elliptic: J – R documentation

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J

Various modular functions

Description

Modular functions including Klein's modular function J (aka Dedekind's Valenz function J, aka the Klein invariant function, aka Klein's absolute invariant), the lambda function, and Delta.

Usage

J(tau, use.theta = TRUE, ...)
lambda(tau, ...)

Arguments

`tau`	tau; it is assumed that `Im(tau)>0`
`use.theta`	Boolean, with default `TRUE` meaning to use the theta function expansion, and `FALSE` meaning to evaluate `g2` and `g3` directly
`...`	Extra arguments sent to either `theta1()` et seq, or `g2.fun()` and `g3.fun()` as appropriate

Author(s)

Robin K. S. Hankin

References

K. Chandrasekharan 1985. Elliptic functions, Springer-Verlag.

Examples

J(2.3+0.23i,use.theta=TRUE)
 J(2.3+0.23i,use.theta=FALSE)

 #Verify that J(z)=J(-1/z):
 z <- seq(from=1+0.7i,to=-2+1i,len=20)
 plot(abs((J(z)-J(-1/z))/J(z)))

 # Verify that lamba(z) = lambda(Mz) where M is a modular matrix with b,c
 # even and a,d odd:

 M <- matrix(c(5,4,16,13),2,2)
 z <- seq(from=1+1i,to=3+3i,len=100)
 plot(lambda(z)-lambda(M %mob% z,maxiter=100))


#Now a nice little plot; vary n to change the resolution:
 n <- 50
 x <- seq(from=-0.1, to=2,len=n)
 y <- seq(from=0.02,to=2,len=n)

 z <- outer(x,1i*y,"+")
 f <- lambda(z,maxiter=40)
 g <- J(z)
 view(x,y,f,scheme=04,real.contour=FALSE,main="try higher resolution")
 view(x,y,g,scheme=10,real.contour=FALSE,main="try higher resolution")

elliptic

Weierstrass and Jacobi Elliptic Functions

v1.4-0

GPL-2

Authors

Robin K. S. Hankin [aut, cre] (<https://orcid.org/0000-0001-5982-0415>)

Initial release

J