Augmenting the residue theorem with boundary terms in finite-density calculations

At zero temperature and finite chemical potential, $d$-dimensional loop integrals with complex-valued integrands in the imaginary-time formalism yield results dependent on the integration order. We observe this even with the simplest one-loop dimensionally regularized integrals. Computing such integrals by evaluating the spatial $\mathrm{d}^{d} p$ integral before the temporal $\mathrm{d} p_0$ integral yields results consistent with those obtained at small but nonvanishing temperatures. Computing the temporal integral first by applying the residue theorem to the integrand yields a different answer. The same holds for general complexified propagators. In this work we aim to understand the theoretical background behind this difference, in order to fully enable the powerful techniques of residue calculus in applications. We cast the difference into the form of a derivative term related to Dirac deltas, and further demonstrate how the difference originates from the zero-temperature limit of the Fermi-Dirac occupation functions treated as complex-valued functions. We also discuss a generalization to propagators raised to non-integer powers.