  Analytic formulas are presented for three flavor neutrino oscillations in
matter in the plane wave approximation. We calculate in particular the time
evolution operator in both mass and flavor bases. We also find the transition
probabilities expressed as functions of the vacuum mass squared differences,
the vacuum mixing angles, and the matter density parameter. The application of
this to neutrino oscillations for both atmospheric and long baseline neutrinos
in a mantle-core-mantle step function model of the Earth's matter density
profile is discussed.
