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Fungrim entry: b1a5e4

H ⁣(aτ+bcτ+d)=(cτ+d)2H ⁣(τ)   where H ⁣(τ)=G2 ⁣(τ)πIm ⁣(τ)H\!\left(\frac{a \tau + b}{c \tau + d}\right) = {\left(c \tau + d\right)}^{2} H\!\left(\tau\right)\; \text{ where } H\!\left(\tau\right) = G_{2}\!\left(\tau\right) - \frac{\pi}{\operatorname{Im}\!\left(\tau\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:
H\!\left(\frac{a \tau + b}{c \tau + d}\right) = {\left(c \tau + d\right)}^{2} H\!\left(\tau\right)\; \text{ where } H\!\left(\tau\right) = G_{2}\!\left(\tau\right) - \frac{\pi}{\operatorname{Im}\!\left(\tau\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
Powab{a}^{b} Power
EisensteinGGk ⁣(τ)G_{k}\!\left(\tau\right) Eisenstein series
ConstPiπ\pi The constant pi (3.14...)
ImIm ⁣(z)\operatorname{Im}\!\left(z\right) Imaginary part
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("b1a5e4"),
    Formula(Where(Equal(H(Div(Add(Mul(a, tau), b), Add(Mul(c, tau), d))), Mul(Pow(Add(Mul(c, tau), d), 2), H(tau))), Equal(H(tau), Sub(EisensteinG(2, tau), Div(ConstPi, Im(tau)))))),
    Variables(tau, a, b, c, d),
    Assumptions(And(Element(tau, HH), Element(Matrix2x2(a, b, c, d), SL2Z))))

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

2019-08-19 14:38:23.809000 UTC