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Fungrim entry: 26fd1b

log ⁣(1q)=πagm ⁣(θ22 ⁣(0,q),θ32 ⁣(0,q))\log\!\left(\frac{1}{q}\right) = \frac{\pi}{\operatorname{agm}\!\left(\theta_{2}^{2}\!\left(0, q\right), \theta_{3}^{2}\!\left(0, q\right)\right)}
Assumptions:q(0,1)q \in \left(0, 1\right)
\log\!\left(\frac{1}{q}\right) = \frac{\pi}{\operatorname{agm}\!\left(\theta_{2}^{2}\!\left(0, q\right), \theta_{3}^{2}\!\left(0, q\right)\right)}

q \in \left(0, 1\right)
Fungrim symbol Notation Short description
Loglog(z)\log(z) Natural logarithm
Piπ\pi The constant pi (3.14...)
AGMagm ⁣(a,b)\operatorname{agm}\!\left(a, b\right) Arithmetic-geometric mean
Powab{a}^{b} Power
OpenInterval(a,b)\left(a, b\right) Open interval
Source code for this entry:
    Formula(Equal(Log(Div(1, q)), Div(Pi, AGM(Pow(JacobiThetaQ(2, 0, q), 2), Pow(JacobiThetaQ(3, 0, q), 2))))),
    Assumptions(Element(q, OpenInterval(0, 1))))

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

2021-03-15 19:12:00.328586 UTC