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Fungrim entry: 830dd4

{λ ⁣(τ):τInterior ⁣(Fλ)}=C((,0][1,))\left\{ \lambda\!\left(\tau\right) : \tau \in \operatorname{Interior}\!\left(\mathcal{F}_{\lambda}\right) \right\} = \mathbb{C} \setminus \left(\left(-\infty, 0\right] \cup \left[1, \infty\right)\right)
This mapping is one-to-one.
References:
  • J. M. Borwein and P. B. Borwein. Pi and the AGM. Wiley, New York, 1987. p. 118.
TeX:
\left\{ \lambda\!\left(\tau\right) : \tau \in \operatorname{Interior}\!\left(\mathcal{F}_{\lambda}\right) \right\} = \mathbb{C} \setminus \left(\left(-\infty, 0\right] \cup \left[1, \infty\right)\right)
Definitions:
Fungrim symbol Notation Short description
SetBuilder{f ⁣(x):P ⁣(x)}\left\{ f\!\left(x\right) : P\!\left(x\right) \right\} Set comprehension
ModularLambdaλ ⁣(τ)\lambda\!\left(\tau\right) Modular lambda function
ModularLambdaFundamentalDomainFλ\mathcal{F}_{\lambda} Fundamental domain of the modular lambda function
CCC\mathbb{C} Complex numbers
OpenClosedInterval(a,b]\left(a, b\right] Open-closed interval
Infinity\infty Positive infinity
ClosedOpenInterval[a,b)\left[a, b\right) Closed-open interval
Source code for this entry:
Entry(ID("830dd4"),
    Formula(Equal(SetBuilder(ModularLambda(tau), tau, Element(tau, Interior(ModularLambdaFundamentalDomain))), SetMinus(CC, Parentheses(Union(OpenClosedInterval(Neg(Infinity), 0), ClosedOpenInterval(1, Infinity)))))),
    Description("This mapping is one-to-one."),
    References("J. M. Borwein and P. B. Borwein. Pi and the AGM. Wiley, New York, 1987. p. 118."))

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

2019-09-15 11:00:55.020619 UTC