Various modular functions

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.

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

References

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

Author

Robin K. S. Hankin

Examples

 J(2.3+0.23i,use.theta=TRUE)
#> [1] 12.01226+8.339351i
 J(2.3+0.23i,use.theta=FALSE)
#> [1] 12.01226+8.339351i

 #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 lambda(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")