Associated Legendre Function of the First Kind/Examples/2, 2

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Example of Associated Legendre Function of the First Kind

Let $\map { {P_n}^m} x$ denote an associated Legendre function of the first kind.

Then:

$\map { {P_2}^2} x = 3 \paren {1 - x^2}$


Proof

\(\ds \map { {P_n}^m} x\) \(=\) \(\ds \dfrac {\paren {1 - x^2}^{m / 2} } {2^n n!} \dfrac {\d^{m + n} } {\d x^{m + n} } \paren {x^2 - 1}^n\) Derivative Form of Associated Legendre Function of the First Kind
\(\ds \leadsto \ \ \) \(\ds \map { {P_2}^2} x\) \(=\) \(\ds \dfrac {\paren {1 - x^2}^{2 / 2} } {2^2 2!} \dfrac {\d^4} {\d x^4} \paren {x^2 - 1}^2\) setting $m = n = 2$
\(\ds \) \(=\) \(\ds \dfrac {1 - x^2} 8 \map {\dfrac {\d^4} {\d x^4} } {x^4 - 2 x^2 + 1}\) Square of Difference
\(\ds \) \(=\) \(\ds \dfrac {1 - x^2} 8 \paren {4!}\) Nth Derivative of Nth Power where $n = 4$
\(\ds \) \(=\) \(\ds 3 \paren {1 - x^2}\) $4! = 24$ and simplifying

$\blacksquare$


Sources

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