Associated Legendre Function of the First Kind/Examples/3, 3

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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:

$\map { {P_3}^3} x = 15 \paren {1 - x^2}^{3 / 2}$


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
\(\ds \leadsto \ \ \) \(\ds \map { {P_3}^3} x\) \(=\) \(\ds \dfrac {\paren {1 - x^2}^{3 / 2} } {2^3 3!} \dfrac {\d^{3 + 3} } {\d x^{3 + 3} } \paren {x^2 - 1}^3\) setting $m = n = 3$
\(\ds \) \(=\) \(\ds \dfrac {\paren {1 - x^2}^{3 / 2} } {8 \times 6} \map {\dfrac {\d^6} {\d x^6} } {x^6 - 3 x^4 + 3 x^2 - 1}\) Cube of Difference
\(\ds \) \(=\) \(\ds \dfrac {\paren {1 - x^2}^{3 / 2} } {48} \paren {6!}\) Nth Derivative of Mth Power where $n = 6$
\(\ds \) \(=\) \(\ds 15 \paren {1 - x^2}^{3 / 2}\) simplifying

$\blacksquare$


Sources

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