= Radial Helmholtz Dirichlet Green function
{title2=$G(x,\xi)=-\xi\sin(kx_<)\sin(k(1-x_>))/(kx\sin k)$}
For $L=d^2/dx^2+(2/x)d/dx+k^2$ on $0<x<1$, impose finiteness at zero and zero value at one. Set $H=xG$; then $H''+k^2H=\xi\delta(x-\xi)$ with ordinary zero-endpoint conditions. The displayed <Green function> follows from the left/right sine solutions, where $x_<=\min(x,\xi)$ and $x_>=\max(x,\xi)$. Its derivative jumps by one. It exists for $k\notin\pi\mathbb Z\setminus\{0\}$; at zero take the removable limit $G=-\xi x_<(1-x_>)/x$. At nonzero integer multiples of pi the radial problem has a nonzero homogeneous solution and a <Fredholm solvability condition>. Integrating against the constant forcing gives $y=k^{-2}[1-\sin(kx)/(x\sin k)]$, with $y=(x^2-1)/6$ at $k=0$.
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