For question 1, is omega_c > 0 the same as "there are no conditions set"?

- no, because it excludes $ \omega_c=0 $. The carrier frequency $ \omega_c=0 $ could be equal to zero, in which case, the signal would simply be multiplied by one. Of course, one can then recover the signal! -pm

For question 1b), should $ e^{-j(\omega_c+\theta_c)} $ be $ e^{-j(\omega_ct+\theta_c)} $ ?

- Yes, you are right. The correction has been made. -pm