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![[tex] \mathcal{F}[f](\omega)=\int_{-\infty}^{\infty}f(x)e^{-i\omega x}dx, [/tex]](images/latex/f386777affc1fdfc2ff170bf8c29a7c7ad75cced_0.png "[tex] \mathcal{F}[f](\omega)=\int_{-\infty}^{\infty}f(x)e^{-i\omega x}dx, [/tex]")
![[tex] \frac{d}{d \omega} \mathcal{F}[e^{-x^{2}}]=\mathcal{F}[-ixe^{-x^2}][/tex]](images/latex/a15c8ac5fed99abaae0b3ac7e1dad7d55af99760_0.png "[tex] \frac{d}{d \omega} \mathcal{F}[e^{-x^{2}}]=\mathcal{F}[-ixe^{-x^2}][/tex]")
![[tex] i \omega \mathcal{F}[e^{-x^{2}}]=\mathcal{F}[(e^{-x^2})^\prime]=\mathcal{F}[-2xe^{-x^2}][/tex]](images/latex/c2b822b0ccf2da04bbafe2a890403f540537260f_0.png "[tex] i \omega \mathcal{F}[e^{-x^{2}}]=\mathcal{F}[(e^{-x^2})^\prime]=\mathcal{F}[-2xe^{-x^2}][/tex]")
![[tex] \frac{d}{d \omega} \mathcal{F}[e^{-x^{2}}]=-\frac{\omega}{2}\mathcal{F}[e^{-x^2}].[/tex]](images/latex/b8fa64232447b0218898063a88e6f243c84877a6_0.png "[tex] \frac{d}{d \omega} \mathcal{F}[e^{-x^{2}}]=-\frac{\omega}{2}\mathcal{F}[e^{-x^2}].[/tex]")
![[tex] \mathcal{F}[e^{-x^{2}}]=K e^{-\frac{\omega^2}{4}}. [/tex]](images/latex/3490cabde9266f4e768aed404840593c6299d325_0.png "[tex] \mathcal{F}[e^{-x^{2}}]=K e^{-\frac{\omega^2}{4}}. [/tex]")
![[tex] K= \mathcal{F}[e^{-x^{2}}](0)=\int_{-\infty}^{\infty} e^{-x^2}dx=\sqrt{\pi},[/tex]](images/latex/7cfcebcfaa32d2582681886403ebe0d2ea977a1f_0.png "[tex] K= \mathcal{F}[e^{-x^{2}}](0)=\int_{-\infty}^{\infty} e^{-x^2}dx=\sqrt{\pi},[/tex]")
![[tex] \mathcal{F}[e^{-x^{2}}]=\sqrt{\pi} e^{-\frac{\omega^2}{4}}. [/tex]](images/latex/b1d3020874b96a8281dedf8305980af7a9965f44_0.png "[tex] \mathcal{F}[e^{-x^{2}}]=\sqrt{\pi} e^{-\frac{\omega^2}{4}}. [/tex]")
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