= Neumann Green function for the modified Helmholtz equation on an interval
{c}
{title2=$G(x,\xi)=-\frac{\cosh(\kappa\min(x,\xi))\cosh(\kappa[1-\max(x,\xi)])}{\kappa\sinh\kappa}$}
For real $\kappa\ne0$, this <Green function> inverts $d^2/dx^2-\kappa^2$ on $[0,1]$ with homogeneous <Neumann boundary conditions>. It is continuous and its first <derivative> has jump one at $x=\xi$. At $\kappa=0$, a Green function with the same unit <Dirac delta> source and zero boundary derivatives does not exist: integrating its defining equation would give zero equal to one. The equation $y''=f$ with these boundary conditions instead requires $\int_0^1f(x)\,dx=0$ and determines $y$ only up to a constant.
Back to article page
