The elliptic package: Weierstrass and Jacobi elliptic functions in R
elliptic: Weierstrass and Jacobi elliptic functions
Overview
An elliptic function is a meromorphic complex function that is periodic
in two directions. That is, there exist two nonzero complex numbers
with
such that
whenever
is defined; note carefully the factors of 2. There are two natural ways
of presenting elliptic functions: that of Weierstrass, and that of
Jacobi. Historically, the Jacobi form was first presented and is the
most practically useful, but the Weierstrass form is more elegant (IMO).
Terminology follows that of Abramowitz and Stegun wherever possible.
Installation
To install the most recent stable version on CRAN, use
install.packages() at the R prompt:
R> install.packages("elliptic")
To install the current development version use devtools:
R> devtools::install_github("RobinHankin/elliptic")
And then to load the package use library():
library("elliptic")The package comes with an extensive and detailed vignette; type
vignette("ellipticpaper") at the R commandline.
The package in use
The Weierstrass elliptic function is evaluated numerically by P(),
which takes the half periods
. Thus
z <- 0.3 + 0.2i
omega1 <- 5+1i; omega2 <- 1+7i # half-periods
f <- function(z){P(z,Omega=c(omega1,omega2))}
c(f(z),f(z + 10+2i), f(z + 2+14i)) # should be equal## [1] 2.958647-7.100563i 2.958647-7.100563i 2.958647-7.100563i
The elliptic functions can be visualised using view():
x <- seq(from=-4, to=4, len=200)
y <- x
z <- outer(x,1i*x, "+")
f <- P(z, c(1+1i,2-3i))
par(pty="s")
view(x,y,f,real.contour=FALSE,drawlabel=FALSE,axes=FALSE,xlab="Re(z)",ylab="Im(z)", main="P(z,1+i,2-3i)")
axis(1,pos = -4)
axis(2,pos = -4)
lines(x=c(-4,4),y=c(4,4))
lines(y=c(-4,4),x=c(4,4))
Related functions include
(
sigma() and the
(
zeta()).
Jacobi forms
Jacobi’s elliptic functions are implemented in the package with their
standard names sn(), cn(), dn() etc. For
example:
view(x,y,sn(z,m=6),real=FALSE,drawlabel=FALSE,axes=FALSE,xlab="Re(z)",ylab="Im(z)", main="The Jacobi sn() function")
axis(1,pos = -4,at=c(-4,-2,0,2,4))
axis(2,pos = -4,at=c(-4,-2,0,2,4))
lines(x=c(-4,4),y=c(4,4))
lines(y=c(-4,4),x=c(4,4))
The Jacobi forms are useful in physics and we can use them to visualise potential flow in a rectangle:
n <- 300
K <- K.fun(1/2) # aspect ratio
f <- function(z){1i*log((z-1.7+3i)*(z-1.7-3i)/(z+1-0.3i)/(z+1+0.3i))} # position of source and sink
x <- seq(from=-K,to=K,len=n)
y <- seq(from=0,to=K,len=n)
z <- outer(x,1i*y,"+")
view(x, y, f(sn(z,m=1/2)), nlevels=44, real.contour=TRUE, drawlabels=FALSE,
main="Potential flow in a rectangle",axes=FALSE,xlab="",ylab="")
rect(-K,0,K,K,lwd=3)
References
- M Abramowitz and IA Stegun (1965). Hankdbook of Mathematical Functions. New York: Dover
- RKS Hankin (2006). “Introducing elliptic, an R package for elliptic and modular functions”. Journal of Statistical Software, 15:7
- K Chandrasekharan (1985). Elliptic functions. Springer-Verlag