# 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"),
Assumptions(And(Element(tau, HH), Element(Matrix2x2(a, b, c, d), PSL2Z))))