Dirichlet modified Helmholtz Green function on an interval ($G(x,\xi )=-\sinh (\kappa x_{<})\sinh (\kappa (L-x_{>}))/(\kappa \sinh (\kappa L))$)

For $\kappa >0$, the Green function of $D_x^2-{\kappa }^2$ on $(0,L)$ with zero Dirichlet boundary conditions is the displayed negative kernel. Here $x_{<}=\min (x,\xi )$ and $x_{>}=\max (x,\xi )$. The left and right homogeneous solutions are $\sinh (\kappa x)$ and $\sinh (\kappa (L-x))$, with Wronskian $-\kappa \sinh (\kappa L)$. The kernel is continuous and its first derivative jumps by $+1$, giving $(D_x^2-{\kappa }^2)G=\delta (x-\xi )$. Reversing the operator sign reverses the kernel sign.