This should give you an idea. The output simplified, used as input, gives your orig. function except with rules. Fourier tables in books are a little infamous for not all being in the same form (and some being incorrect "in ways").
In: FourierTransform[(
E^((a x)/(2 \[Pi])) HeavisideTheta[-x Sign[Re[a]]] Sign[Re[a]])/
Sqrt[2 \[Pi]], x, w] // Simplify
Out: 1/(a + 2 I \[Pi] w) /; Re[a] != 0
