Eigenfunction property for LTI sinusoidal and the sinusoidal steady-state response
It can be shown mathematically that the output of an LTI system when a complex exponential is applied is just a scaled version of the input, with no transient terms.


that's for an input that has been going on since forever ago (and will continue forever to the future):

if
$$ x(t) = e^{j \omega_0 t} \qquad \forall t \in \mathbb{R} $$

then

$$ y(t) = H(j \omega_0) e^{j \omega_0 t} = H(j \omega_0) x(t) $$


  What’s going on here?


problem is, sometimes we have to apply an input at some time we might call "$t=0$", and before the LTI settles down to the $y(t)$ shown above, there are transients.  but if the LTI is stable (and not "marginally stable"), the transients will go away eventually.