 Radial Helmholtz Dirichlet Green function

For $L=d^2/d{x}^2+(2/x)d/dx+k^2$ on $0<x<1$, impose finiteness at zero and zero value at one. Set $H=xG$; then $H^{\prime \prime }+k^{2}H=\xi \delta (x-\xi )$ 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 $k=0$.