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I am trying to understand why and when $ e^{-|x|^2} = \int d \omega e^{i \omega x} e^{- |\omega| \gamma }$. I tried the most basic substitution of $cis \theta = e^{i\theta}= \cos \theta + i \sin \theta$:

$$e^{- |x|^2} = \int e^{i \omega x} e^{-|\omega|^2 \gamma} d\omega = \int [\cos \omega x + i sin \omega x ] e^{- |\omega|^2 \gamma} d\omega$$ $$ \int e^{- |\omega|^2 \gamma} \cos \omega x d\omega+ \int i sin \omega x e^{- |\omega|^2 \gamma} d\omega $$

from here on I could try say integration by parts but that technique seems to make matters much worse specially because $e^{x}$ won't disappear with differentiation or integration operations and cos/sin also seem to not go anywhere since they just alternate to -sin/-cos etc when differentiation and integration are used. Furthermore, I can't fathom how on earth I am going to get rid of the imaginary number. Maybe I have not tried enough (might be true) but at this point I don't seen anything useful to even attempt to try.

Anyone has any help? Advice? Direction or anything that can show this equality to be true?

[it feels that this should be common knowledge? I might be wrong, but if I am not wrong suggestions of topics to study up are appreciated, my tags just suggest where I believe the answer might be lying at though I have not a very clear idea if its even right.]

Charlie Parker
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    See e.g. http://mathworld.wolfram.com/FourierTransformGaussian.html – Winther Nov 16 '16 at 02:45
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    Leaving rigour aside, the simplest way I know to solving this is to complete the square in the exponent, i.e. write $e^{-\gamma \omega^2 + i\omega x} = e^{-\gamma(\omega - \frac{ix}{2\gamma})^2 - \frac{x^2}{4\gamma}}$ and then change variables $u = \omega - \frac{ix}{2\gamma}$ (this needs to be justified as $ix/2\gamma$ is complex so we really have a contour integral but the final result becomes the same anyway). In the end we can finish it up by using the well-known result $\int_{-\infty}^\infty e^{-at^2}{\rm d}t = \sqrt{\frac{\pi}{a}}$. – Winther Nov 16 '16 at 02:59
  • Finally found some related MSE questions (which has some simpler methods): http://math.stackexchange.com/questions/270566/how-to-calculate-the-fourier-transform-of-a-gaussian-function?rq=1 ; http://math.stackexchange.com/questions/1267007/inverse-fourier-transform-of-gaussian – Winther Nov 16 '16 at 03:01
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    @Winther Otherwise, once you have $\int_{-\infty}^\infty e^{2b x-x^2} dx = \sqrt{\pi}e^{b^2}$ for $b \in \mathbb{R}$, you can extend it to $b \in \mathbb{C}$ by analytic continuation or by the identity theorem – reuns Nov 16 '16 at 03:04

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