Fungrim home page

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:τHandRe(τ)=1}\tau \in \operatorname{Interior}(\mathcal{F}_{\lambda}) \cup \left\{ \tau : \tau \in \mathbb{H} \,\mathbin{\operatorname{and}}\, \operatorname{Re}(\tau) = 1 \right\}
TeX:
\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\}
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
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:
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), Set(tau, For(tau), And(Element(tau, HH), Equal(Re(tau), 1)))))))

Topics using this entry

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