  The fermion self-energy is calculated from the rainbow-ladder truncation of
the Dyson-Schwinger equation (DSE) in quantum electrodynamics (QED) for
spacelike momenta and in the complex momentum plane close to the timelike
region, both using Pauli-Villars regularization. Specifically, the DSE is
solved in the complex momentum plane by rotating either the energy component of
the four-momentum or the magnitude of Euclidean four-momentum to reach the
timelike region in Minkowski space. The coupling constant is appropriately
chosen to ensure the singularities of the fermion propagator are located in the
timelike region while producing significant differences from the perturbative
solutions. For simplicity, we choose Feynman gauge, but the method is
applicable in other covariant gauges as well. We demonstrate that the
approximate spectral representation based on the fermion self-energy near the
timelike region is consistent with the solution of the DSE directly in the
Euclidean space.
