R/elliptic_PI.R

In Carlson: Carlson Elliptic Integrals and Incomplete Elliptic Integrals

#' Incomplete elliptic integral of the third kind
#' @description Evaluate the incomplete elliptic integral of the third kind.
#'
#' @param phi amplitude, real or complex number/vector
#' @param n characteristic, real or complex number/vector
#' @param m parameter, real or complex number/vector
#' @param minerror the bound on the relative error passed to
#'   \code{\link{Carlson_RF}} and \code{\link{Carlson_RJ}}
#'
#' @return A complex number or vector, the value(s) of the incomplete elliptic
#'   integral
#'   \ifelse{html}{\out{Π(φ,n,m)}}{\eqn{\Pi(\phi,n,m)}{PI(phi,n,m)}}.
#' @export
#'
#' @examples elliptic_PI(1, 0.8, 0.2)
#' gsl::ellint_P(1, sqrt(0.2), -0.8)
elliptic_PI <- function(phi, n, m, minerror = 1e-15) {
  phi <- as.complex(phi)
  n   <- as.complex(n)
  m   <- as.complex(m)
  lgths <- c(length(phi), length(n), length(m))
  L <- max(lgths)
  if(!all(lgths %in% c(1L, L))) {
    stop("Incompatible lengths of the arguments.")
  }
  if(L != 1L && any(lgths == 1L)) {
    if(length(phi) == 1L) {
      phi <- rep(phi, L)
    }
    if(length(n) == 1L) {
      n <- rep(n, L)
    }
    if(length(m) == 1L) {
      m <- rep(m, L)
    }
  }
  ellPIcpp(phi, n, m, minerror)
  # if(phi == 0 || (is.infinite(Re(m)) && Im(m) == 0) ||
  #    (is.infinite(Re(n)) && Im(n) == 0)){
  #   as.complex(0)
  # }else if(phi == pi/2 && m == 1 && Im(n) == 0 && n != 1){
  #   ifelse(Re(n) > 1, -Inf, Inf)
  # }else if(phi == pi/2 && n == 1){
  #   NaN
  # }else if(phi == pi/2 && m == 0){
  #   pi/2/sqrt(as.complex(1-n))
  # }else if(phi == pi/2 && n == m){
  #   elliptic_E(pi/2, m, minerror) / (1-m)
  # }else if(phi == pi/2 && n == 0){
  #   elliptic_F(pi/2, m, minerror)
  # }else if(Re(phi) >= -pi/2 && Re(phi) <= pi/2){
  #   sine <- sin(phi)
  #   if(is.infinite(Re(sine)) || is.infinite(Im(sine))){
  #     stop("`sin(phi)` is not finite.")
  #   }
  #   sine2 <- sine*sine
  #   cosine2 <- 1 - sine2
  #   oneminusmsine2 <- 1 - m*sine2
  #   sine * (Carlson_RF(cosine2, oneminusmsine2, 1, minerror) +
  #             n * sine2 * Carlson_RJ(cosine2, oneminusmsine2, 1, 1-n*sine2,
  #                                    minerror) / 3)
  # }else if(Re(phi) > pi/2){
  #   k <- ceiling(Re(phi)/pi - 0.5)
  #   phi <- phi - k*pi
  #   2*k*elliptic_PI(pi/2, n, m, minerror) + elliptic_PI(phi, n, m, minerror)
  # }else{
  #   k <- -floor(0.5 - Re(phi)/pi)
  #   phi <- phi - k*pi
  #   2*k*elliptic_PI(pi/2, n, m, minerror) + elliptic_PI(phi, n, m, minerror)
  # }
}