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Fungrim entry: 29d9ab

η24 ⁣(aτ+bcτ+d)=(cτ+d)12η24 ⁣(τ)\eta^{24}\!\left(\frac{a \tau + b}{c \tau + d}\right) = {\left(c \tau + d\right)}^{12} \eta^{24}\!\left(\tau\right)
Assumptions:τHand(abcd)SL2(Z)\tau \in \mathbb{H} \,\mathbin{\operatorname{and}}\, \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \operatorname{SL}_2(\mathbb{Z})
TeX:
\eta^{24}\!\left(\frac{a \tau + b}{c \tau + d}\right) = {\left(c \tau + d\right)}^{12} \eta^{24}\!\left(\tau\right)

\tau \in \mathbb{H} \,\mathbin{\operatorname{and}}\, \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \operatorname{SL}_2(\mathbb{Z})
Definitions:
Fungrim symbol Notation Short description
Powab{a}^{b} Power
DedekindEtaη(τ)\eta(\tau) Dedekind eta function
HHH\mathbb{H} Upper complex half-plane
Matrix2x2(abcd)\begin{pmatrix} a & b \\ c & d \end{pmatrix} Two by two matrix
SL2ZSL2(Z)\operatorname{SL}_2(\mathbb{Z}) Modular group
Source code for this entry:
Entry(ID("29d9ab"),
    Formula(Equal(Pow(DedekindEta(Div(Add(Mul(a, tau), b), Add(Mul(c, tau), d))), 24), Mul(Pow(Add(Mul(c, tau), d), 12), Pow(DedekindEta(tau), 24)))),
    Variables(tau, a, b, c, d),
    Assumptions(And(Element(tau, HH), Element(Matrix2x2(a, b, c, d), SL2Z))))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-10-05 13:11:19.856591 UTC