eta

0th

Percentile

Dedekind's eta function

Dedekind's \(\eta\) function

Keywords
math
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

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

Aliases
  • eta
  • eta.series
Examples
# NOT RUN {
 z <- seq(from=1+1i,to=10+0.06i,len=999)
 plot(eta(z))

max(abs(eta(z)-eta.series(z)))
# }
Documentation reproduced from package elliptic, version 1.4-0, License: GPL-2

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