Dirichlet Helmholtz Green function on an interval ($G(x,\xi )=-\sin (k{x}_{<})\sin (k(\ell -x_{>}))/(k\sin (k\ell ))$)

For ${\partial }_x^2+k^2$ on $(0,\ell )$ with zero endpoint values and $\sin (k\ell )\ne 0$, let $x_{<}=\min (x,\xi )$ and $x_{>}=\max (x,\xi )$. The displayed Green function is continuous at $x=\xi$ and its first derivative has jump one, giving unit Dirac delta forcing. The formula follows from the left solution $\sin (kx)$ and right solution $\sin (k(\ell -x))$, whose Wronskian is $-k\sin (k\ell )$. At $k=0$ its limit is $-x_{<}(\ell -x_{>})/\ell$, although zero is excluded by a condition that $k$ is not an integer in the length-$\pi$ version.