Definition:Associated Legendre Function of the First Kind

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Definition

Let $m, n \in \Z_{\ge 0}$ be non-negative integers.

The associated Legendre functions of the first kind are the real functions defined and denoted as:

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

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


Examples

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: ${P_3}^3$

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


Also known as

Sources which do not delve deeply into this area often call the associated Legendre functions of the first kind as just the associated Legendre functions.

However, the definition of the associated Legendre function is considerably more general than this.


Also see

  • Results about the associated Legendre functions can be found here.


Source of Name

This entry was named for Adrien-Marie Legendre.


Sources

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