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Fungrim entry: 737f2b

Fλ={τ:τHand((Re ⁣(τ)(1,1)andmin ⁣(τ12,z+12)>12)orRe ⁣(τ)=1orτ+12=12)}\mathcal{F}_{\lambda} = \left\{ \tau : \tau \in \mathbb{H} \,\mathbin{\operatorname{and}}\, \left(\left(\operatorname{Re}\!\left(\tau\right) \in \left(-1, 1\right) \,\mathbin{\operatorname{and}}\, \min\!\left(\left|\tau - \frac{1}{2}\right|, \left|z + \frac{1}{2}\right|\right) > \frac{1}{2}\right) \,\mathbin{\operatorname{or}}\, \operatorname{Re}\!\left(\tau\right) = -1 \,\mathbin{\operatorname{or}}\, \left|\tau + \frac{1}{2}\right| = \frac{1}{2}\right) \right\}
References:
  • J. M. Borwein and P. B. Borwein. Pi and the AGM. Wiley, New York, 1987. p. 113.
TeX:
\mathcal{F}_{\lambda} = \left\{ \tau : \tau \in \mathbb{H} \,\mathbin{\operatorname{and}}\, \left(\left(\operatorname{Re}\!\left(\tau\right) \in \left(-1, 1\right) \,\mathbin{\operatorname{and}}\, \min\!\left(\left|\tau - \frac{1}{2}\right|, \left|z + \frac{1}{2}\right|\right) > \frac{1}{2}\right) \,\mathbin{\operatorname{or}}\, \operatorname{Re}\!\left(\tau\right) = -1 \,\mathbin{\operatorname{or}}\, \left|\tau + \frac{1}{2}\right| = \frac{1}{2}\right) \right\}
Definitions:
Fungrim symbol Notation Short description
ModularLambdaFundamentalDomainFλ\mathcal{F}_{\lambda} Fundamental domain of the modular lambda function
SetBuilder{f ⁣(x):P ⁣(x)}\left\{ f\!\left(x\right) : P\!\left(x\right) \right\} Set comprehension
HHH\mathbb{H} Upper complex half-plane
ReRe ⁣(z)\operatorname{Re}\!\left(z\right) Real part
OpenInterval(a,b)\left(a, b\right) Open interval
Absz\left|z\right| Absolute value
Source code for this entry:
Entry(ID("737f2b"),
    Formula(Equal(ModularLambdaFundamentalDomain, SetBuilder(tau, tau, And(Element(tau, HH), Or(And(Element(Re(tau), OpenInterval(-1, 1)), Greater(Min(Abs(Sub(tau, Div(1, 2))), Abs(Add(z, Div(1, 2)))), Div(1, 2))), Equal(Re(tau), -1), Equal(Abs(Add(tau, Div(1, 2))), Div(1, 2))))))),
    References("J. M. Borwein and P. B. Borwein. Pi and the AGM. Wiley, New York, 1987. p. 113."))

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2019-09-15 11:00:55.020619 UTC