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Does anyone know where to find a good resource for solving for the Green's Function of the Laplacian in 3D or tips on where to start? $$\nabla^2G(\boldsymbol{x,x_0})=\delta (\boldsymbol{x,x_0})$$

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Any book on mathematical methods should have that. I used Arfken's book that is reasonably good. You may want to check chapter 9.7.

You can also solve it by hand by fourier transform :).

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