  We compute the complete O(alpha^2) QED corrections to the electron energy
spectrum in unpolarized muon decay, including the full dependence on the
electron mass. Our calculation reduces the theoretical uncertainty on the
electron energy spectrum well below 10^{-4}, the precision anticipated by the
TWIST experiment at TRIUMF, which is currently performing this measurement. For
this calculation, we extend techniques we have recently developed for
performing next-to-next-to-leading order computations to handle the decay
spectra of massive particles. Such an extension enables further applications to
precision predictions for b, t, and Higgs differential decay rates.
