Fourier transform of Gaussian?

Question

For the Fourier transform defined as $$\frac {1}{\sqrt{2\pi}} \int_{-\infty}^\infty f(x) e^{-i\alpha x}\,dx$$ I know there is simple formula for the Fourier transformation and inverse transformation of $$f(x)=e^{-x^2/a}$$ but I can't remember it or derive it on my own. Does anyone have a formula for this?

Answer 1

Assume $a$ is a real number such that $a>0$. Set $$F(\alpha):=\frac {1}{\sqrt{2\pi}} \int_{-\infty}^{+\infty}f(x) e^{-i\alpha x}dx $$ with $\displaystyle f(x):=e^{-x^2/a}$. You may differentiate $F$ giving $$F'(\alpha)=-\frac {i}{\sqrt{2\pi}} \int_{-\infty}^{+\infty}x e^{-x^2/a}e^{-i\alpha x}dx $$ then integrate by parts $$ \begin{align} F'(\alpha)=\left.\frac {i}{2a\sqrt{2\pi}}e^{-i\alpha x}e^{-x^2/a}\right|_{-\infty}^{+\infty}-\frac {\alpha}{2a\sqrt{2\pi}} \int_{-\infty}^{+\infty} e^{-x^2/a}e^{-i\alpha x}dx =-\frac {\alpha}{2a}F(\alpha) \end{align}$$ observing that $\displaystyle F(0)=\frac{\sqrt{\pi }}{\sqrt{a}}$ you get $$ F(\alpha)=\frac{\sqrt{\pi }}{\sqrt{a}}e^{-\Large \frac{\alpha ^2}{4 a}}. $$