Dedekind eta function

Named after the German mathematician Richard Dedekind η - function is a holomorphic function on the upper half-plane.

It plays an important role in the theory of elliptic functions, and the theta functions.

Definition

The η - function is usually defined as follows as an infinite product:

From the definition it follows immediately that has no zeros.

The η - function is closely related to the discriminant, it is

For the calculation of the function η Pentagonalzahlensatz can be used.

Transformation behavior

Its importance is the η - function of their transformation properties under the substitutions of the generators of the modular group

It is namely:

And