Some properties of WKB series

We investigate some properties of the WKB series for arbitrary analytic potentials and then specifically for potentials x(N) (N even), where more explicit formulae for the WKB terms are derived. Our main new results are: (i) we find the explicit functional form for the general WKB terms sigma(k)', where one has only to solve a general recursion relation for the rational coefficients; (ii) we give a systematic algorithm for an essential simplification of the integrated WKB terms phi sigma(k)' dx that enter the energy eigenvalue equation; and (iii) we derive almost explicit formulae for the WKB terms for the energy eigenvalues of the homogeneous power law potentials V(x) = x(N), where N is even.