  We develop an exact analytical formulation of neutrino oscillations in matter
within the framework of the Standard Neutrino Model assuming 3 Dirac Neutrinos.
Our Hamiltonian formulation, which includes CP violation, leads to expressions
for the partial oscillation probabilities that are linear combinations of
spherical Bessel functions in the eigenvalue differences. The coefficients of
these Bessel functions are polynomials in the neutrino CKM matrix elements, the
neutrino mass differences squared, the strength of the neutrino interaction
with matter, and the neutrino mass eigenvalues in matter. We give exact
closed-form expressions for all partial oscillation probabilities in terms of
these basic quantities. Adopting the Standard Neutrino Model, we then examine
how the exact expressions for the partial oscillation probabilities might
simplify by expanding in one of the small parameters {\alpha} and sin{\theta}13
of this model. We show explicitly that for small {\alpha} and sin{\theta}13
there are branch points in the analytic structure of the eigenvalues that lead
to singular behavior of expansions near the solar and atmospheric resonances.
We present numerical calculations that indicate how to use the small-parameter
expansions in practice.
