  Improved Wentzel-Kramers-Brillouin (WKB)-type approximations are presented in
order to study cosmological perturbations beyond the lowest order. Our methods
are based on functions which approximate the true perturbation modes over the
complete range of the independent (Langer) variable, from sub-horizon to
super-horizon scales, and include the region near the turning point. We employ
both a perturbative Green's function technique and an adiabatic (or
``semiclassical'') expansion (for a linear turning point) in order to compute
higher order corrections. Improved general expressions for the WKB scalar and
tensor power spectra are derived for both techniques. We test our methods on
the benchmark of power-law inflation, which allows comparison with exact
expressions for the perturbations, and find that the next-to-leading order
adiabatic expansion yields the amplitude of the power spectra with excellent
accuracy, whereas the next-to-leading order with the perturbative Green's
function method does not improve the leading order result significantly.
However, in more general cases, either or both methods may be useful.
