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Possible Duplicate:
Fourier Transform of complicated product: $(1+x)^2 e^{-x^2/2}$

I calculate the Fourier Transform of $f(x)$ by $\mathbb{F}(t) =\int_{-\infty}^{\infty}e^{-x^2} \cdot e^{-ixt}dx,$ but my result is not equal to the Mathematica result. I tried to integrate by parts it, and next do an equal with the integral above.

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    That should have been $\mathbb{F}(0) = \sqrt{\pi}$ of course.2011-11-20

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The simplest way to do this is: $\int_{-\infty}^{\infty} e^{-(x^2+ixt)}dx=\int_{-\infty}^{\infty} e^{-(x+it/2)^2}e^{-t^2/4}dx=e^{-t^2/4} \sqrt{\pi}$ because $\int_{-\infty}^{\infty}e^{-(x+it/2)^2}dx=\int_{-\infty}^{\infty}e^{-(x+it/2)^2}d(x+it/2)=\int_{-\infty+it/2}^{\infty+it/2}e^{-z^2}dz=\sqrt{\pi}$.