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Fungrim entry: 171724

E6 ⁣(τ)=1+63m=12cos4 ⁣(πmτ)+11cos2 ⁣(πmτ)+2sin6 ⁣(πmτ)E_{6}\!\left(\tau\right) = 1 + 63 \sum_{m=1}^{\infty} \frac{2 \cos^{4}\!\left(\pi m \tau\right) + 11 \cos^{2}\!\left(\pi m \tau\right) + 2}{\sin^{6}\!\left(\pi m \tau\right)}
Assumptions:τH\tau \in \mathbb{H}
TeX:
E_{6}\!\left(\tau\right) = 1 + 63 \sum_{m=1}^{\infty} \frac{2 \cos^{4}\!\left(\pi m \tau\right) + 11 \cos^{2}\!\left(\pi m \tau\right) + 2}{\sin^{6}\!\left(\pi m \tau\right)}

\tau \in \mathbb{H}
Definitions:
Fungrim symbol Notation Short description
EisensteinEEk ⁣(τ)E_{k}\!\left(\tau\right) Normalized Eisenstein series
Sumnf ⁣(n)\sum_{n} f\!\left(n\right) Sum
Powab{a}^{b} Power
ConstPiπ\pi The constant pi (3.14...)
Sinsin ⁣(z)\sin\!\left(z\right) Sine
Infinity\infty Positive infinity
HHH\mathbb{H} Upper complex half-plane
Source code for this entry:
Entry(ID("171724"),
    Formula(Equal(EisensteinE(6, tau), Add(1, Mul(63, Sum(Div(Add(Add(Mul(2, Pow(Cos(Mul(Mul(ConstPi, m), tau)), 4)), Mul(11, Pow(Cos(Mul(Mul(ConstPi, m), tau)), 2))), 2), Pow(Sin(Mul(Mul(ConstPi, m), tau)), 6)), Tuple(m, 1, Infinity)))))),
    Variables(tau),
    Assumptions(And(Element(tau, HH))))

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2019-08-21 11:44:15.926409 UTC