Half-line Green function for the first cylindrical harmonic ($G(x,s)=-\frac{s}{2}\frac{\min (x,s)}{\max (x,s)}$)

For $Ly=y^{\prime \prime }+y^{\prime }/x-y/x^2$ on $x>0$, impose $y(0)=0$ by a continuous limit and $y(x)\to 0$ 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 $L_{x}G(x,s)=\delta (x-s)$ with ordinary source measure $ds$. For continuous $f$ with ${\int }_0^{\infty }(1+s^2)|f(s)|ds<\infty$,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 $(x{y}^{\prime }{)}^{\prime }-y/x$, explaining the weighted reciprocity $xG(x,s)=sG(s,x)$.