91Ó°ÊÓ

The given function is$f\left(x\right)=e^{−x^2/\left(2{\mathrm{σ}}^2\right)}$.

The Fourier transform is a mathematical technique for expressing a function as the summation of sines and cosines functions.

The fourier transform is given below.

$g\left(\mathrm{Î\pm }\right)=\frac{1}{2\mathrm{Ï€}}\underset{−\mathrm{∞}}{\overset{\mathrm{∞}}{∫}}{e}^{−x^2/\left(2{\mathrm{σ}}^2\right)}{e}^{−i\mathrm{Î\pm }x}dx$

Put $u=\frac{x}{\mathrm{σ}\sqrt{2}}$

Differentiate with respect to x.

$du=\frac{dx}{\mathrm{σ}\sqrt{2}}$

$\begin{array}{l}g\left(\mathrm{Î\pm }\right)=\frac{1}{2\mathrm{Ï€}}\underset{−\mathrm{∞}}{\overset{\mathrm{∞}}{∫}}{e}^{−u^2−i\mathrm{Î\pm }\sqrt{2}\mathrm{σ}u}\sqrt{2}\mathrm{σ}du\\ g\left(\mathrm{Î\pm }\right)=\frac{\mathrm{σ}}{\mathrm{Ï€}\sqrt{2}}\underset{−\mathrm{∞}}{\overset{\mathrm{∞}}{∫}}{e}^{−u^2−i\mathrm{Î\pm }\sqrt{2}\mathrm{σ}u}du\\ g\left(\mathrm{Î\pm }\right)=\frac{\mathrm{σ}}{\mathrm{Ï€}\sqrt{2}}\underset{−\mathrm{∞}}{\overset{\mathrm{∞}}{∫}}{e}^{−{(u+i\mathrm{Î\pm }\mathrm{σ}/\left(\sqrt{2}\right))}^2−{\mathrm{σ}}^2{\mathrm{Î\pm }}^2/2}du\end{array}$

Simplify further

$g\left(\mathrm{Î\pm }\right)=\frac{\mathrm{σ}}{\mathrm{Ï€}\sqrt{2}}{e}^{−{\mathrm{σ}}^2{\mathrm{Î\pm }}^2/2}\underset{−\mathrm{∞}}{\overset{\mathrm{∞}}{∫}}{e}^{−{(u+i\mathrm{Î\pm }\mathrm{σ}/\left(\sqrt{2}\right))}^2}d(u+i\mathrm{σ}\mathrm{Î\pm }/\sqrt{2})$

Put$u+i\mathrm{σ}\mathrm{Î\pm }/\sqrt{2}=\mathrm{η}$

$g\left(\mathrm{Î\pm }\right)=\frac{\mathrm{σ}}{\mathrm{Ï€}\sqrt{2}}{e}^{−{\mathrm{σ}}^2{\mathrm{Î\pm }}^2/2}\underset{−\mathrm{∞}}{\overset{\mathrm{∞}}{∫}}{e}^{−{\mathrm{η}}^2}d\mathrm{η}$

But the integral $\underset{−\mathrm{∞}}{\overset{\mathrm{∞}}{∫}}{e}^{−{\mathrm{η}}^2}d\mathrm{η}$is Euler-poisson integral and its value is$\sqrt{\mathrm{π}}$.

$g\left(\mathrm{Î\pm }\right)=\frac{\mathrm{σ}}{\sqrt{2\mathrm{Ï€}}}{e}^{−{\mathrm{σ}}^2{\mathrm{Î\pm }}^2/2}$

The fourier transform of $f\left(x\right)=e^{−x^2/\left(2{\mathrm{σ}}^2\right)}$is$g\left(\mathrm{Î\pm }\right)=\frac{\mathrm{σ}}{\sqrt{2\mathrm{Ï€}}}{e}^{−{\mathrm{σ}}^2{\mathrm{Î\pm }}^2/2}$.