A direct method for modifying certain phase-integral approximations of arbitrary order

The approximate solution of the differential equation $\text{d}{}^2\text{ψ}\text{dz}{}^2\text{+ Q}{}^2\text{(z)ψ = 0}$ by a general modification of certain phase-integral approximations of arbitrary order is considered. A consistent modification of these higher-order phase-integral approximations is derived on the assumption that one has found a function Q_mod²(z) which makes the modified first-order approximation good at certain singular points of Q²(z), where the unmodified approximation would break down.