Past exam of the mathematics course of the University of Cambridge / 2006 / ii / Paper 4 / 31B (Asymptotic Methods)

Choose an analytic square-root branch of $f$ near infinity in a sector. The leading Liouville–Green approximation iswith successive inverse-power corrections; the two signs give an asymptotic basis in suitable sectors. Since $\sqrt{f}=\sqrt{f_0}+f_1/(2\sqrt{f_0}z)+\cdots$, the exponent contains a linear term and a logarithm. For the specified $f$, $\sqrt{f}=1+1/(2z)+7/(8z^2)+\cdots$, explaining $e^z{z}^{1/2}$ and $e^{-z}{z}^{-1/2}$ on the positive real axis.

To determine the requested coefficient from the differential equation itself, put $w_1=e^z{z}^{1/2}h(z)$. Substitution givesFor $h=1+a_1/z+\cdots$, the order-$z^{-2}$ coefficient is $-2a_1-9/4$. It must vanish, giving

Under the usual smoothness assumptions for WKB approximation, the strict positive lower bound on $g$ excludes turning point. Put $S(x)={\int }_0^x\sqrt{g(s)}ds$. The oscillatory approximation is $g(x{)}^{-1/4}[A\sin (\lambda S(x))+B\cos (\lambda S(x))]$. The zero boundary value at the origin forces $B=0$, and the one at the right end gives the quantization conditionHere $n$ is large and $C$ is an arbitrary normalization constant. Nonzero solutions exist only at the eigenvalues, not for every arbitrary large $\lambda$. The approximation concerns the leading phase and envelope.