Fungrim entry: 737f2b

\mathcal{F}_{\lambda} = \left\{ \tau : \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; \left(\left(\operatorname{Re}(\tau) \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}(\tau) = -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}(\tau) \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}(\tau) = -1 \;\mathbin{\operatorname{or}}\; \left|\tau + \frac{1}{2}\right| = \frac{1}{2}\right) \right\}

Definitions:

Fungrim symbol	Notation	Short description
`ModularLambdaFundamentalDomain`	$\mathcal{F}_{\lambda}$	Fundamental domain of the modular lambda function
`HH`	$\mathbb{H}$	Upper complex half-plane
`Re`	$\operatorname{Re}(z)$	Real part
`OpenInterval`	$\left(a, b\right)$	Open interval
`Abs`	$\left\|z\right\|$	Absolute value

Source code for this entry:

Entry(ID("737f2b"),
    Formula(Equal(ModularLambdaFundamentalDomain, Set(tau, For(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."))

Topics using this entry

Modular lambda function