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I am working on finding a solution to Poisson equation through Green's function in both 2D and 3D. For the equation: [tex]\nabla^2 D = f[/tex], in 3D the solution is:

[tex]D(\mathbf x) = \frac{1}{4\pi} \int_V \frac{f(\mathbf x')}{|\mathbf x - \mathbf x'|} d\mathbf{x}'[/tex], and in 2D the solution is:

[tex]D(\mathbf x) = \frac{1}{2\pi}\int_V \log(|\mathbf x - \mathbf x'|) f(\mathbf x') d\mathbf{x}'[/tex].

Now, my question is that where these solutions hold true only for infinite domains?

If I have a small rectangular domain, can I still use these equations to solve the Poisson's equationwithoutany boundary conditions ?

Can someone help me with this, or point me to a reference which I should read ?

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# Green's function for Poisson Equation

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