Infrared singularities in can be avoided by including at least one massless source which couples to the photon. Once that is arranged it is possible to obtain non-zero mass solutions for the source Green functions in scalar and (doubled) spinor electrodynamics, by using the Gauge Technique. We find the asymptotic results for the source spectral $\rho$ functions to be typically, $\rho(p) \simeq (p^2 - m^2)^{-1-e^2/c\pi^2}$ as $p \rightarrow m$, and $\rho(p) \simeq p^{-2} \exp(c'e^2/\pi p)$ as $p \rightarrow \infty$. The constants $c$ and $c'$ are respectively related to the number of charged massless particles and their ``spin'' character.