  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.
