= Dirichlet Green kernel for minus second derivative plus one
{c}
{title2=$G(x,s)=\sinh(1+\min(x,s))\sinh(1-\max(x,s))/\sinh2$}
For $L=-d^2/dx^2+1$ 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 $(\cosh1-1)/(2\cosh1)$, obtained by integrating $\sinh1\sinh(1-s)/\sinh2$ over $0\le s\le1$. The derivative-jump sign changes if the operator is written with the opposite overall sign.
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