Dedekind's eta function

Dedekind's \(\eta\) function

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

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.

Author

Robin K. S. Hankin

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)))
#> [1] 1.445008e-14