Newtonian potential

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The Newtonian potential is an operator in vector calculus that acts as the inverse to the negative Laplacian, on functions that are smooth and decay rapidly enough at infinity. In its general nature, it is a singular integral operator, defined by convolution with a function having a mathematical singularity at the origin, the Newtonian kernel G.

Stating its property in another way, the Newtonian potential applied to a function f satisfies Poisson's equation with f as RHS. If we denote the Newtonian potential of a function f by

\mathcal{G} * f

(where the star denotes the convolution of f and G) then this statement means

\mathcal{G} * (-\nabla^2 f) = f.

The Newtonian kernel in d dimensions is defined by

\mathcal{G}(x) = \left\{ \begin{matrix} c_1 \left| x \right| & : & d=1 \\ c_2 \log{ \left\| x \right\| } & : & d=2 \\ c_d \left\| x \right\| ^{2-d} & : & d>2 \end{matrix} \right.

Here cd denotes a normalization constant which depends on the dimension d.

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