legendre: Legendre Functions (Matlab Style)

Description

Calculate the values of (associated) Legendre functions.

Usage

legendre(n, x)

Arguments

degree of the Legendre polynomial involved.

real points to evaluate Legendre's functions at.

Value

Returns a matrix of size (n+1)-by-N where N=length(x).

Details

legendre(n,x) computes the associated Legendre functions of degree n and order m=0,1,...,n, evaluated for each element of x where x must contain real values in [-1,1].

If x is a vector, then L=legendre(n,x) is an (n+1)-by-N matrix, where N=length(x). Each element L[m+1,i] corresponds to the associated Legendre function of degree legendre(n,x) and order m evaluated at x[i].

Note that the first row of L is the Legendre polynomial evaluated at x.

Examples

Run this code

x <- c(0.0, 0.1, 0.2)
legendre(2, x)
#      [,1]       [,2]       [,3]
# [1,] -0.5 -0.4850000 -0.4400000
# [2,]  0.0 -0.2984962 -0.5878775
# [3,]  3.0  2.9700000  2.8800000

## Not run: 
# x <- seq(0, 1, len = 50)
# L <- legendre(2, x)
# plot(x, L[1, ], type = "l", col = 1, ylim = c(-2, 3), ylab = "y",
#                 main = "Legendre Functions of degree 2")
# lines(x, L[2, ], col = 2)
# lines(x, L[3, ], col = 3)
# grid()## End(Not run)
## Generate Legendre's Polynomial as function
# legendre_P <- function(n, x) {
#     L <- legendre(n, x)
#     return(L[1, ])
# }