Dedekind’s Eta Function and Modular Forms

Abstract

Throughout this monograph we use the notation

$$e(z) = e^{2\pi iz}$$

where z is a complex number. We define the Dedekind eta function by the infinite product

$$\eta(z) = e\bigl( {\tfrac{z}{24}}\bigr) \prod_{n=1}^{\infty} (1 - q^n) \qquad \mbox{with} \qquad q = e(z) .$$

(1.1)

The product converges normally for q in the unit disc or, equivalently, for z in the upper half plane ℍ={z∈ℂ∣Im(z)>0}. This means that the product of the absolute values |1−q ⁿ| converges uniformly for z in every compact subset of ℍ. The normal convergence of the product implies that η is a holomorphic function on ℍ and that η(z)≠0 for all z∈ℍ.