FIG. 13.

Analytical structure of a causal (retarded) response function on the complex plane of ω. When calculating the inverse Fourier transform such as Eq. (A5), the original integration contour over the real axis can be closed in the infinite semicircle with Im ω > 0 (light blue dashed line) when t < 0, according to the Jordan’s lemma. Causality then requires that no singularities are present in the upper half of the complex plane, in which case the integration contour can be shrunk to a point. For t > 0, the contour can be closed where Im ω < 0 (light red dashed line). This contour can be shrunk to encircle the singularities of the transformed function (red solid lines).