Associated Legendre Function of the First Kind/Examples/1, 1

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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:

$\map { {P_1}^1} x = \paren {1 - x^2}^{1/2} = \sqrt {1 - x^2}$


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
\(\ds \leadsto \ \ \) \(\ds \map { {P_1}^1} x\) \(=\) \(\ds \dfrac {\paren {1 - x^2}^{1 / 2} } {2^1 1!} \dfrac {\d^2} {\d x^2} \paren {x^2 - 1}^1\) setting $m = n = 1$
\(\text {(1)}: \quad\) \(\ds \) \(=\) \(\ds \dfrac {\paren {1 - x^2}^{1 / 2} } 2 \dfrac {\d^2} {\d x^2} \paren {x^2 - 1}\) simplifying

Then we have:

\(\ds \map {\dfrac {\d^2} {\d x^2} } {x^2 - 1}\) \(=\) \(\ds \map {\dfrac \d {\d x} } {2 x}\) Derivative of Power
\(\text {(2)}: \quad\) \(\ds \) \(=\) \(\ds 2\) Derivative of Power again


Hence:

\(\ds \map { {P_n}^m} x\) \(=\) \(\ds \dfrac {\paren {1 - x^2}^{1 / 2} } 2 \dfrac {\d^2} {\d x^2} \paren {x^2 - 1}\) from $(1)$ above
\(\ds \) \(=\) \(\ds \dfrac {\paren {1 - x^2}^{1 / 2} } 2 \times 2\) from $(2)$ above
\(\ds \) \(=\) \(\ds \paren {1 - x^2}^{1 / 2}\) simplifying
\(\ds \) \(=\) \(\ds \sqrt {1 - x^2}\) simplifying

$\blacksquare$


Sources

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