= Dirichlet modified Helmholtz Green function on an interval
{c}
{title2=$G(x,\xi)=-\sinh(\kappa x_<)\sinh(\kappa(L-x_>))/(\kappa\sinh(\kappa L))$}
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.
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