Fungrim entry: 26fd1b

\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 \in \left(0, 1\right)

TeX:

\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)

Definitions:

Fungrim symbol	Notation	Short description
`Log`	$\log(z)$	Natural logarithm
`Pi`	$\pi$	The constant pi (3.14...)
`AGM`	$\operatorname{agm}\!\left(a, b\right)$	Arithmetic-geometric mean
`Pow`	${a}^{b}$	Power
`OpenInterval`	$\left(a, b\right)$	Open interval

Source code for this entry:

Entry(ID("26fd1b"),
    Formula(Equal(Log(Div(1, q)), Div(Pi, AGM(Pow(JacobiThetaQ(2, 0, q), 2), Pow(JacobiThetaQ(3, 0, q), 2))))),
    Variables(q),
    Assumptions(Element(q, OpenInterval(0, 1))))

Topics using this entry

Arithmetic-geometric mean

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