Associated Legendre functions
P = legendre(n,X)
S = legendre(n,X,'sch')
N = legendre(n,X,'norm')
P = legendre(n,X) computes the associated Legendre functions of degree n and order m = 0,1,...,n, evaluated for each element of X. Argument n must be a scalar integer, and X must contain real values in the domain −1 ≤ x ≤ 1.
If X is a vector, then P is an (n+1)-by-q matrix, where q = length(X). Each element P(m+1,i) corresponds to the associated Legendre function of degree n and order m evaluated at X(i).
In general, the returned array P has one more dimension than X, and each element P(m+1,i,j,k,...) contains the associated Legendre function of degree n and order m evaluated at X(i,j,k,...). Note that the first row of P is the Legendre polynomial evaluated at X, i.e., the case where m = 0.
S = legendre(n,X,'sch') computes the Schmidt Seminormalized Associated Legendre Functions.
N = legendre(n,X,'norm') computes the Fully Normalized Associated Legendre Functions.
The statement legendre(2,0:0.1:0.2) returns the matrix
|x = 0||x = 0.1||x = 0.2|
m = 0
m = 1
m = 2
X = rand(2,4,5); n = 2; P = legendre(n,X)
size(P) ans = 3 2 4 5
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
The Legendre functions are defined by
is the Legendre polynomial of degree n:
The Schmidt seminormalized associated Legendre functions are related to the nonnormalized associated Legendre functions by
The fully normalized associated Legendre functions are normalized such that
and are related to the unnormalized associated Legendre functions by
legendre uses a three-term backward recursion relationship in m. This recursion is on a version of the Schmidt seminormalized associated Legendre functions , which are complex spherical harmonics. These functions are related to the standard Abramowitz and Stegun  functions by
They are related to the Schmidt form given previously by