Nanometer-scale mapping of complex optical constants by scattering-type near-field microscopy has been suffering from quantitative discrepancies between the theory and experiments. To resolve this problem, a novel analytical model is presented here. The comparison with experimental data demonstrates that the model quantitatively reproduces approach curves on a Au surface and yields an unprecedented agreement with amplitude and phase spectra recorded on a phonon-polariton resonant SiC sample. The simple closed-form solution derived here should enable the determination of the local complex dielectric function on an unknown sample, thereby identifying its nanoscale chemical composition, crystal structure and conductivity.