ellipke,ellipj {pracma}

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

u

numeric vector.

m

input vector, all input elements must satisfy 0 <= x <= 1.

tol

tolerance; default is machine precision.

Details

ellipke computes the complete elliptic integrals to accuracy tol, based on the algebraic-geometric mean.

ellipj computes the Jacobi elliptic integrals sn, cn, and dn. For instance, sn is the inverse function for

u = \int_0^\phi dt / \sqrt{1 - m \sin^2 t}

with sn(u) = \sin(\phi).

Some definitions of the elliptic functions use the modules k instead of the parameter m. They are related by k^2=m=sin(a)^2 where a is the ‘modular angle’.

Value

ellipke returns list with two components, k the values for the first kind, e the values for the second kind.

ellipj returns a list with components the three Jacobi elliptic integrals sn, cn, and dn.

References

Abramowitz, M., and I. A. Stegun (1965). Handbook of Mathematical Functions. Dover Publications, New York.

See Also

elliptic::sn,cn,dn

Examples

x <- linspace(0, 1, 20)
ke <- ellipke(x)

## Not run: 
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()
## End(Not run)

## ellipse circumference with axes a, b
ellipse_cf <- function(a, b) {
    return(4*a*ellipke(1 - (b^2/a^2))$e)
}
print(ellipse_cf(1.0, 0.8), digits = 10)
# [1] 5.672333578

## Jacobi elliptic integrals
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