legendre

Associated Legendre functions

Syntax

P = legendre(n,X) S = legendre(n,X,'sch') N = legendre(n,X,'norm')

Description

P = legendre(n,X)computes theassociated Legendre functionsof degreenand orderm = 0,1,...,n, evaluated for each element ofX。Argumentnmust be a positive integer, andXmust contain real values in the domain−1 ≤x≤ 1。

IfXis a vector, thenPis an(n+1)-by-qmatrix, whereq = length(X)。Each elementP(m+1,i)corresponds to the associated Legendre function of degreenand ordermevaluated atX(i)。

In general, the returned arrayPhas one more dimension thanX, and each elementP(m+1,i,j,k,...)contains the associated Legendre function of degreenand ordermevaluated atX(i,j,k,...)。注意第一行Pis the Legendre polynomial evaluated atX, i.e., the case wherem= 0.

S = legendre(n,X,'sch')computes theSchmidt Seminormalized Associated Legendre Functions。

N = legendre(n,X,'norm')computes theFully Normalized Associated Legendre Functions。

Examples

Example 1

The statementlegendre(2,0:0.1:0.2)returns the matrix

	x = 0	x = 0.1	x = 0.2
`m = 0`	`-0.5000`	`-0.4850`	`-0.4400`
`m = 1`	`0`	`-0.2985`	`-0.5879`
`m = 2`	`3.0000`	`2.9700`	`2.8800`

Example 2

Given,

X = rand(2,4,5); n = 2; P = legendre(n,X)

then

size(P) ans = 3 2 4 5

and

P(:,1,2,3) ans = -0.2475 -1.1225 2.4950

is the same as

legendre(n,X(1,2,3)) ans = -0.2475 -1.1225 2.4950l

More About

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Associated Legendre Functions

The Legendre functions are defined by

$P_{n}^{m} (x) = {(- 1)}^{m} {(1 - x^{2})}^{m / 2} \frac{d^{m}}{d x^{m}} P_{n} (x),$

where

$P_{n} (x)$

is the Legendre polynomial of degreen:

$P_{n} (x) = \frac{1}{2^{n} n!} [\frac{d^{n}}{d x^{n}} {(x^{2} - 1)}^{n}]$

Schmidt Seminormalized Associated Legendre Functions

The Schmidt seminormalized associated Legendre functions are related to the nonnormalized associated Legendre functionsby

$\begin{array}{l} P_{n} (x) for m = 0, \\ S_{n}^{m} (x) = (^{- 1) m} \sqrt{\frac{2 (n - m)!}{(n + m)!}} P_{n}^{m} (x) for m > 0. \end{array}$

Fully Normalized Associated Legendre Functions

The fully normalized associated Legendre functions are normalized such that

$\int_{- 1}^{1} {(N_{n}^{m} (x))}^{2} d x = 1$

and are related to the unnormalized associated Legendre functionsby

$N_{n}^{m} = {(- 1)}^{m} \sqrt{\frac{(n + \frac{1}{2}) (n - m)!}{(n + m)!}} P_{n}^{m} (x)$

Algorithms

legendreuses a three-term backward recursion relationship inm。This recursion is on a version of the Schmidt seminormalized associated Legendre functions $Q_{n}^{m} (x)$ , which are complex spherical harmonics. These functions are related to the standard Abramowitz and Stegun[1]functions $P_{n}^{m} (x)$ by

$P_{n}^{m} (x) = \sqrt{\frac{(n + m)!}{(n - m)!}} Q_{n}^{m} (x)$

They are related to the Schmidt form given previously by

$\begin{array}{l} m = 0 : S_{n}^{m} (x) = Q_{n}^{0} (x) \\ m > 0 : S_{n}^{m} (x) = {(- 1)}^{m} \sqrt{2} Q_{n}^{m} (x) \end{array}$

References

[1] Abramowitz, M. and I. A. Stegun,Handbook of Mathematical Functions, Dover Publications, 1965, Ch.8.

[2] Jacobs, J. A.,Geomagnetism, Academic Press, 1987, Ch.4.

Introduced before R2006a

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