legendre (MATLAB Function Reference)

Associated Legendre functions

Syntax

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

Definition

The Legendre functions are defined by:

where

is the Legendre polynomial of degree n:

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

by:

where m > 0.

Description

P = legendre(n,X) computes the associated Legendre functions of degree n and order m = 0,1,...,n, evaluated at X. Argument n must be a scalar integer less than 256, and X must contain real values in the domain -1 x 1.

The returned array P has one more dimension than X, and each element P(m+1,d1,d2...) contains the associated Legendre function of degree n and order m evaluated at X(d1,d2...).

If X is a vector, then P is a matrix of the form:

S = legendre(...,'sch') computes the Schmidt seminormalized associated Legendre functions .

Examples

The statement legendre(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

	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`

Note that this matrix is of the form shown at the bottom of the previous page.

Given,

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

Then size(P) is 3-by-2-by-4-by-5, and P(:,1,2,3) is the same as legendre(n,X(1,2,3)).

legend length