# Modular transformations

Symbol: SL2Z $\operatorname{SL}_2(\mathbb{Z})$ Modular group
Symbol: PSL2Z $\operatorname{PSL}_2(\mathbb{Z})$ Modular group (canonical representatives)
Symbol: ModularGroupAction $\gamma \circ \tau$ Action of modular group
Symbol: ModularGroupFundamentalDomain $\mathcal{F}$ Fundamental domain for action of the modular group

## Basic formulas

$\mathbb{H} = \left\{ \tau : \tau \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \operatorname{Im}\!\left(\tau\right) \gt 0 \right\}$
$\operatorname{SL}_2(\mathbb{Z}) = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} : a \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, b \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, c \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, d \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, a d - b c = 1 \right\}$
$\operatorname{PSL}_2(\mathbb{Z}) = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} : \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \operatorname{SL}_2(\mathbb{Z}) \,\mathbin{\operatorname{and}}\, \left(c \gt 0 \,\mathbin{\operatorname{or}}\, \left(c = 0 \,\mathbin{\operatorname{and}}\, d \gt 0\right)\right) \right\}$
$\begin{pmatrix} a & b \\ c & d \end{pmatrix} \circ \tau = \frac{a \tau + b}{c \tau + d}$
$\left(\gamma \eta\right) \circ \tau = \gamma \circ \left(\eta \circ \tau\right)$

## Fundamental domain

$\mathcal{F} = \left\{ \tau : \tau \in \mathbb{H} \,\mathbin{\operatorname{and}}\, \operatorname{Re}\!\left(\tau\right) \in \left[-\frac{1}{2}, \frac{1}{2}\right) \,\mathbin{\operatorname{and}}\, \left(\left|\tau\right| \gt 1 \,\mathbin{\operatorname{or}}\, \left(\left|\tau\right| = 1 \,\mathbin{\operatorname{and}}\, \operatorname{Re}\!\left(\tau\right) \le 0\right)\right) \right\}$
$i \in \mathcal{F}$
${e}^{2 \pi i / 3} = \frac{-1 + \sqrt{3} i}{2} \in \mathcal{F}$
$\left|{e}^{2 \pi i \tau}\right| \lt 0.004334$
$\left\{ \gamma \circ \tau : \tau \in \mathcal{F} \,\mathbin{\operatorname{and}}\, \gamma \in \operatorname{PSL}_2(\mathbb{Z}) \right\} = \mathbb{H}$

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

2019-06-18 07:49:59.356594 UTC