Special functions MOC

Associated Legendre function

To every Legendre polynomial 𝑃ℓ exist a number of associated Legendre functions π‘ƒπ‘šβ„“ defined (by one convention1) for π‘š β‰₯0 as fun

π‘ƒπ‘šβ„“(π‘₯)=(βˆ’1)π‘š(1βˆ’π‘₯2)π‘š/2(𝑑𝑑π‘₯)π‘šπ‘ƒβ„“(π‘₯)

and

π‘ƒβˆ’π‘šβ„“(π‘₯)=(βˆ’1)π‘š(β„“βˆ’π‘š)!(β„“+π‘š)!π‘ƒπ‘šβ„“(π‘₯)

where naturally π‘ƒπ‘šβ„“(π‘₯) =0 for |π‘š| >β„“.

Mathematica

The Associated Legendre polynomial π‘ƒπ‘šβ„“(π‘₯) may be generated in Wolfram Mathematica with LegendreP[β„“, m, x].

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Footnotes

  1. 2018. Introduction to quantum mechanics, Β§4.1, p. 135 ↩

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