Laplace expansion (potential)

The Laplace expansion is in fact the expansion of the inverse distance between two points. Let the points have position vectors \textbf{r} and \textbf{r}' , then the Laplace expansion is \frac{1}{\|\mathbf{r}-\mathbf{r}'\|} = \sum_{\ell=0}^\infty \frac{4\pi}{2\ell+1} \sum_{m=-\ell}^{\ell} (-1)^m \frac{r_^\ell }{r_{\scriptscriptstyle>}^{\ell+1} } Y^{-m}_\ell(\theta, \varphi) Y^m_\ell(\theta', \varphi'). Here \textbf{r} has the spherical polar coordinates (r, \theta, \varphi) and \textbf{r}' has (r', \theta', \varphi') with homogeneous polynomials of degree \ell . Further r< is min(r, r′) and r> is max(r, r′). The function Y^m_\ell is a normalized spherical harmonic function. The expansion takes a simpler form when written in terms of solid harmonics, \frac{1}{\|\mathbf{r}-\mathbf{r}'\|} = \sum_{\ell=0}^\infty \sum_{m=-\ell}^\ell (-1)^m I^{-m}_\ell(\mathbf{r}) R^{m}_\ell(\mathbf{r}')\quad\text{with}\quad \|\mathbf{r}\| > \|\mathbf{r}'\|. ==Derivation==

The derivation of this expansion is simple. By the law of cosines, \frac{1}{\|\mathbf{r}-\mathbf{r}'\|} = \frac{1}{\sqrt{r^2 + (r')^2 - 2 r r' \cos\gamma}} = \frac{1}{r\sqrt{1 + h^2 - 2 h \cos\gamma}} \quad\hbox{with}\quad h := \frac{r'}{r} . We find here the generating function of the Legendre polynomials P_\ell(\cos\gamma): \frac{1}{\sqrt{1 + h^2 - 2 h \cos\gamma}} = \sum_{\ell=0}^\infty h^\ell P_\ell(\cos\gamma). Use of the spherical harmonic addition theorem P_{\ell}(\cos \gamma) = \frac{4\pi}{2\ell + 1} \sum_{m=-\ell}^\ell (-1)^m Y^{-m}_\ell(\theta, \varphi) Y^m_\ell (\theta', \varphi') gives the desired result. ==Neumann expansion==

A similar equation has been derived by Carl Gottfried Neumann that allows expression of 1/r in prolate spheroidal coordinates as a series: \frac{1} = \frac{4\pi}{a} \sum_{\ell=0}^\infty \sum_{m=-\ell}^\ell (-1)^m \frac{(\ell-|m|)!}{(\ell+|m|)!} \mathcal{P}_\ell^(\sigma_{}) Y_\ell^m(\arccos\tau,\varphi) Y_\ell^{m*}(\arccos\tau',\varphi') where \mathcal{P}_\ell^{m}(z) and \mathcal{Q}_\ell^{m}(z) are associated Legendre functions of the first and second kind, respectively, defined such that they are real for z\in(1, \infty). In analogy to the spherical coordinate case above, the relative sizes of the radial coordinates are important, as \sigma_{ and \sigma_{>}=\max(\sigma, \sigma'). ==References==