20.2.9 Prove that \frac{h}{2 \pi i} \int_{-\infty}^{\infty} \frac{e^{-i \omega t} d \omega}{E_{0}-i \Gamma / 2-\hbar \omega}=\left\{\begin{array}{ll} \exp \left(-\frac{\Gamma t}{2 \hbar}\right) \exp \left(-i \frac{E_{0} t}{\hbar}\right), & t>0 \\ 0, & t<0

\end{array}\right. This Fourier integral appears in a variety of problems in quantum mechanics: barrier penetration, scattering, time-dependent perturbation theory, and so on.