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# legendre

Associated Legendre functions

## Syntax

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

## 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 −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.

## Examples

### Example 1

The statement legendre(2,0:0.1:0.2) returns the matrix

x = 0x = 0.1x = 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
2.4950l```

expand all

### Associated Legendre Functions

The Legendre functions are defined by

where

is the Legendre polynomial of degree n:

### Schmidt Seminormalized Associated Legendre Functions

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

### 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

### Algorithms

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

## 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.