# Fungrim entry: b7174d

$\tau = i \frac{K\!\left(1 - \lambda\!\left(\tau\right)\right)}{K\!\left(\lambda\!\left(\tau\right)\right)}$
Assumptions:$\tau \in \operatorname{Interior}\!\left(\mathcal{F}_{\lambda}\right) \cup \left\{ \tau : \tau \in \mathbb{H} \,\mathbin{\operatorname{and}}\, \operatorname{Re}\!\left(\tau\right) = 1 \right\}$
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
ConstI$i$ Imaginary unit
EllipticK$K\!\left(m\right)$ Complete elliptic integral of the first kind
ModularLambda$\lambda\!\left(\tau\right)$ Modular lambda function
ModularLambdaFundamentalDomain$\mathcal{F}_{\lambda}$ Fundamental domain of the modular lambda function
SetBuilder$\left\{ f\!\left(x\right) : P\!\left(x\right) \right\}$ Set comprehension
HH$\mathbb{H}$ Upper complex half-plane
Re$\operatorname{Re}\!\left(z\right)$ 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)))))))

## Topics using this entry

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