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Concept
Dedekind eta function
Summary
In mathematics, the Dedekind eta function, named after Richard Dedekind, is a modular form of weight 1/2 and is a function defined on the upper half-plane of complex numbers, where the imaginary part is positive. It also occurs in bosonic string theory.
For any complex number τ with Im(τ) > 0, let q = e2πiτ; then the eta function is defined by,
Raising the eta equation to the 24th power and multiplying by (2π)12 gives
where Δ is the modular discriminant. The presence of 24 can be understood by connection with other occurrences, such as in the 24-dimensional Leech lattice.
The eta function is holomorphic on the upper half-plane but cannot be continued analytically beyond it.
The eta function satisfies the functional equations
In the second equation the branch of the square root is chosen such that = 1 when τ = i.
More generally, suppose a, b, c, d are integers with ad − bc = 1, so that
is a transformation belonging to the modular group. We may assume that either c > 0, or c = 0 and d = 1. Then
where
Here s(h,k) is the Dedekind sum
Because of these functional equations the eta function is a modular form of weight 1/2 and level 1 for a certain character of order 24 of the metaplectic double cover of the modular group, and can be used to define other modular forms. In particular the modular discriminant of Weierstrass can be defined as
and is a modular form of weight 12. Some authors omit the factor of (2π)12, so that the series expansion has integral coefficients.
The Jacobi triple product implies that the eta is (up to a factor) a Jacobi theta function for special values of the arguments:
where χ(n) is "the" Dirichlet character modulo 12 with χ(±1) = 1 and χ(±5) = −1. Explicitly,
The Euler function
has a power series by the Euler identity:
Because the eta function is easy to compute numerically from either power series, it is often helpful in computation 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.
Official source
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