Let $f(x) = e^{-\alpha\pi|x|^4}, x \in \mathbb{R}^n$, where $\alpha > 0$ is a constant. I know that, taking $g(x) = e^{-\pi \alpha|x|^2}$, we get $$\hat{g}(\xi) = \alpha^{-n/2}e^{-\pi|\xi|^2/\alpha}.$$ Although the functions $f$ and $g$ sounds similar, to calculate $\hat{f}$ sounds trickier. In fact, I have heard that there's no representation with elementary functions for $\hat{f}$.
The context: I have the integral, $$g(\xi) = \int_{\mathbb{R}^n} e^{-i \xi \cdot y - |y|^4} dy = \int_{\mathbb{R}^n} e^{- |y|^4} e^{-i \xi \cdot y } dy = \hat{f}(\xi),$$ where $f(x) = e^{-|x|^4}$. This is why I'm trying to calculate that Fourier Transform.
Anyone know how to rewrite that Fourier transform in terms of non elementary functions? I would be very thankful for any reference with this things.