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- Thread starter med17k
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[tex]\int_{\mathrm{S}_1} \mathrm{d} \Omega \mathrm{Y}_{lm}^*(\vartheta,\varphi)\mathrm{Y}_{l'm'}(\vartheta,\varphi)=\int_0^{\pi} \mathrm{d} \vartheta \int_0^{2 \pi} \mathrm{d} \varphi \sin \vartheta \mathrm{Y}_{lm}^*(\vartheta,\varphi)\mathrm{Y}_{l'm'}(\vartheta,\varphi)=\delta_{ll'} \delta_{mm'}.[/tex]

Any harmonic function on [itex]\mathbb{R}^3[/itex] then can be expanded in the sense of convergence with respect to the corresponding Hilbert-space norm by

[tex]f(\vec{x})=\sum_{l=0}^{\infty} \sum_{m=-l}^{l} \left [f_{1lm}(r) r^l + f_{2lm} \frac{1}{r^{l+1}} \right ]\mathrm{Y}_{lm}.[/tex]

In arbitrary dimensions the analogues of the spherical harmonics are known as Gegenbauer functions.

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Ben Niehoff

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[tex]G(\vec r, \vec r') = \frac{1}{\lvert \vec r - \vec r' \rvert}[/tex]

and in n dimensions, it is easy to show that the Green function is

[tex]G(\vec r, \vec r') = \frac{1}{\lvert \vec r - \vec r' \rvert^{n-2}}[/tex]

To show this, write the Laplace equation in spherical coordinates, and assume your solution is a function of r alone. Then the Green function follows by translational invariance.

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