Multipole expansion of $1 / 𝔯$

Let $⃗ 𝖗 = ⃗ 𝐫 - ⃗ 𝐫'$ and let $𝛼$ be the angle between $⃗ 𝐫$ and $⃗ 𝐫'$ , so that $⃗ 𝐫 \cdot ⃗ 𝐫' = 𝑟 𝑟' c o s 𝛼$ .¹ Then we have the multipole expansion

1 𝔯 = 1 𝑟 \infty \sum 𝑛 = 0 (𝑟' 𝑟) 𝑛 𝑃 𝑛 (c o s 𝛼)

where $𝑃 𝑛 (𝑥)$ is a Legendre polynomial. #m/thm/anal/vec

Proof

#missing/proof

#state/develop | #lang/en | #SemBr

Presented here is the multipole expansion as it pertains to Electrodynamics MOC. ↩

Multipole expansion of 1/𝔯

Footnotes

Multipole expansion of $1 / 𝔯$