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Fungrim entry: 3bfced

E6(τ)=2πi(E2 ⁣(τ)E6 ⁣(τ)E42 ⁣(τ)2)E'_{6}(\tau) = 2 \pi i \left(\frac{E_{2}\!\left(\tau\right) E_{6}\!\left(\tau\right) - E_{4}^{2}\!\left(\tau\right)}{2}\right)
Assumptions:τH\tau \in \mathbb{H}
References:
  • B. C. Berndt and A. J. Yee (2002) Ramanujan's Contributions to Eisenstein Series, Especially in His Lost Notebook. In: Kanemitsu S., Jia C. (eds) Number Theoretic Methods. Developments in Mathematics, vol 8. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-3675-5_3
TeX:
E'_{6}(\tau) = 2 \pi i \left(\frac{E_{2}\!\left(\tau\right) E_{6}\!\left(\tau\right) - E_{4}^{2}\!\left(\tau\right)}{2}\right)

\tau \in \mathbb{H}
Definitions:
Fungrim symbol Notation Short description
ComplexDerivativeddzf ⁣(z)\frac{d}{d z}\, f\!\left(z\right) Complex derivative
EisensteinEEk ⁣(τ)E_{k}\!\left(\tau\right) Normalized Eisenstein series
ConstPiπ\pi The constant pi (3.14...)
ConstIii Imaginary unit
Powab{a}^{b} Power
HHH\mathbb{H} Upper complex half-plane
Source code for this entry:
Entry(ID("3bfced"),
    Formula(Equal(ComplexDerivative(EisensteinE(6, tau), tau, tau), Mul(Mul(Mul(2, ConstPi), ConstI), Parentheses(Div(Sub(Mul(EisensteinE(2, tau), EisensteinE(6, tau)), Pow(EisensteinE(4, tau), 2)), 2))))),
    Variables(tau),
    Assumptions(Element(tau, HH)),
    References("B. C. Berndt and A. J. Yee (2002) Ramanujan's Contributions to Eisenstein Series, Especially in His Lost Notebook. In: Kanemitsu S., Jia C. (eds) Number Theoretic Methods. Developments in Mathematics, vol 8. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-3675-5_3"))

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2019-09-22 15:43:45.410764 UTC