# 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

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

2021-03-15 19:12:00.328586 UTC