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Fungrim entry: 5161ab

G2 ⁣(aτ+bcτ+d)=(cτ+d)2G2 ⁣(τ)2πic(cτ+d)G_{2}\!\left(\frac{a \tau + b}{c \tau + d}\right) = {\left(c \tau + d\right)}^{2} G_{2}\!\left(\tau\right) - 2 \pi i c \left(c \tau + d\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})
TeX:
G_{2}\!\left(\frac{a \tau + b}{c \tau + d}\right) = {\left(c \tau + d\right)}^{2} G_{2}\!\left(\tau\right) - 2 \pi i c \left(c \tau + d\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
EisensteinGGk ⁣(τ)G_{k}\!\left(\tau\right) Eisenstein series
Powab{a}^{b} Power
ConstPiπ\pi The constant pi (3.14...)
ConstIii Imaginary unit
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:
Entry(ID("5161ab"),
    Formula(Equal(EisensteinG(2, Div(Add(Mul(a, tau), b), Add(Mul(c, tau), d))), Sub(Mul(Pow(Add(Mul(c, tau), d), 2), EisensteinG(2, tau)), Mul(Mul(Mul(Mul(2, ConstPi), ConstI), c), Add(Mul(c, tau), d))))),
    Variables(tau, a, b, c, d),
    Assumptions(And(Element(tau, HH), Element(Matrix2x2(a, b, c, d), SL2Z))))

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2019-08-19 14:38:23.809000 UTC