  When considering NLO corrections to thermal particle production in the
"relativistic" regime, in which the invariant mass squared of the produced
particle is K^2 ~ (pi T)^2, then the production rate can be expressed as a sum
of a few universal "master" spectral functions. Taking the most complicated
2-loop master as an example, a general strategy for obtaining a convergent
2-dimensional integral representation is suggested. The analysis applies both
to bosonic and fermionic statistics, and shows that for this master the
non-relativistic approximation is only accurate for K^2 > (8 pi T)^2, whereas
the zero-momentum approximation works surprisingly well. Once the simpler
masters have been similarly resolved, NLO results for quantities such as the
right-handed neutrino production rate from a Standard Model plasma or the
dilepton production rate from a QCD plasma can be assembled for K^2 ~ (pi T)^2.
