Use this tag for questions about a particular function defined on the upper half-plane of complex numbers and that is a modular form of weight one-half.

For any complex number $\tau$ with Im $\tau > 0,$ let $q = e^{2\pi i \tau}.$ Then the *Dedekind eta function* is
$$\eta(\tau ) = e^{\frac{\pi i\tau}{12}} \prod_{n=1}^{\infty} \left(1 -e ^{2 n \pi i \tau}\right) = q^{\frac1{24}} \prod_{n=1}^{\infty} \left(1 - q^n\right).$$

The eta function is holomorphic on the upper half-plane but cannot be continued analytically beyond it.

Because the eta function is easy to compute numerically from power series, it is often helpful in computation to express other functions in terms of it when possible. Products and quotients of eta functions, called eta quotients, can be used to express a great variety of modular forms.