I am trying to solve this integral, but I cannot think of an easy way. Please help! So the initial problem is this
Show that if $H^1(t) = -e \varepsilon X/[1+(t/z)^2]$, then, to the first order,
$P_{0 \rightarrow 1} = \frac{e^2 (\varepsilon)^2 (\pi)^2 z^2}{2m\omega \hbar} e^{-2\omega z} $
$X$ is a matrix
So in order to fully solve the problem you need to reduce the integral below.
$$\int_{-\infty}^{\infty} \frac{e^{i\omega t}}{1 + \left(\frac{t^2}{z^2}\right)} dt$$
where $i$ is an imaginary number and $\omega$ and $z$ are constants.