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A calculation of the real part of the refractive index near the fundamental absorption edge is given, which is based on a quantum mechanical calculation of the complex dielectric constant using the quantum density matrix method. An analytical expression is obtained in terms of experimentally known quantities for a given semiconductor and compared with available experimental data. The band structure of the Kane theory, which applies to direct gap III-V and II-VI compounds, is assumed. The expression obtained is a function of the band-gap energy, the effective electron and heavy hole masses at the bandedge, the spin orbit splitting energy, the carrier concentration for n-type or p-type materials, the temperature, and the frequency of the incident radiation. The temperature dependence occurs through the dependence of the bandgap energy and the effective mass on temperature for degenerate n-type or p-type materials, and there is an additional temperature-dependent factor for nondegenerate materials. The expression also involves the value of at the absorption edge which is not accessible to measurement. However, an equation for at the absorption edge can be found in terms of experimentally obtainable values of near the absorption edge and solved to give the desired quantity. This can then be used to predict the refractive index to a high degree of accuracy over the entire frequency spectrum up to the bandedge. Within limits of the above statement regarding at the absorption edge, there are no adjustable parameters involved, and this constitutes a significant improvement over previous theories of the refractive index of a semiconductor. In particular, the theory predicts the dispersion near the fundamental absorption edge which has been observed experimentally for a number of III-V and II-VI compounds and enables its precise calculations as a function of frequency. This fact is expected to be of considerable importance in technological applications involving integrated optics. Theory is compared with experimental results for a number of III-V and II-VI compounds.