# J: Various modular functions In elliptic: Weierstrass and Jacobi Elliptic 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

 ```1 2``` ```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

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25``` ``` 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 documentation built on May 2, 2019, 9:37 a.m.