  In this note we extend the Differential Transfer Matrix Method (DTMM) for a
second-order linear ordinary differential equation to the complex plane. This
is achieved by separation of real and imaginary parts, and then forming a
system of equations having a rank twice the size of the real-valued problem.
The method discussed in this paper also successfully removes the problem of
dealing with essential singularities, which was present in the earlier
formulations. Then we simplify the result for real-valued problems and obtain a
new set of basis functions, which may be used instead of the WKB solutions.
These basis functions not only satisfy the initial conditions perfectly, but
also, may approach the turning points without the divergent behavior, which is
observed in WKB solutions. Finally, an analytical transformation in the form of
a matrix exponential is presented for improving the accuracy of solutions.
