A weakly stationary, continuous-time stochastic process has spectral density

\[s(\omega)=\sum_{k=1}^{n} \frac{\alpha_{k}}{\omega^{2}+\beta_{k}^{2}}, \quad \alpha_{k}>0\]

Prove that its covariance function is given by

\[C(\tau)=\pi \sum_{k=1}^{n} \frac{\alpha_{k}}{\beta_{k}} e^{-\beta_{k}|\tau|}, \quad \alpha_{k}>0\]