I'm want to understand Green's functions, but not just the definition. In particular, consider the equation $-ΔG_j(x)=δ(x-x_j)$ for $x\in Ω\subset\mathbb{C}$ with boundary condition $G_j(x)=0$ for $x\in\partial Ω$ for some fixed point $x_j$.
I know the limit of $G_j$ while $x_j\to\infty$ should exist (possibly under some additional assumptions) and this is what we call Green's function of $\partial Ω$ with a pole at infinity. (Please, correct me if I'm wrong.)
Can anybody elaborate over this or suggest a good book/paper for reference?
Thank you.