A weakly stationary continuous-time process has covariance function [C(tau)=sigma^{2} e^{-alpha|tau|}left(cos beta tau+frac{alpha}{beta} sin beta|tau| ight)] Prove that

A weakly stationary continuous-time process has covariance function

\[C(\tau)=\sigma^{2} e^{-\alpha|\tau|}\left(\cos \beta \tau+\frac{\alpha}{\beta} \sin \beta|\tau|\right)\]

Prove that its spectral density is given by

\[s(\omega)=\frac{2 \sigma^{2} \alpha\left(\alpha^{2}+\beta^{2}\right)}{\pi\left(\omega^{2}+\alpha^{2}-\beta^{2}+4 \alpha^{2} \beta^{2}\right)}\]