Associated Legendre function

To every Legendre polynomial $𝑃 ℓ$ exist a number of associated Legendre functions $𝑃 𝑚 ℓ$ defined (by one convention¹) for $𝑚 \geq 0$ as #m/def/fun

𝑃 𝑚 ℓ (𝑥) = (- 1) 𝑚 (1 - 𝑥 2) 𝑚 / 2 (𝑑 𝑑 𝑥) 𝑚 𝑃 ℓ (𝑥)

and

𝑃 - 𝑚 ℓ (𝑥) = (- 1) 𝑚 (ℓ - 𝑚)! (ℓ + 𝑚)! 𝑃 𝑚 ℓ (𝑥)

where naturally $𝑃 𝑚 ℓ (𝑥) = 0$ for $| 𝑚 | > ℓ$ .

Mathematica

The Associated Legendre polynomial $𝑃 𝑚 ℓ (𝑥)$ may be generated in Wolfram Mathematica with LegendreP[ℓ, m, x].

#state/tidy | #lang/en | #SemBr

2018. Introduction to quantum mechanics, §4.1, p. 135 ↩

Associated Legendre function

Footnotes