According to wiki: <https://en.wikipedia.org/wiki/Dedekind_eta_function>, Dedekind eta function is defined in many equivalent forms. But none of them is an explicit description (say in algorithmic format) on how to computing it. Where to find such one? Thanks!



How to compute Dedekind eta function efficiently?
Euler's formula


$$
\sum\limits\_{n \in \mathbb{Z}} {( - 1)^n q^{\frac{{(3n^2 - n)}}
{2}} } = \prod\limits\_{n = 1}^\infty {(1 - q^n ),}
$$


(which can be proven from Jacobi’s triple product identity by using the fact that $\prod\limits\_{n = 1}^\infty {(1 - q^{3n} )(1 - q^{3n - 2} )} (1 - q^{3n - 1} ) = \prod\limits\_{n = 1}^\infty {(1 - q^n )}
$)
provides a good way of numerically computing 


$$
\eta (\tau )=e^{\frac {\pi {\rm {{i}\tau }}}{12}}\prod \_{n=1}^{\infty }(1-e^{2n\pi {\rm {{i}\tau }}})=q^{\frac {1}{24}}\prod \_{n=1}^{\infty }(1-q^{n}).
$$


I hope this answers your question.


