Dirichlet Green kernel for minus second derivative plus one ($G(x,s)=\sinh (1+\min (x,s))\sinh (1-\max (x,s))/\sinh 2$)

For $L=-d^2/d{x}^2+1$ on $[-1,1]$ with zero endpoint values, the displayed Green kernel is positive and symmetric. Each side solves the homogeneous equation; the kernel is continuous at $x=s$ and its first derivative has jump $-1$. Hence $LG(\cdot ,s)={\delta }_s$. Integrating against a forcing function solves $Ly=f$. For the indicator of the right half-interval, the value at zero is $(\cosh 1-1)/(2\cosh 1)$, obtained by integrating $\sinh 1\sinh (1-s)/\sinh 2$ over $0\le s\le 1$. The derivative-jump sign changes if the operator is written with the opposite overall sign.