[[Infinite series]]
# Multipole expansion of $1/\Sr$

Let $\vab \SR = \vab r - \vab r'$ and let $\alpha$ be the angle between $\vab r$ and $\vab r'$,
so that $\vab r \cdot \vab r' = rr' \cos \alpha$.[^m]
Then we have the [[multipole expansion]]
$$
\begin{align*}
\frac{1}{\Sr} = \frac{1}{r} \sum_{n=0}^\infty \left( \frac{r'}{r} \right)^n P_{n}(\cos\alpha)
\end{align*}
$$ 
where $P_{n}(x)$ is a [[Legendre polynomial]]. #m/thm/anal/vec 

  [^m]: Presented here is the multipole expansion as it pertains to [[Electrodynamics MOC]].

> [!missing]- Proof
> #missing/proof

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