Dedekind eta function

The Dedekind eta function is a function defined on the upper half plane of complex numbers whose imaginary part is positive. For any such complex number z, we may set q = e^2πzi, and define the eta function by

$\eta(z) = q^{1/24} \prod_{n=1}^{\infty} (1-q^n)$

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

The eta function satisfies the functional equations

η(z + 1) = exp(2πi / 24)η(z)

$\eta(-1/z) = \sqrt {-iz} \eta(z)$

Because of these functional equations the eta function is a modular form of weight 1/2, and can be used to define other modular forms. In particular the modular discriminant of Weierstrass can be defined as

Δ(z) = (2π)¹²η(z)²⁴

and is a modular form of weight 12. Because the eta function is easy to compute, it is often helpful to express other functions in terms of it when possible, and products and quotients of eta functions, called eta quotients, can be used to express a great variety of modular forms.

Categories: Modular forms | Special functions

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