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:
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 
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
|
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)).
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