New basis functions for wave equation

The Di erential Transfer Matrix Method is extended to the complex plane, which allows dealing with singularities at turning points. The results for real-valued systems are simpli ed and a pair of basis functions are 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 ndings of this paper allow explicit evaluation of eigenvalues of con ned modes with high precision, as demonstrated by few examples.