Extension of the WKB Equation

The WKB form suitable for the classical region of potential energy less than the total energy, $\psi \mathrm{}\left(x\right)={\left(\frac{\mathrm{dz}}{\mathrm{dx}}\right)}^{-\frac{1}{2}}sinz$, is used to obtain a function $z\left(x\right)\mathrm{}$ that reproduces the time-independent Schrödinger equation solutions $\psi \mathrm{}\left(x\right)\mathrm{}$, to any desired degree of accuracy through the turning point and into the nonclassical region.