Confused about the Fourier Transform of $e^{at}u(t)$ This is the problem at hand:

I'm unaware of why we didn't have to say anything about $\omega$ like that it should be also greater than $0$, I know it's variable...but it's multiplied by $t$ whatsoever.
Also, if $a$ is complex, I don't understand why do we require that $Re \{a\} &gt;0$ only, without mentioning anything about the imaginary part (it's mentioned further down in the book).
Finally, why isn't it $Re\{a\}&gt;=0$? is it because that may reduce to the Heaviside step function if the imaginary part of $a$ is also 0?
Thank you.
$\omega$'s value does nothing for the convergence of the integral. $a$'s does.

$a$ is not complex, it's real only.

If $a$ is 0, the integral doesn't converge.