I have a Lorentzian function:
$$f(t)=\frac{a}{\pi}\frac{1}{t^2+a^2}$$
To take the Fourier transform, one would compute
$$\mathscr{F}[f(t)]=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\frac{a}{\pi}\frac{e^{-i\omega t}}{t^2+a^2}dt$$
According to Mathematica, this evaluates to
$$ \mathscr{F}[f(t)]=\sqrt{\frac{\pi}{2a^2}}e^{-a\omega} $$
Several online integral calculators confirm this as well. However, it is not immediately clear to me how this integral would be evaluated analytically. This does not seem like a "standard" integral. So, my question is how would this integral be evaluated explicitly?