legendre

Purpose

Associated Legendre functions.

Synopsis

P = legendre(n,x)

Description

P = legendre(n,x) computes the associated Legendre functions of degree n and order m = 0,1,...,n, evaluated at x. n is an integer less than or equal to 256. x is a vector whose elements are real and have absolute value less than or equal to one. P is a matrix with n+1 rows and length(x) columns. Each element of the returned matrix, P(i,j), corresponds to the associated Legendre function of degree n and order (i-1) evaluated at x(j). The first row of P is the Legendre polynomial evaluated at x.

Examples

legendre(2,0:0.1:0.2)
          
ans =
    -0.5000    -0.4850    -0.4400
          0    -0.2985    -0.5879
     3.0000     2.9700     2.8800
Note that this answer is of the form

   P(2,0;0)   P(2,0;0.1)   P(2,0;0.2)
   P(2,1;0)   P(2,1;0.1)   P(2,1;0.2)
   P(2,2;0)   P(2,2;0.1)   P(2,2;0.2)

Algorithm

The mathematical definition is

where

is the Legendre polynomial of degree n.

(c) Copyright 1994 by The MathWorks, Inc.