Laplace Expansion (potential)

Derivation

The derivation of this expansion is simple. One writes

$\frac{1}{|\mathbf{r}-\mathbf{r}'|} = \frac{1}{\sqrt{r^2 + (r')^2 - 2 r r' \cos\gamma}} = \frac{1}{r_{{\scriptscriptstyle>}} \sqrt{1 + h^2 - 2 h \cos\gamma}} \quad\hbox{with}\quad h \equiv \frac{r_{{\scriptscriptstyle<}}}{r_{{\scriptscriptstyle>}}} .$

We find here the generating function of the Legendre polynomials :

$\frac{1}{\sqrt{1 + h^2 - 2 h \cos\gamma}} = \sum_{\ell=0}^\infty h^\ell P_\ell(\cos\gamma).$

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.

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Laplace Expansion (potential) - Derivation