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Fungrim entry: 7cda09

E2(τ)=2πi(E22 ⁣(τ)E4 ⁣(τ)12)E'_{2}(\tau) = 2 \pi i \left(\frac{E_{2}^{2}\!\left(\tau\right) - E_{4}\!\left(\tau\right)}{12}\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'_{2}(\tau) = 2 \pi i \left(\frac{E_{2}^{2}\!\left(\tau\right) - E_{4}\!\left(\tau\right)}{12}\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("7cda09"),
    Formula(Equal(ComplexDerivative(EisensteinE(2, tau), tau, tau), Mul(Mul(Mul(2, ConstPi), ConstI), Parentheses(Div(Sub(Pow(EisensteinE(2, tau), 2), EisensteinE(4, tau)), 12))))),
    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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2019-09-19 20:12:49.583742 UTC