legendre (MATLAB Functions)

Associated Legendre functions

Syntax

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

Definitions

Associated Legendre Functions. The Legendre functions are defined by

where

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 .

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)-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 .

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

	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
    2.4950

Algorithm

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 [1] functions by

They are related to the Schmidt form given previously by

for .

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.

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