= Bessel inverse-power expansion at infinity
{c}
{title2=$w\sim e^{\pm iz}\sum_{k\geq0}c_kz^{-k}$}
= Bessel inverse-power expansions at infinity
{c}
{synonym}
For $w''+[1-A/z^2]w=0$, 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 $c_1=sAc_0/2$. 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.
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