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Fungrim entry: 9f19c1

η ⁣(aτ+bcτ+d)=ε ⁣(a,b,c,d)(cτ+d)1/2η(τ)\eta\!\left(\frac{a \tau + b}{c \tau + d}\right) = \varepsilon\!\left(a, b, c, d\right) {\left(c \tau + d\right)}^{1 / 2} \eta(\tau)
Assumptions:τH  and  (abcd)PSL2(Z)\tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \operatorname{PSL}_2(\mathbb{Z})
\eta\!\left(\frac{a \tau + b}{c \tau + d}\right) = \varepsilon\!\left(a, b, c, d\right) {\left(c \tau + d\right)}^{1 / 2} \eta(\tau)

\tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \operatorname{PSL}_2(\mathbb{Z})
Fungrim symbol Notation Short description
DedekindEtaη(τ)\eta(\tau) Dedekind eta function
DedekindEtaEpsilonε ⁣(a,b,c,d)\varepsilon\!\left(a, b, c, d\right) Root of unity in the functional equation of the Dedekind eta function
Powab{a}^{b} Power
HHH\mathbb{H} Upper complex half-plane
Matrix2x2(abcd)\begin{pmatrix} a & b \\ c & d \end{pmatrix} Two by two matrix
PSL2ZPSL2(Z)\operatorname{PSL}_2(\mathbb{Z}) Modular group (canonical representatives)
Source code for this entry:
    Formula(Equal(DedekindEta(Div(Add(Mul(a, tau), b), Add(Mul(c, tau), d))), Mul(Mul(DedekindEtaEpsilon(a, b, c, d), Pow(Add(Mul(c, tau), d), Div(1, 2))), DedekindEta(tau)))),
    Variables(tau, a, b, c, d),
    Assumptions(And(Element(tau, HH), Element(Matrix2x2(a, b, c, d), PSL2Z))))

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2021-03-15 19:12:00.328586 UTC