Sunyaev-Zel'dovich effect

The thermal Sunyaev-Zel’dovich effect is usuall derived for Compton-interactions of the CMB photons with a thermal gas of electrons - is the assumption of thermality necessary?

bonus: What’s happening if the electrons become relativistic?

extra bonus: Can you express \(\mathrm{i}^{+\mathrm{i}}\pm \mathrm{i}^{-\mathrm{i}}\) in terms of trigonometric functions?