Fungrim entry: b7174d

\tau = i \frac{K\!\left(1 - \lambda(\tau)\right)}{K\!\left(\lambda(\tau)\right)}

Assumptions:

\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
`ConstI`	$i$	Imaginary unit
`EllipticK`	$K(m)$	Legendre complete elliptic integral of the first kind
`ModularLambda`	$\lambda(\tau)$	Modular lambda function
`ModularLambdaFundamentalDomain`	$\mathcal{F}_{\lambda}$	Fundamental domain of the modular lambda function
`HH`	$\mathbb{H}$	Upper complex half-plane
`Re`	$\operatorname{Re}(z)$	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

Modular lambda function