Exact expressions are found for the electromagnetic field arising from
the instant acceleration of a charged particle, its subsequent motion
with the velocity exceeding the light velocity in the medium and the
instant transition into the state of rest. It turns out that these exp
ressions have definite advantages over the usual ones grounded on the
use of the Fourier transform. In particular, they clearly show when an
d where the Cerenkov radiation should be observed in order to discrimi
nate it from the bremsstrahlung.