 Landau contour

The velocity-integration contour implementing causal continuation of a kinetic response from the upper complex phase velocity half-plane. As a pole approaches the real axis, the contour passes below it. With denominator $v-u$, the real-axis limit is $\mathrm{PV}\int f_0^{\prime }(v)/(v-u)dv+i\pi f_0^{\prime }(u)$. Continuing the pole below the axis keeps this causal prescription rather than simply replacing the contour by an undeformed real-axis integral.