 260
 2
 Problem Statement

Find 1D Green function for charged layer with distribution:
\begin{equation}
\phi=
\begin{cases}
0, x<a \\
\rho_0, a<x<0 \\
\rho_0, 0<x<a \\
0, x>a
\end{cases}
\end{equation}
With boundary condition ##\phi(x \to \infty) = 0##.
 Relevant Equations
 ##\nabla^2\phi = 4\pi\rho## and ##\nabla^2G(x,x') = 4\pi\delta(xx')##
I came across an example of a solution to finding the potential of a charged layer using the Green function (here, pdf). The standard algorithm for finding the Green function by boundary conditions for many problems is understandable:
\begin{align*}
G_\mathrm{Left} = Ax+ B \\
G_\mathrm{Right} = Cx + D \\
G_\mathrm{Right}'  G_\mathrm{Left}' = 4\pi
\end{align*}
since the Green function is linear in 1D problems, then using boundary conditions we find constants that are functions of primed coordinates:
But in this case I cannot understand how to find the Green function (how to determine constants) by the boundary condition ##\phi(x \to \infty) = 0##, which, as indicated in the link, is equal ##G(x,x') = 4\pi x_{<}##. After all, the linear function does not disappear when striving to infinity. So, I need help.
\begin{align*}
G_\mathrm{Left} = Ax+ B \\
G_\mathrm{Right} = Cx + D \\
G_\mathrm{Right}'  G_\mathrm{Left}' = 4\pi
\end{align*}
since the Green function is linear in 1D problems, then using boundary conditions we find constants that are functions of primed coordinates:
But in this case I cannot understand how to find the Green function (how to determine constants) by the boundary condition ##\phi(x \to \infty) = 0##, which, as indicated in the link, is equal ##G(x,x') = 4\pi x_{<}##. After all, the linear function does not disappear when striving to infinity. So, I need help.