legendre {pracma}

Legendre Functions (Matlab Style)

Description

Calculate the values of (associated) Legendre functions.

Usage

legendre(n, x)

Arguments

n

degree of the Legendre polynomial involved.

x

real points to evaluate Legendre's functions at.

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.

Value

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

Note

Legendre functions are solutions to Legendre's differential equation (it occurs when solving Laplace's equation in spherical coordinates).

See Also

chebPoly

Examples

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, ])
# }