= Half-line Green function for the first cylindrical harmonic
{title2=$G(x,s)=-\frac s2\frac{\min(x,s)}{\max(x,s)}$}
For $Ly=y''+y'/x-y/x^2$ on $x>0$, impose $y(0)=0$ by a continuous limit and $y(x)\to0$ at infinity. The adapted homogeneous solutions are $x$ at zero and $1/x$ at infinity, with <Wronskian> $-2/x$. The displayed <Green function> is continuous and its first <derivative> has jump one at $x=s$, so $L_xG(x,s)=\delta(x-s)$ with ordinary source measure $ds$. For continuous $f$ with $\int_0^\infty(1+s^2)|f(s)|ds<\infty$,
$$
y(x)=\int_0^\infty G(x,s)f(s)ds=-\frac1{2x}\int_0^xs^2f(s)ds-\frac x2\int_x^\infty f(s)ds.
$$
This solves $Ly=f$ with the stated endpoint limits. No nonzero combination of $x,1/x$ meets both homogeneous endpoint conditions, proving <uniqueness>. Multiplying $L$ by $x$ gives the self-adjoint expression $(xy')'-y/x$, explaining the weighted reciprocity $xG(x,s)=sG(s,x)$.
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