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Modular transformations

Table of contents: Basic formulas - Fundamental domain - Ford circles

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Symbol: SL2Z SL2(Z)\operatorname{SL}_2(\mathbb{Z}) Modular group
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Symbol: PSL2Z PSL2(Z)\operatorname{PSL}_2(\mathbb{Z}) Modular group (canonical representatives)
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Symbol: ModularGroupAction γτ\gamma \circ \tau Action of modular group
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Symbol: ModularGroupFundamentalDomain F\mathcal{F} Fundamental domain for action of the modular group

Basic formulas

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H={τ:τCandIm(τ)>0}\mathbb{H} = \left\{ \tau : \tau \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \operatorname{Im}(\tau) > 0 \right\}
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SL2(Z)={(abcd):aZ  and  bZ  and  cZ  and  dZ  and  adbc=1}\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\}
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PSL2(Z)={(abcd):(abcd)SL2(Z)  and  (c>0  or  (c=0  and  d>0))}\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\}
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(abcd)τ=aτ+bcτ+d\begin{pmatrix} a & b \\ c & d \end{pmatrix} \circ \tau = \frac{a \tau + b}{c \tau + d}
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(γη)τ=γ(ητ)\left(\gamma \eta\right) \circ \tau = \gamma \circ \left(\eta \circ \tau\right)

Fundamental domain

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F={τ:τH  and  Re(τ)[12,12)  and  (τ>1  or  (τ=1  and  Re(τ)0))}\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\}
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iFi \in \mathcal{F}
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e2πi/3=1+3i2F{e}^{2 \pi i / 3} = -\frac{1 + \sqrt{3} i}{2} \in \mathcal{F}
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e2πiτ<0.004334\left|{e}^{2 \pi i \tau}\right| < 0.004334
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{γτ:τF  and  γPSL2(Z)}=H\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

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π4q=1φ(q)q4=π4ζ ⁣(3)ζ ⁣(4)=45ζ ⁣(3)2π3\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