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the integral is over all of space. Note here is where we observe the field, where we are solving for. And integrating is equivalent to integrating all the "sources". In a sense that we integrate all the effects of sources at to get what's seen at (with the addition of point spread of source). A general exposition of the Green's function for Poisson's equation is given in the article on the screened Poisson equation. There are various methods for numerical solution, such as the relaxation method,

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