Assumptions:
TeX:
\lambda\!\left(\frac{a \tau + b}{c \tau + d}\right) = \begin{cases} \lambda(\tau), & \left(a, b, c, d\right) \equiv \left(1, 0, 0, 1\right) \pmod {2}\\1 - \lambda(\tau), & \left(a, b, c, d\right) \equiv \left(0, 1, 1, 0\right) \pmod {2}\\\frac{1}{\lambda(\tau)}, & \left(a, b, c, d\right) \equiv \left(1, 0, 1, 1\right) \pmod {2}\\\frac{1}{1 - \lambda(\tau)}, & \left(a, b, c, d\right) \equiv \left(0, 1, 1, 1\right) \pmod {2}\\\frac{\lambda(\tau) - 1}{\lambda(\tau)}, & \left(a, b, c, d\right) \equiv \left(1, 1, 1, 0\right) \pmod {2}\\\frac{\lambda(\tau)}{\lambda(\tau) - 1}, & \left(a, b, c, d\right) \equiv \left(1, 1, 0, 1\right) \pmod {2}\\ \end{cases}
\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 | Modular lambda function | |
| HH | Upper complex half-plane | |
| Matrix2x2 | Two by two matrix | |
| SL2Z | Modular group |
Source code for this entry:
Entry(ID("099301"),
Formula(Equal(ModularLambda(Div(Add(Mul(a, tau), b), Add(Mul(c, tau), d))), Cases(Tuple(ModularLambda(tau), CongruentMod(Tuple(a, b, c, d), Tuple(1, 0, 0, 1), 2)), Tuple(Sub(1, ModularLambda(tau)), CongruentMod(Tuple(a, b, c, d), Tuple(0, 1, 1, 0), 2)), Tuple(Div(1, ModularLambda(tau)), CongruentMod(Tuple(a, b, c, d), Tuple(1, 0, 1, 1), 2)), Tuple(Div(1, Sub(1, ModularLambda(tau))), CongruentMod(Tuple(a, b, c, d), Tuple(0, 1, 1, 1), 2)), Tuple(Div(Sub(ModularLambda(tau), 1), ModularLambda(tau)), CongruentMod(Tuple(a, b, c, d), Tuple(1, 1, 1, 0), 2)), Tuple(Div(ModularLambda(tau), Sub(ModularLambda(tau), 1)), CongruentMod(Tuple(a, b, c, d), Tuple(1, 1, 0, 1), 2))))),
Variables(tau, a, b, c, d),
Assumptions(And(Element(tau, HH), Element(Matrix2x2(a, b, c, d), SL2Z))))