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Consider the Lagendre expansion of the potential at x due to a unit charge 1/|x-x'|, where x' is the position of the unit point charge.

To do the expansion, we need to consider a volume in space where the potential satisfies the Laplace equation. I can see that the potential satisfies Laplaces equation anywhere in space except at x=x', but what are the boundary conditions? Don't we have to know the potential at x=x', which is a boundary?