  The Differential Transfer Matrix Method is extended to the complex plane,
which allows dealing with singularities at turning points. The result for
real-valued systems are simplified and a pair of basis functions is found.
These bases are a bit less accurate than WKB solutions but much easier to work
with because of their algebraic form. Furthermore, these bases exactly satisfy
the initial conditions and may go over the turning points without the divergent
behavior of WKB solutions. The findings of this paper allow explicit evaluation
of eigenvalues of confined modes with high precision, as demonstrated by few
examples.
