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Fungrim entry: 7f4c85

E2 ⁣(aτ+bcτ+d)=(cτ+d)2E2 ⁣(τ)6iπc(cτ+d)E_{2}\!\left(\frac{a \tau + b}{c \tau + d}\right) = {\left(c \tau + d\right)}^{2} E_{2}\!\left(\tau\right) - \frac{6 i}{\pi} 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:
E_{2}\!\left(\frac{a \tau + b}{c \tau + d}\right) = {\left(c \tau + d\right)}^{2} E_{2}\!\left(\tau\right) - \frac{6 i}{\pi} 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
EisensteinEEk ⁣(τ)E_{k}\!\left(\tau\right) Normalized Eisenstein series
Powab{a}^{b} Power
ConstIii Imaginary unit
ConstPiπ\pi The constant pi (3.14...)
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("7f4c85"),
    Formula(Equal(EisensteinE(2, Div(Add(Mul(a, tau), b), Add(Mul(c, tau), d))), Sub(Mul(Pow(Add(Mul(c, tau), d), 2), EisensteinE(2, tau)), Mul(Mul(Div(Mul(6, ConstI), ConstPi), 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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2019-10-05 13:11:19.856591 UTC