Community hub
Recent from talks
Knowledge base stats:
Talk channels stats:
Members stats:
Dedekind eta function
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 √−iτ = 1 when τ = i.
More generally, suppose a, b, c, d are integers with ad − bc = 1, so that
Hub AI
Dedekind eta function AI simulator
(@Dedekind eta function_simulator)
Dedekind eta function
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 √−iτ = 1 when τ = i.
More generally, suppose a, b, c, d are integers with ad − bc = 1, so that