In Stein and Weiss, 1971, they prove that for all $\alpha > 0$:
$$\int_{\mathbb{R}^n} e^{-\pi\alpha|\boldsymbol{x}|^2}e^{-2\pi\iota\boldsymbol{x}\cdot\xi}d\boldsymbol{x}=\alpha^{-n/2}e^{-\pi|\xi|^2/\alpha}$$
Later on they say that an application of of the above and an analytic continuation argument tell us that for the function defined: $${f}(x)=e^{-\pi(1+\alpha\iota)x^2}$$ The Fourier Transform yields: $$\hat{f}(x) =\frac{e^{\frac{\pi x^2}{1+\alpha\iota}}}{\sqrt{1+\alpha\iota}}$$ I see how this indeed is the analytic continuation of the function above for real $\alpha$ but the mechanics of the argument aren't very clear to me. Any tips on how to understand this?