WeierstrassP: Weierstrass P and related functions

In elliptic: Weierstrass and Jacobi Elliptic Functions

Description

Weierstrass elliptic function and its derivative, Weierstrass sigma function, and the Weierstrass zeta function

Usage

P(z, g=NULL, Omega=NULL, params=NULL, use.fpp=TRUE, give.all.3=FALSE, ...)
Pdash(z, g=NULL, Omega=NULL, params=NULL, use.fpp=TRUE, ...)
sigma(z, g=NULL, Omega=NULL, params=NULL, use.theta=TRUE, ...)
zeta(z, g=NULL, Omega=NULL, params=NULL, use.fpp=TRUE, ...)

Arguments

z	Primary complex argument
g	Invariants g=c(g2,g3). Supply exactly one of (g, Omega, params)
Omega	Half periods
params	Object with class “parameters” (typically provided by parameters())
use.fpp	Boolean, with default TRUE meaning to calculate P(z^C) where z^C is congruent to z in the period lattice. The default means that accuracy is greater for large z but has the deficiency that slight discontinuities may appear near parallelogram boundaries
give.all.3	Boolean, with default FALSE meaning to return P(z) and TRUE meaning to return the other forms given in equation 18.10.5, p650. Use TRUE to check for accuracy
use.theta	Boolean, with default TRUE meaning to use theta function forms, and FALSE meaning to use a Laurent expansion. Usually, the theta function form is faster, but not always
...	Extra parameters passed to theta1() and theta1dash()

Note

In this package, function sigma() is the Weierstrass sigma function. For the number theoretic divisor function also known as “sigma”, see divisor().

Author(s)

Robin K. S. Hankin

References

R. K. S. Hankin. Introducing Elliptic, an R package for Elliptic and Modular Functions. Journal of Statistical Software, Volume 15, Issue 7. February 2006.

Examples

## Example 8, p666, RHS:
P(z=0.07 + 0.1i,g=c(10,2))

## Example 8, p666, RHS:
P(z=0.1 + 0.03i,g=c(-10,2))
## Right answer!

## Compare the Laurent series, which also gives the Right Answer (tm):
 P.laurent(z=0.1 + 0.03i,g=c(-10,2))


## Now a nice little plot of the zeta function:
x <- seq(from=-4,to=4,len=100)
z <- outer(x,1i*x,"+")
view(x,x,limit(zeta(z,c(1+1i,2-3i))),nlevels=6,scheme=1)


#now figure 18.5, top of p643:
p <- parameters(Omega=c(1+0.1i,1+1i))
n <- 40

f <- function(r,i1,i2=1)seq(from=r+1i*i1, to=r+1i*i2,len=n)
g <- function(i,r1,r2=1)seq(from=1i*i+r1,to=1i*i+2,len=n)

solid.lines <-
  c(
    f(0.1,0.5),NA,
    f(0.2,0.4),NA,
    f(0.3,0.3),NA,
    f(0.4,0.2),NA,
    f(0.5,0.0),NA,
    f(0.6,0.0),NA,
    f(0.7,0.0),NA,
    f(0.8,0.0),NA,
    f(0.9,0.0),NA,
    f(1.0,0.0)
    )
dotted.lines <-
  c(
    g(0.1,0.5),NA,
    g(0.2,0.4),NA,
    g(0.3,0.3),NA,
    g(0.4,0.2),NA,
    g(0.5,0.0),NA,
    g(0.6,0.0),NA,
    g(0.7,0.0),NA,
    g(0.8,0.0),NA,
    g(0.9,0.0),NA,
    g(1.0,0.0),NA
    )

plot(P(z=solid.lines,params=p),xlim=c(-4,4),ylim=c(-6,0),type="l",asp=1)
lines(P(z=dotted.lines,params=p),xlim=c(-4,4),ylim=c(-6,0),type="l",lty=2)

Example output

Attaching package: 'elliptic'

The following objects are masked from 'package:stats':

    sd, sigma

The following object is masked from 'package:base':

    is.primitive

[1] -22.9745-63.05323i
[1] 76.58833-50.50379i
[1] 76.58833-50.50379i
Warning message:
In parameters(Omega = c(1 + (0+0.1i), 1 + (0+1i))) :
  Omega supplied not a primitive pair of half periods.  Function converting Omega to a primitive pair 

elliptic documentation built on May 2, 2019, 9:37 a.m.