Complex Gaussian Fourier transform (${\hat{g}}_{\tau }(u)=(-i\tau {)}^{-n/2}{e}^{-\pi i|u{|}^2/\tau }$)

With kernel $e^{-2\pi i⟨u,x⟩}$ and $\mathrm{Im}\tau >0$, the Fourier transform of $e^{\pi i\tau |x{|}^2}$ is $(-i\tau {)}^{-n/2}{e}^{-\pi i|u{|}^2/\tau }$. Choose the logarithm of $-i\tau$ on the right half-plane. The ordinary real Gaussian Fourier transform proves the identity at $\tau =it$; holomorphic continuation proves the complex formula.