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Fungrim entry: 73427b

λ ⁣(aτ+bcτ+d){λ ⁣(τ),1λ ⁣(τ),1λ ⁣(τ),11λ ⁣(τ),λ ⁣(τ)1λ ⁣(τ),λ ⁣(τ)λ ⁣(τ)1}\lambda\!\left(\frac{a \tau + b}{c \tau + d}\right) \in \left\{\lambda\!\left(\tau\right), 1 - \lambda\!\left(\tau\right), \frac{1}{\lambda\!\left(\tau\right)}, \frac{1}{1 - \lambda\!\left(\tau\right)}, \frac{\lambda\!\left(\tau\right) - 1}{\lambda\!\left(\tau\right)}, \frac{\lambda\!\left(\tau\right)}{\lambda\!\left(\tau\right) - 1}\right\}
Assumptions:τHand(abcd)SL2(Z)\tau \in \mathbb{H} \,\mathbin{\operatorname{and}}\, \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \operatorname{SL}_2(\mathbb{Z})
\lambda\!\left(\frac{a \tau + b}{c \tau + d}\right) \in \left\{\lambda\!\left(\tau\right), 1 - \lambda\!\left(\tau\right), \frac{1}{\lambda\!\left(\tau\right)}, \frac{1}{1 - \lambda\!\left(\tau\right)}, \frac{\lambda\!\left(\tau\right) - 1}{\lambda\!\left(\tau\right)}, \frac{\lambda\!\left(\tau\right)}{\lambda\!\left(\tau\right) - 1}\right\}

\tau \in \mathbb{H} \,\mathbin{\operatorname{and}}\, \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \operatorname{SL}_2(\mathbb{Z})
Fungrim symbol Notation Short description
ModularLambdaλ ⁣(τ)\lambda\!\left(\tau\right) Modular lambda function
HHH\mathbb{H} Upper complex half-plane
Matrix2x2(abcd)\begin{pmatrix} a & b \\ c & d \end{pmatrix} Two by two matrix
SL2ZSL2(Z)\operatorname{SL}_2(\mathbb{Z}) Modular group
Source code for this entry:
    Formula(Element(ModularLambda(Div(Add(Mul(a, tau), b), Add(Mul(c, tau), d))), Set(ModularLambda(tau), Sub(1, ModularLambda(tau)), Div(1, ModularLambda(tau)), Div(1, Sub(1, ModularLambda(tau))), Div(Sub(ModularLambda(tau), 1), ModularLambda(tau)), Div(ModularLambda(tau), Sub(ModularLambda(tau), 1))))),
    Variables(tau, a, b, c, d),
    Assumptions(And(Element(tau, HH), Element(Matrix2x2(a, b, c, d), SL2Z))))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-09-19 20:12:49.583742 UTC