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.

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 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’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)
```

- 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

RobinHankin/elliptic documentation built on Feb. 21, 2020, 3:43 p.m.

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