MATLAB Function Reference  legendre

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 • ` ` 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`

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  functions by

• They are related to the Schmidt form given previously by ` ` for • ` `for .

References

  Abramowitz, M. and I. A. Stegun, Handbook of Mathematical Functions, Dover Publications, 1965, Ch.8.

  Jacobs, J. A., Geomagnetism, Academic Press, 1987, Ch.4.

© 1994-2005 The MathWorks, Inc.