Infinite series

Multipole expansion of 1/𝔯

Let ⃗𝖗 =⃗𝐫 −⃗𝐫′ and let 𝛼 be the angle between ⃗𝐫 and ⃗𝐫′, so that ⃗𝐫 ⋅⃗𝐫′ =𝑟𝑟′cos⁡𝛼.¹ Then we have the multipole expansion

1𝔯=1𝑟∞∑𝑛=0(𝑟′𝑟)𝑛𝑃𝑛(cos⁡𝛼)

where 𝑃𝑛(𝑥) is a Legendre polynomial. vec

Proof

proof

---

develop | en | SemBr

Footnotes

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