The Fourier transform F(k) of a function f(x) is defined as \(F(k) = \int_{ - \infty }^\infty {dx\,\,f\left( x \right)\,\exp \,\left( {ikx} \right)}\). Then, F(k)
for
f(x) = exp (-x2) is
[Given \(\int_{ - \infty }^\infty {\exp \,( - {x^2})dx = \sqrt \pi }\)]
1
π exp (-k)
2
\(\sqrt{\pi} \,\,exp \left(\frac{-k^2}{4}\right)\)
3
\(\frac{\sqrt{\pi}}{2} \,\,exp \left(\frac{-k^2}{2}\right)\)
4
\(\sqrt{2\pi}\,\,exp\,\,(-k^2)\)