Fourier transform of a rapidly decreasing function

Let

$$\widehat{f}(\xi)=\int_{-\infty}^{+\infty}f(x)e^{-ix\xi}dx$$

be the Fourier transform of a function in Lesbegue space $ \mathcal{L}^1(R) $
For function $ f(x)=e^{-x^2} $ we have

$$\widehat{f}(\xi)=\int_{-\infty}^{+\infty}e^{-x^2}e^{-ix\xi}dx=e^{-\frac{\xi^2}{4}}\int_{-\infty}^{+\infty}e^{-x^2-ix\xi-\frac{i^2\xi^2}{4}}dx<br />
=e^{-\frac{\xi^2}{4}}\int_{-\infty}^{+\infty}e^{-(x+i\frac{\xi}{2})^2}dx=\sqrt{\pi}e^{-\frac{\xi^2}{4}}$$
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