Weierstrass P and related functions

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

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 \(\wp(z^C)\) 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 \(\wp(z)\) 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()

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.

Author

Robin K. S. Hankin

Note

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

Examples

## Example 8, p666, RHS:
P(z=0.07 + 0.1i,g=c(10,2))
#> [1] -22.9745-63.05323i

## Example 8, p666, RHS:
P(z=0.1 + 0.03i,g=c(-10,2))
#> [1] 76.58833-50.50379i
## Right answer!

## Compare the Laurent series, which also gives the Right Answer (tm):
 P.laurent(z=0.1 + 0.03i,g=c(-10,2))
#> [1] 76.58833-50.50379i


## 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))
#> Warning: Omega supplied not a primitive pair of half periods.  Function converting Omega to a primitive pair 
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)