Fungrim entry: 9f19c1

\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:

\tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \operatorname{PSL}_2(\mathbb{Z})

TeX:

\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})

Definitions:

Fungrim symbol	Notation	Short description
`DedekindEta`	$\eta(\tau)$	Dedekind eta function
`DedekindEtaEpsilon`	$\varepsilon\!\left(a, b, c, d\right)$	Root of unity in the functional equation of the Dedekind eta function
`Pow`	${a}^{b}$	Power
`HH`	$\mathbb{H}$	Upper complex half-plane
`Matrix2x2`	$\begin{pmatrix} a & b \\ c & d \end{pmatrix}$	Two by two matrix
`PSL2Z`	$\operatorname{PSL}_2(\mathbb{Z})$	Modular group (canonical representatives)

Source code for this entry:

Entry(ID("9f19c1"),
    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))))

Topics using this entry

Dedekind eta function