I need help with the inverse fourier transform of the following expression
$$\frac{ae^{-w^2/2}}{ae^{-w^2/2}+b}$$
where $a > 0,b > 0$ and $w$ is the angular frequency.
Top term is simply a Gaussian. It looks very simple except the additive term in denominator which complicates things.
At least for $|a| < |b|$, you can express your function as a convergent series in powers of $a$:
$$ \sum_{n=1}^\infty (-1)^{n+1} \frac{a^n}{b^n} e^{-nw^2/2}$$
and then transform term-by-term.