R/alp.R

In obreschkow/cooltools: Practical Tools for Scientific Computations and Visualizations

#' Associated Legendre Polynomials
#'
#' @description Compute associated Legendre polynomials P_l^m(x), defined as the canonical solutions of the general Legendre equation. These polynomials are used, for instance, to compute the spherical harmonics.
#'
#' @param x real argument between -1 and +1, can be a vector
#' @param l degree of the polynomial (0,1,2,...); accurate to about 25
#' @param m order of the polynomial (-l,-l+1,...,l); for negative values the standard convention is used: if m>0, then P(x,l,-m) = P(x,l,m) (-1)^m*factorial(l-m)/factorial(l+m).
#'
#' @return Returns a vector with the same number of elements as \code{x}
#'
#' @author Danail Obreschkow
#'
#' @seealso \code{\link{sphericalharmonics}}
#'
#' @export

alp = function(x, l=0, m=0) {

  if (m<0) {
    m = -m
    f = (-1)^m*factorial(l-m)/factorial(l+m)
  } else {
    f = 1
  }
  out = 0
  logx = log(abs(x))

  for (k in seq(l,m)) {
    alpha = (l+k-1)/2
    a = alpha-seq(0,l-1)
    exponent = -lfactorial(l-k)-lfactorial(k-m)+(k-m)*logx+sum(log(abs(a)))
    out = out+exp(exponent)*prod(sign(a))
  }
  out = f*(-1)^m*2^l*(1-x^2)^(m/2)*out
  out[x<0] = out[x<0]*(-1)^(l+m)
  return(out)
}