Fungrim entry: 5d550c

\tau = i \frac{K\!\left(1 - \lambda(\tau)\right)}{K\!\left(\lambda(\tau)\right)} + 2 \left\lceil \frac{1}{2} \operatorname{Re}(\tau) - \frac{1}{2} \right\rceil

Assumptions:

\tau \in \left\{ {\tau}_{1} + n : {\tau}_{1} \in \operatorname{Interior}(\mathcal{F}_{\lambda}) \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z} \right\}

TeX:

\tau = i \frac{K\!\left(1 - \lambda(\tau)\right)}{K\!\left(\lambda(\tau)\right)} + 2 \left\lceil \frac{1}{2} \operatorname{Re}(\tau) - \frac{1}{2} \right\rceil

\tau \in \left\{ {\tau}_{1} + n : {\tau}_{1} \in \operatorname{Interior}(\mathcal{F}_{\lambda}) \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z} \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
`Re`	$\operatorname{Re}(z)$	Real part
`ModularLambdaFundamentalDomain`	$\mathcal{F}_{\lambda}$	Fundamental domain of the modular lambda function
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("5d550c"),
    Formula(Equal(tau, Add(Mul(ConstI, Div(EllipticK(Sub(1, ModularLambda(tau))), EllipticK(ModularLambda(tau)))), Mul(2, Ceil(Sub(Mul(Div(1, 2), Re(tau)), Div(1, 2))))))),
    Variables(tau),
    Assumptions(Element(tau, Set(Add(Subscript(tau, 1), n), For(Tuple(Subscript(tau, 1), n)), And(Element(Subscript(tau, 1), Interior(ModularLambdaFundamentalDomain)), Element(n, ZZ))))))

Topics using this entry

Modular lambda function