I am trying to calculate some Fourie-Transformation. In the answer, I do not understand why this split works. The abs t '|t|' confuses me.
$$= \int_{-\infty}^{+\infty} e^{-a|t|}e^{-j\omega t} \, dt$$
$$ = \int_{-\infty}^{0}e^{t(a - j \omega)} + \int_{0}^{\infty}e^{t(-a - j \omega)} $$
In my calculation, this would look more like this:
$$= \int_{-\infty}^{+\infty} e^{-a|t|}e^{-j\omega t} \, dt$$ $$ = \int_{-\infty}^{0}e^{-at}e^{-j\omega t} + \int_{0}^{\infty}e^{-at}e^{-j\omega t} $$
$$ = \int_{-\infty}^{0}e^{t(-a - j \omega)} + \int_{0}^{\infty}e^{t(-a - j \omega)} $$
Can someone help?
For $t \in (-\infty, 0]$ $$-\alpha |t| = \alpha t$$ because $|t| = -t$ there.