I was just given this question from Folland's real analysis second edition dealing with tempered distributions and their Fourier transforms Exercise 26 on page 300 :
On $ R^n \times R $ let $ G(x,t) = (4t\pi)^{(-n/2)} e^{{-|x|^2}/{4t}} \chi_{(0,\infty)}(t)$
We are to show that $ \widetilde{G}(\xi,\tau) = (2\pi\tau i + 4\pi | \xi |^2)^{-1} $ For this task we are instructed to look at the proposition that for $ f(x) = e^{-\pi a |x|^2 }$ (a>0) then $ \widetilde{f}( \xi ) = a^{-n/2}e^{{-\pi | \xi |^2 }/2} $. Also we may use the previous exercise stating that the Fourier transform of a multi-variable function is product of its partial Fourier transforms covering all variables
I do not know how to use partial Fourier transforms and the given hint to complete the proof. Help appreciated thanks