= Radial modified Helmholtz Green function on the half-line
{title2=$G(x,\xi)=\frac\xi x\sinh(\min(x,\xi))e^{-\max(x,\xi)}$}
For $Ly=-x^{-2}(x^2y')'+y$ on $x>0$, require boundedness at zero and at infinity. The substitution $z=xy$ reduces the homogeneous equation to $-z''+z=0$. The adapted solutions are $\sinh x/x$ at zero and $e^{-x}/x$ at infinity, with <Wronskian> $-x^{-2}$. The displayed <Green function> is continuous and has first-derivative jump $-1$, so $L_xG=\delta(x-\xi)$ with ordinary source measure $d\xi$. It satisfies weighted reciprocity $x^2G(x,\xi)=\xi^2G(\xi,x)$. For forcing equal to one on $[0,R]$ and zero outside, integration gives $y=1-(R+1)e^{-R}\sinh x/x$ inside and $y=(R\cosh R-\sinh R)e^{-x}/x$ outside. Limits define the regular value at zero.
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