FT1

Find the Fourier transform of $f(t) = e^{-|t|}\,$

The Fourier transform is given by

$\hat{f}(\omega) = \int_R f(t) e^{-i \omega t}\,dt\,$

In this problem,

$\hat{f}(\omega) = \int_R e^{-|t|} e^{-i \omega t}\,dt\,$

$= \int_{-\infty}^0 e^{(1-i\omega)t}\,dt + \int_0^\infty e^{-(1+i\omega)t}\,dt\,$

$= \frac{1}{1+i\omega} + \frac{1}{1-i\omega} = \frac{2}{1+\omega^2}\,$