  Using the main results of the Kuramoto theory of globally coupled phase
oscillators combined with methods from probability and generalized function
theory in a geometric analysis, we extend Kuramoto's results and obtain a
mathematical description of the instantaneous frequency (phase-velocity)
distribution. Our result is validated against numerical simulations, and we
illustrate it in cases where the natural frequencies have normal and Beta
distributions. In both cases, we vary the coupling strength and compare
systematically the distribution of time-averaged frequencies (a known result of
Kuramoto theory) to that of instantaneous frequencies, focussing on their
qualitative differences near the synchronized frequency and in their tails. For
a class of natural frequency distributions with power-law tails, which includes
the Cauchy-Lorentz distribution, we analyze rare events by means of an
asymptotic formula obtained from a power series expansion of the instantaneous
frequency distribution.
