= Singular-endpoint Green function with a double-zero right solution
{title2=$G(x,s)=-\min(x,s)\bigl(e^{\max(x,s)}-e\max(x,s)\bigr)$}
For $0<x<1$, consider the <self-adjoint differential operator> $L=(p\,d/dx)'+q$ with $p=e^{-x}/(1-x)$ and $q=-e^{-x}/(1-x)^2$. The homogeneous solutions $u=x$ and $v=e^x-ex$ satisfy the left and right zero <Dirichlet boundary conditions>. Their <Wronskian> is $(x-1)e^x$, and $pW=-1$. Thus the displayed <Dirichlet Green function> is continuous, zero at both endpoints, and has unit flux jump $p(s)[G_x(s^+,s)-G_x(s^-,s)]=1$.
The right solution has a double zero, $v(s)\sim e(1-s)^2/2$. Consequently $G(x,s)F(s)$ is integrable at $s=1$ for fixed interior $x$ when $F(s)=-e^{-s}/(1-s)^2$, even though $F$ itself is not integrable there. The endpoint zero of the <Wronskian> causes no loss of this construction because the weighted <Wronskian> $pW$ stays nonzero. This illustrates the need to retain the <integrating factor> when constructing a Green function at a singular endpoint.
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