# Fungrim entry: 5d550c

$\tau = i \frac{K\!\left(1 - \lambda(\tau)\right)}{K\!\left(\lambda(\tau)\right)} + 2 \left\lceil \frac{1}{2} \operatorname{Re}(\tau) - \frac{1}{2} \right\rceil$
Assumptions:$\tau \in \left\{ {\tau}_{1} + n : {\tau}_{1} \in \operatorname{Interior}(\mathcal{F}_{\lambda}) \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z} \right\}$
TeX:
\tau = i \frac{K\!\left(1 - \lambda(\tau)\right)}{K\!\left(\lambda(\tau)\right)} + 2 \left\lceil \frac{1}{2} \operatorname{Re}(\tau) - \frac{1}{2} \right\rceil

\tau \in \left\{ {\tau}_{1} + n : {\tau}_{1} \in \operatorname{Interior}(\mathcal{F}_{\lambda}) \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z} \right\}
Definitions:
Fungrim symbol Notation Short description
ConstI$i$ Imaginary unit
EllipticK$K(m)$ Legendre complete elliptic integral of the first kind
ModularLambda$\lambda(\tau)$ Modular lambda function
Re$\operatorname{Re}(z)$ Real part
ModularLambdaFundamentalDomain$\mathcal{F}_{\lambda}$ Fundamental domain of the modular lambda function
ZZ$\mathbb{Z}$ Integers
Source code for this entry:
Entry(ID("5d550c"),
Formula(Equal(tau, Add(Mul(ConstI, Div(EllipticK(Sub(1, ModularLambda(tau))), EllipticK(ModularLambda(tau)))), Mul(2, Ceil(Sub(Mul(Div(1, 2), Re(tau)), Div(1, 2))))))),
Variables(tau),
Assumptions(Element(tau, Set(Add(Subscript(tau, 1), n), For(Tuple(Subscript(tau, 1), n)), And(Element(Subscript(tau, 1), Interior(ModularLambdaFundamentalDomain)), Element(n, ZZ))))))

## Topics using this entry

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