J {elliptic} | R Documentation |
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 |
|
use.theta |
Boolean, with default |
... |
Extra arguments sent to either |
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")
[Package elliptic version 1.4-0 Index]