Legendre Functions (Matlab Style)
Calculate the values of (associated) Legendre functions.
legendre(n, x)
n |
degree of the Legendre polynomial involved. |
x |
real points to evaluate Legendre's functions at. |
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.
Returns a matrix of size (n+1)-by-N where N=length(x).
Legendre functions are solutions to Legendre's differential equation (it occurs when solving Laplace's equation in spherical coordinates).
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, ])
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