Exact derivative interband dielectric function at Van Hove singularities

Interpretations of derivative spectra obtained by wavelength or thermal modulation have been based on universal lineshape functions that include only the resonant term, $\text{(ω}{}_{\mathrm{g}}\text{? ω ?iη)}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}$, obtained from differentiating ε(ω) with respect to ω or ω_g, and that neglect the (ω + iη)^?2 prefactor. We find the following differences between the approximate universal lineshape functions and the exact results obtained by including the nonresonant terms. dε/dω and dε₂/dω are no longer simply related to each other; two entirely different functions are obtained. Wavelength and temperature modulation no longer have identical lineshapes. Either dε₁/dω or dε₂/dω has a resonant derivative lineshape according to whether j is even or odd, respectively, in contrast to the approximate universal function, which has no zeros, dε₁/dω for a pure M₁ singularity has positive and negative peaks of approximately equal strength, which would agree with the experimental results without invoking hyperbolic excitons (admixtures of M_j and m_j+1 lineshapes). On the other hand, the nonexcitonic exact result fails to reproduce other experimental features. At the very least, our results call into question the extraction of numbers from detailed curve fits to the approximate functions.