Radial modified Helmholtz Green function on the half-line ($G(x,\xi )=\frac{\xi }{x}\sinh (\min (x,\xi ))e^{-\max (x,\xi )}$)

For $Ly=-x^{-2}(x^2{y}^{\prime }{)}^{\prime }+y$ on $x>0$, require boundedness at zero and at infinity. The substitution $z=xy$ reduces the homogeneous equation to $-z^{\prime \prime }+z=0$. 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 $L_{x}G=\delta (x-\xi )$ with ordinary source measure $d\xi$. It satisfies weighted reciprocity $x^{2}G(x,\xi )={\xi }^{2}G(\xi ,x)$. 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.