \left({\frac {1}{2}}\right)}{2{\frac {a}{\tau _{0}}}}}\cdot {}_{1}F_{1}\left({\frac {1}{2}};{\frac {1}{2}};{\frac {-\omega ^{2}}{4\left({\frac {a}{\tau _{0}}}\right)^{2}}}\right)={\frac {{\sqrt {\pi }}\tau _{0}}{\pi 2a}}\sum _{n=0}^{\infty }{\frac {\left({\frac {-\omega ^{2}}{4\left({\frac {a}{\tau _{0}}}\right)^{2}}}\right)^{n}}{n!}}\,}$

${\displaystyle ={\frac {\tau _{0}}{{\sqrt {\pi }}2a}}\exp \left({\frac {-\omega ^{2}\tau _{0}^{2}}{4a^{2}}}\right)\,}$