Question

Is this a valid way of evaluating the Fourier transform of a Gaussian?

Answer 1

This is true, no problem: $$\sqrt{\frac{\pi}{a}} = \int_{-\infty}^\infty e^{-at^2}dt$$

Now notice that it doesn't matter where it's centered on the real line so: $$\int_{-\infty}^\infty e^{-at^2}dt=\int_{-\infty}^\infty e^{-a(t+b)^2}dt$$

Expand that quadratic and notice that we can pull a constant term out:
$$\int_{-\infty}^\infty e^{-a(t+b)^2}dt = e^{-ab^2}\int_{-\infty}^\infty e^{-at^2-2abt}dt $$

So we have:
$$e^{ab^2}\sqrt{\frac{\pi}{a}} = \int_{-\infty}^\infty e^{-at^2-2abt}dt$$

Plug in \(a=\frac{1}{2}\) and \(b=i \omega\) and we have:

\[\sqrt{2 \pi} e^{-\frac{\omega^2}{2}} = \int_{-\infty}^\infty e^{-\frac{t^2}{2}} e^{-\omega t}dt\]

The question I have is, I'm shifting into the complex plane instead of along the real line, so even though I think this is the correct answer, I don't think applying that shift of \(t \to t+b\) is valid since \(b = i \omega\) which is a complex number not real!

Answer 2

Found the answer here: http://math.stackexchange.com/questions/1297096/gaussian-integral-with-a-shift-in-the-complex-plane