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Fungrim entry: 5d550c

τ=iK ⁣(1λ(τ))K ⁣(λ(τ))+212Re(τ)12\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:τ{τ1+n:τ1Interior(Fλ)andnZ}\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
ConstIii Imaginary unit
EllipticKK(m)K(m) Complete elliptic integral of the first kind
ModularLambdaλ(τ)\lambda(\tau) Modular lambda function
ReRe(z)\operatorname{Re}(z) Real part
ModularLambdaFundamentalDomainFλ\mathcal{F}_{\lambda} Fundamental domain of the modular lambda function
ZZZ\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))))))

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

2019-12-30 15:00:46.909060 UTC