Based on resonant-convergent-and thereby complex-electric-dipole polarizabilities, formulae for differential cross sections in resonant Raman spectroscopy are presented. In absorptive regions of the spectrum, there are phase delays in the induced polarization that enter the expressions for the cross sections. The theory is exemplified by calculations employing the adiabatic time-dependent density functional theory approximation, with applications on the low-lying, dipole-allowed, electronic states of a sample of molecules, including hydrogen peroxide, pyridine, and trinitrotoluene. Results obtained with the Coulomb attenuated B3LYP exchange-correlation functional are found to be more accurate than those obtained with the conventional B3LYP functional-an observation that holds not only for trinitrotoluene with its transitions of charge-transfer character but also for the other cases. The qualitative features of the resonant Raman spectra for a given molecule vary strongly from one resonance wavelength to another, which is a fact that could further facilitate the use of this spectroscopy in applications of stand-off detection of gaseous samples in ultra-low concentrations.