Dielectric constant and the semiconductor energy gap the effect on '' (w) of an energy gap w

Dielectric constant and the semiconductor energy gap the effect on ε'' (w) of an energy gap w_g in a semiconductor may be approximated very roughly by substituting ½ δ(w – w_g) for δ (w) in the response function (16); that is, we take ε"(w) = (2πne²/mw) εδ (w – w_g). This is crude because it puts all the absorption at the gall frequency. The factor 1W enters as soon as we move the delta function away from the origin, because the integral in the sum rule of Problem 2 starts at the origin. Show that the real part of the dielectric constant on this model is ε'(w) = 1 + w²_p/(w²_g – w²), w²p ≡ 4πne²/m

It follows that the static dielectric constant is ε' (0) = 1 + w²_p/w²_g, widely used as a rule of thumb.