Q. Fourier Transformation of $f(t) = e^{-|t|}\cos(2t)$?
Why can't I say $f(t) = e^{-(-t)}\cos(2t)u(-t) + e^{-t}\cos(2t)u(t)$ and then use scaling property?
\begin{align*} \begin{array}{|c|c|c|} \hline \text{Scaling ($a$ real)} & f(at) & \frac{1}{|a|} F(\frac{\omega}{a}) \\ \hline \hspace{4em} \vdots \hspace{4em} & \hspace{6em} \vdots \hspace{6em} & \hspace{4em} \vdots \hspace{4em} \\ \hline \text{15} & e^{-at}\cos(\omega_0 t) u(t), \ a > 0 & \frac{a+j\omega}{(a+j\omega)^2+\omega_0^2} \\ \hline \end{array} \end{align*}
After doing that why doesn't it work when I use #15 and then the scaling property.
$$ F(\omega) = \frac{1}{1+(\omega-2)^2} + \frac{1}{1+(\omega+2)^2} $$
The answer is above and below is what I'm getting
