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

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

Fundamental domain

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

Ford circles

Area of Ford circles on the unit square

$\frac{\pi}{4} \sum_{q=1}^{\infty} \frac{\varphi(q)}{{q}^{4}} = \frac{\pi}{4} \frac{\zeta\!\left(3\right)}{\zeta\!\left(4\right)} = \frac{45 \zeta\!\left(3\right)}{2 {\pi}^{3}}$

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

2020-04-08 16:14:44.404316 UTC