MATLAB Function Reference 
Syntax
Definitions
Associated Legendre Functions. The Legendre functions are defined by
is the Legendre polynomial of degree .
Schmidt Seminormalized Associated Legendre Functions. The Schmidt seminormalized associated Legendre functions are related to the nonnormalized associated Legendre functions by
for
for .
Fully Normalized Associated Legendre Functions. The fully normalized associated Legendre functions are normalized such that
and are related to the unnormalized associated Legendre functions by
Description
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 .
If X
is a vector, then P
is an (n+1)
byq
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 .
Examples
Example 1. 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 
Algorithm
legendre
uses a threeterm 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 [1] functions by
They are related to the Schmidt form given previously by
for
for .
References
[1] Abramowitz, M. and I. A. Stegun, Handbook of Mathematical Functions, Dover Publlications, 1965, Ch.8.
[2] Jacobs, J. A., Geomagnetism, Academic Press, 1987, Ch.4.
legend  length 