Dedekind Eta Database

Introduction

The Dedekind eta function, \( \eta(\tau) \), is a modular form with the following properties:

• Holomorphic on the upper half-plane: \( \operatorname{Im}(\tau) > 0 \).
• \[ \eta(\tau) = q^{1/24} \prod_{n=1}^{\infty} (1 - q^n), \quad q = e^{2\pi i \tau} \]

\[ \eta(\tau + 1) = e^{\pi i /12} \eta(\tau), \qquad \eta(-1/\tau) = \sqrt{-i\tau}\, \eta(\tau) \]