The general WKB approximation formula states that $$ \int_a^b\sqrt{E_k-V(x)} = (k+1)\pi \text{ with } x \in [a,b] $$ for a regular Schrödinger equation (without the $\hbar$ and such). However, in the special case of a radial equation, there exists a Langer correction $$ \ell(\ell+1) \rightarrow \left(\ell+\frac{1}{2}\right)^2 $$ as explained for example at http://en.wikipedia.org/wiki/Langer_correction and in more detail at http://arxiv.org/abs/quant-ph/0205122 , but I can't seem to find an analogue easy explained formula as above in the special case of a radial Schrödinger equation.
Edit
The duplicate link explains why there is a shift, but I still don't know if there is a formula analogue to the WKB formula. Do we have to change the right hand side? Does applying the shift makes the formula work? Is there need of a transformation?
