Complex Gaussian Fourier transform
= Complex Gaussian Fourier transform
{title2=$\widehat g_\tau(u)=(-i\tau)^{-n/2}e^{-\pi i|u|^2/\tau}$}
With kernel $e^{-2\pi i\langle u,x\rangle}$ and $\operatorname{Im}\tau>0$, 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.