Bessel inverse-power expansion at infinity ($w\sim e^{\pm iz}{\sum }_{k\ge 0}{c}_k{z}^{-k}$)

For $w^{\prime \prime }+[1-A/z^2]w=0$, with $A=n^2-1/4$, substitution gives $s=\pm i$, exponent zero and $c_{k+1}=[k(k+1)-A]c_k/[2s(k+1)]$. Thus the first correction is $c_1=sA{c}_0/2$. For integer $n$ no coefficient truncation occurs and coefficient ratios grow as $k/(2s)$; this is an asymptotic expansion, not a convergent series at a finite $z$. The leading Liouville–Green approximation $q^{-1/4}{e}^{\pm i\int \sqrt{q}}$ has the same $1/z$ correction.