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Fungrim entry: b7174d

τ=iK ⁣(1λ ⁣(τ))K ⁣(λ ⁣(τ))\tau = i \frac{K\!\left(1 - \lambda\!\left(\tau\right)\right)}{K\!\left(\lambda\!\left(\tau\right)\right)}
Assumptions:τInterior ⁣(Fλ){τ:τHandRe ⁣(τ)=1}\tau \in \operatorname{Interior}\!\left(\mathcal{F}_{\lambda}\right) \cup \left\{ \tau : \tau \in \mathbb{H} \,\mathbin{\operatorname{and}}\, \operatorname{Re}\!\left(\tau\right) = 1 \right\}
TeX:
\tau = i \frac{K\!\left(1 - \lambda\!\left(\tau\right)\right)}{K\!\left(\lambda\!\left(\tau\right)\right)}

\tau \in \operatorname{Interior}\!\left(\mathcal{F}_{\lambda}\right) \cup \left\{ \tau : \tau \in \mathbb{H} \,\mathbin{\operatorname{and}}\, \operatorname{Re}\!\left(\tau\right) = 1 \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
ModularLambdaFundamentalDomainFλ\mathcal{F}_{\lambda} Fundamental domain of the modular lambda function
SetBuilder{f ⁣(x):P ⁣(x)}\left\{ f\!\left(x\right) : P\!\left(x\right) \right\} Set comprehension
HHH\mathbb{H} Upper complex half-plane
ReRe ⁣(z)\operatorname{Re}\!\left(z\right) Real part
Source code for this entry:
Entry(ID("b7174d"),
    Formula(Equal(tau, Mul(ConstI, Div(EllipticK(Sub(1, ModularLambda(tau))), EllipticK(ModularLambda(tau)))))),
    Variables(tau),
    Assumptions(Element(tau, Union(Interior(ModularLambdaFundamentalDomain), SetBuilder(tau, tau, And(Element(tau, HH), Equal(Re(tau), 1)))))))

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

2019-09-15 14:14:26.267625 UTC