Associated Legendre Function of the First Kind/Examples/n, 0

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Example of Associated Legendre Function of the First Kind

$\map { {P_n}^0} x = \map {P_n} x$

where:

$\map { {P_n}^0} x$ denotes the associated Legendre function of the first kind
$\map {P_n} x$ denotes the Legendre polynomial.


Proof

From the definition of the associated Legendre function of the first kind:

$\map { {P_n}^m} x = \paren {1 - x^2}^{m / 2} \dfrac {\d^m} {\d x^m} \map {P_n} x$

where $\map {P_n} x$ is the Legendre polynomial of order $n$.


When $m = 0$ we have:

\(\ds \map { {P_n}^0} x\) \(=\) \(\ds \paren {1 - x^2}^{0 / 2} \dfrac {\d^0} {\d x^0} \map {P_n} x\)
\(\ds \) \(=\) \(\ds \map {P_n} x\) as $\paren {1 - x^2}^{0 / 2} = 1$ and $\dfrac {\d^0} {\d x^0}$ is the identity mapping

$\blacksquare$


Sources

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