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?

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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.