I need to find the Fourier transform of the following Gaussian-like function:
$$f(x)=\frac{1}{\sqrt{2\pi\sigma^2(x)}}\,e^{-\frac{x^2}{2\sigma^2(x)}}$$
where $\sigma(x) = \sigma_0 \sqrt{1+\left(\frac{x}{a}\right)^2}$ and $a>0$.
I know the FT of a Gaussian function, but how can I use it for calculating the FT of $f(x)$.