Fourier transform of a Generalized Gaussian I've got a family of functions called Generalized Gaussians.
They're given by:
$f(x) = \exp(-ax^{2p})$
Where $p \in \{1,2,3,\ldots\}$
Could anyone tell me how to find their Fourier transforms?

Here is a method: we define $g(t):=\int_{\mathbb R}e^{itx}e^{-x^{2p}}\mathrm dx$. Then, taking the derivative under the integral and integrating by parts, we derive the differential equation
$$g^{(2p-1)}(t)=(-1)^p\frac t{2p}g(t).$$
The solutions of this equation are analytic, hence we can find a recurrence relation between the coefficients. 
