  Within a conventional Hamiltonian description, we find accurate closed-form
expressions for the oscillation probabilities of three coupled neutrinos
propagating in matter. Subtle cancelations that occur in coefficients of our
formulation are avoided for all transitions $\nu_a \to \nu_b$ by transforming
to a different set of coefficients presented in an appendix of this paper. The
neutrino mass eigenvalues are easily obtained numerically as the solution of a
cubic equation. Our methods are illustrated for flavor-changing transitions in
the $(\nu_e,\nu_\mu)$ sector. The resulting analytic expressions oscillation
probabilities, which are particularly simple, are also accurate to a few
percent over all regions of interest at present and the envisioned future
neutrino facilities. While somewhat less accurate than numerical simulations,
our approximate expressions are sufficiently accuracy to obviate the need for
exact computer simulations in many circumstances.
