elliptic (version 1.5-0)

eta: Dedekind's eta function

Description

Dedekind's \(\eta\) function

Usage

eta(z, ...)
eta.series(z, maxiter=300)

Arguments

z

Complex argument

...

In function eta(), extra arguments sent to theta3()

maxiter

In function eta.series(), maximum value of iteration

Author

Robin K. S. Hankin

Details

Function eta() uses Euler's formula, viz $$\eta(z)=e^{\pi iz/12}\theta_3\left(\frac{1}{2}+\frac{z}{2},3z\right)$$

Function eta.series() is present for validation (and interest) only; it uses the infinite product formula: $$\eta(z)= e^{\pi iz/12}\prod_{n=1}^\infty\left(1-e^{2\pi inz}\right)$$

References

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

See Also

farey

Examples

 z <- seq(from=1+1i, to=10+0.06i, len=999)
 plot(eta(z))

max(abs(eta(z) - eta.series(z)))