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Green function of the Laplacian ( $G (x, x^{'})$ )

In three dimensions,

- \nabla^{2} \frac{1}{4 π ∣ x - x ^{'} ∣} = δ^{(3)} (x - x^{'}) .

Consequently the decaying solution of

- \nabla^{2} u = f

in free space is

u (x) = \frac{1}{4 π} \int_{R^{3}} \frac{f ( x ^{'} )}{∣ x - x ^{'} ∣} d^{3} x^{'} .

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Solution

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Laplace's equation