= Dirichlet half-line Green function for minus d2 plus 1
{title2=$G(x,\xi)=\sinh(x_<)e^{-x_>}$}
On the positive half-line, the <Green function> for $-d^2/dx^2+1$ with a zero <Dirichlet boundary condition> at the origin and decay at infinity is
$$
G(x,\xi)=\sinh(\min(x,\xi))e^{-\max(x,\xi)}=\frac12\left(e^{-|x-\xi|}-e^{-(x+\xi)}\right).
$$
It is continuous, has <derivative> jump $G_x(\xi+,\xi)-G_x(\xi-,\xi)=-1$, and is nonnegative. It is the negative of the <Dirichlet half-line Green function for d2 minus 1>; the sign of the leading differential term matters.
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