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Fungrim entry: 5d550c

τ=iK ⁣(1λ ⁣(τ))K ⁣(λ ⁣(τ))+212Re ⁣(τ)12\tau = i \frac{K\!\left(1 - \lambda\!\left(\tau\right)\right)}{K\!\left(\lambda\!\left(\tau\right)\right)} + 2 \left\lceil \frac{1}{2} \operatorname{Re}\!\left(\tau\right) - \frac{1}{2} \right\rceil
Assumptions:τ{τ1+n:τ1Interior ⁣(Fλ)andnZ}\tau \in \left\{ {\tau}_{1} + n : {\tau}_{1} \in \operatorname{Interior}\!\left(\mathcal{F}_{\lambda}\right) \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z} \right\}
TeX:
\tau = i \frac{K\!\left(1 - \lambda\!\left(\tau\right)\right)}{K\!\left(\lambda\!\left(\tau\right)\right)} + 2 \left\lceil \frac{1}{2} \operatorname{Re}\!\left(\tau\right) - \frac{1}{2} \right\rceil

\tau \in \left\{ {\tau}_{1} + n : {\tau}_{1} \in \operatorname{Interior}\!\left(\mathcal{F}_{\lambda}\right) \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z} \right\}
Definitions:
Fungrim symbol Notation Short description
ConstIii Imaginary unit
EllipticKK ⁣(m)K\!\left(m\right) Complete elliptic integral of the first kind
ModularLambdaλ ⁣(τ)\lambda\!\left(\tau\right) Modular lambda function
ReRe ⁣(z)\operatorname{Re}\!\left(z\right) Real part
SetBuilder{f ⁣(x):P ⁣(x)}\left\{ f\!\left(x\right) : P\!\left(x\right) \right\} Set comprehension
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, SetBuilder(Add(Subscript(tau, 1), n), Tuple(Subscript(tau, 1), n), And(Element(Subscript(tau, 1), Interior(ModularLambdaFundamentalDomain)), Element(n, ZZ))))))

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2019-09-15 14:14:26.267625 UTC