I was wondering if there exists a closed form solution for the Fourier transform of
$$ \exp{(ax(1-\log x ))}, $$
where $a$ is a positive constant, and clearly $x > 0$ (we consider the function to be zero for $x<0$). For larger values of $a$ (say 25 or so) the function is quite similar to a shifted Gaussian.
If there exists a closed form solution, how do I calculate it?
Any help is greatly appreciated!