Fungrim entry: 03ad5a

E_{2}\!\left(\tau\right) = -\frac{12 i}{\pi} \frac{\eta'(\tau)}{\eta(\tau)}

Assumptions:

\tau \in \mathbb{H}

TeX:

E_{2}\!\left(\tau\right) = -\frac{12 i}{\pi} \frac{\eta'(\tau)}{\eta(\tau)}

\tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`ConstI`	$i$	Imaginary unit
`Pi`	$\pi$	The constant pi (3.14...)
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`DedekindEta`	$\eta(\tau)$	Dedekind eta function
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("03ad5a"),
    Formula(Equal(EisensteinE(2, tau), Neg(Mul(Div(Mul(12, ConstI), Pi), Div(ComplexDerivative(DedekindEta(tau), For(tau, tau)), DedekindEta(tau)))))),
    Variables(tau),
    Assumptions(Element(tau, HH)))

Topics using this entry

Eisenstein series

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

2021-03-15 19:12:00.328586 UTC