Fungrim entry: 73427b

\lambda\!\left(\frac{a \tau + b}{c \tau + d}\right) \in \left\{\lambda(\tau), 1 - \lambda(\tau), \frac{1}{\lambda(\tau)}, \frac{1}{1 - \lambda(\tau)}, \frac{\lambda(\tau) - 1}{\lambda(\tau)}, \frac{\lambda(\tau)}{\lambda(\tau) - 1}\right\}

Assumptions:

\tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \operatorname{SL}_2(\mathbb{Z})

TeX:

\lambda\!\left(\frac{a \tau + b}{c \tau + d}\right) \in \left\{\lambda(\tau), 1 - \lambda(\tau), \frac{1}{\lambda(\tau)}, \frac{1}{1 - \lambda(\tau)}, \frac{\lambda(\tau) - 1}{\lambda(\tau)}, \frac{\lambda(\tau)}{\lambda(\tau) - 1}\right\}

\tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \operatorname{SL}_2(\mathbb{Z})

Definitions:

Fungrim symbol	Notation	Short description
`ModularLambda`	$\lambda(\tau)$	Modular lambda function
`HH`	$\mathbb{H}$	Upper complex half-plane
`Matrix2x2`	$\begin{pmatrix} a & b \\ c & d \end{pmatrix}$	Two by two matrix
`SL2Z`	$\operatorname{SL}_2(\mathbb{Z})$	Modular group

Source code for this entry:

Entry(ID("73427b"),
    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))))

Topics using this entry

Modular lambda function