I know that the idea of the Fourier Transform is to break a function into a sum of trigonometric functions. Consider the following function: $$ f_{\alpha}(t) = e^{-\alpha|t|}$$
The Fourier Transform of this is $$\tilde{f}_{\alpha}(\omega) = \frac{1}{2 \pi} \int_{-\infty}^{\infty} e^{-i \omega t} e^{-\alpha |t|} \ dt $$
$$ = \frac{\alpha}{\pi(\alpha^{2}+\omega^{2})}$$
What precisely does this mean? How does did relate to the question of breaking a function into a sum of trigonometric functions?