Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 337 / 1 / vii / Solution

The Drude responsehas a pole at $\omega =-i/t_0$. Analyticity in the upper half plane therefore requiresFor real frequency,The sign condition from part ii then requiresso with a nonzero causal relaxation time the convention used in this question has ${\sigma }_0<0$. In conventions where the physical conductivity is defined with an additional minus sign, its static value is positive. The parameter $t_0$ is the relaxation time: after forcing is removed, the corresponding current or response decays as $e^{-t/t_0}$.

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