I'm attempting to find a Green's function for the Laplacian. I was looking at examples for finding the Green's function for the 2D Laplacian on the unit disk and the upper half plane. In both examples when finding the Green's function they find an image point $x_0^*$ that is outside the domain of definition for the Laplacian and use it to find the Green's function.
My question is why must this image point be found for finding the Laplacian and why can't it be taken as a point inside the domain of definition? Thanks.