Fungrim entry: 7cda09

E'_{2}(\tau) = 2 \pi i \left(\frac{E_{2}^{2}\!\left(\tau\right) - E_{4}\!\left(\tau\right)}{12}\right)

Assumptions:

\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
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`Pow`	${a}^{b}$	Power
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("7cda09"),
    Formula(Equal(ComplexDerivative(EisensteinE(2, tau), For(tau, tau)), Mul(Mul(Mul(2, Pi), 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"))

Topics using this entry

Eisenstein series