Associated Legendre Function of the First Kind/Examples

Examples of Associated Legendre Functions of the First Kind

Example: ${P_n}^0$

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

Example: ${P_n}^m$ where $m > n$

Let $m > n$.

Then:

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

Example: ${P_1}^1$

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

Example: ${P_2}^1$

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

Example: ${P_2}^2$

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

Example: ${P_3}^1$

$\map { {P_3}^1} x = \dfrac 3 2 \paren {5 x^2 - 1} \paren {1 - x^2}^{1 / 2}$

Example: ${P_3}^2$

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

Example: ${P_3}^3$

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