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Fungrim entry: b7174d

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

\tau \in \operatorname{Interior}(\mathcal{F}_{\lambda}) \cup \left\{ \tau : \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(\tau) = 1 \right\}
Fungrim symbol Notation Short description
ConstIii Imaginary unit
EllipticKK(m)K(m) Legendre complete elliptic integral of the first kind
ModularLambdaλ(τ)\lambda(\tau) Modular lambda function
ModularLambdaFundamentalDomainFλ\mathcal{F}_{\lambda} Fundamental domain of the modular lambda function
HHH\mathbb{H} Upper complex half-plane
ReRe(z)\operatorname{Re}(z) Real part
Source code for this entry:
    Formula(Equal(tau, Mul(ConstI, Div(EllipticK(Sub(1, ModularLambda(tau))), EllipticK(ModularLambda(tau)))))),
    Assumptions(Element(tau, Union(Interior(ModularLambdaFundamentalDomain), Set(tau, For(tau), And(Element(tau, HH), Equal(Re(tau), 1)))))))

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2021-03-15 19:12:00.328586 UTC