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## Homework Statement

This comes up in the context of Poisson's equation

Solve for ##\mathbf{x} \in \mathbb{R}^n ## $$ \nabla^2 G(\mathbf{x}) = \delta(\mathbf{x})$$

## Homework Equations

$$\int_0^\pi \sin\theta e^{ikr \cos\theta}\mathop{dk} = \int_{-1}^1 e^{ikr \cos\theta}\mathop{d\cos \theta

}$$

## The Attempt at a Solution

Attempt by using Fourier Transforms

$$ \nabla^2 G(\mathbf{x}) = \delta(\mathbf{x}) \Rightarrow -\lVert k\rVert^2 \tilde{G}(\mathbf{k}) = 1$$

$$ G(\mathbf{x}) = -\mathcal{F}^{-1}\left[ \lVert k\rVert^{-2} \right]$$

I reasoned that we could use N-dimensional spherical coordinates and be left with an integral integral over one angle in the plane between ##\mathbf{k}## and ##\mathbf{x}## and a radial integral which would have an element ##k^{n-1}\,dk ## where ##n## is the number of dimensions.

$$G\left(\mathbf{x}\right) = \frac{-1}{(2\pi)^n}\int k^{n-1}F(\phi_1,\dots,\phi_{n-3})\sin(\phi_{n-2})\mathop{dk d\phi_1 \dots d\phi_{n-2}d\phi_{n-1}} \frac{e^{ikr\cos \phi_{n-2}}}{k^2} $$

with ##u = \cos \phi_{n-2}##

$$G\left(\mathbf{x}\right) = \frac{-1}{(2\pi)^n}\int F(\phi_1,\dots,\phi_{n-3})\mathop{d\phi_1 \dots d\phi_{n-3}d\phi_{n-1} }\int_{-1}^{1} du\int_0^\infty\mathop{dk} k^{n-3} e^{ikru}$$

Maybe I'm not thinking clearly but the last two integrals don't seem like they are going to converge to anything nice. I don't know if I've made a mistake or I need to take these integrals in a particular order. I know I can show the Green's function is proportional to ##|| r||^\alpha ## for ##n>2## by using a test function, but the the idea here is to calculate it directly without assuming that much about its form.

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