ellip.Rd
\name{ellipke,ellipj}
\alias{ellipke}
\alias{ellipj}
\title{
Elliptic and Jacobi Elliptic Integrals
}
\description{
Complete elliptic integrals of the first and second kind, and
Jacobi elliptic integrals.
}
\usage{
ellipke(m, tol = .Machine$double.eps)
ellipj(u, m, tol = .Machine$double.eps)
}
\arguments{
\item{u}{numeric vector.}
\item{m}{input vector, all input elements must satisfy \code{0 <= x <= 1}.}
\item{tol}{tolerance; default is machine precision.}
}
\details{
\code{ellipke} computes the complete elliptic integrals to accuracy
\code{tol}, based on the algebraic-geometric mean.
\code{ellipj} computes the Jacobi elliptic integrals \code{sn}, \code{cn},
and \code{dn}. For instance, \eqn{sn} is the inverse function for
\deqn{u = \int_0^\phi dt / \sqrt{1 - m \sin^2 t}}
with \eqn{sn(u) = \sin(\phi)}.
Some definitions of the elliptic functions use the modules \code{k} instead
of the parameter \code{m}. They are related by \code{k^2=m=sin(a)^2} where
\code{a} is the `modular angle'.
}
\value{
\code{ellipke} returns list with two components, \code{k} the values for
the first kind, \code{e} the values for the second kind.
\code{ellipj} returns a list with components the three Jacobi elliptic
integrals \code{sn}, \code{cn}, and \code{dn}.
}
\references{
Abramowitz, M., and I. A. Stegun (1965).
Handbook of Mathematical Functions. Dover Publications, New York.
}
\seealso{
\code{elliptic::sn,cn,dn}
}
\examples{
x <- linspace(0, 1, 20)
ke <- ellipke(x)
\dontrun{
plot(x, ke$k, type = "l", col ="darkblue", ylim = c(0, 5),
main = "Elliptic Integrals")
lines(x, ke$e, col = "darkgreen")
legend( 0.01, 4.5,
legend = c("Elliptic integral of first kind",
"Elliptic integral of second kind"),
col = c("darkblue", "darkgreen"), lty = 1)
grid()}
u <- c(0, 1, 2, 3, 4, 5)
m <- seq(0.0, 1.0, by = 0.2)
je <- ellipj(u, m)
# $sn 0.0000 0.8265 0.9851 0.7433 0.4771 0.9999
# $cn 1.0000 0.5630 -0.1720 -0.6690 -0.8789 0.0135
# $dn 1.0000 0.9292 0.7822 0.8176 0.9044 0.0135
je$sn^2 + je$cn^2 # 1 1 1 1 1 1
je$dn^2 + m * je$sn^2 # 1 1 1 1 1 1
}
\keyword{ math }