Fungrim entry: 18a4d1

E_{2}\!\left(\tau\right) = 1 + 6 \sum_{m=1}^{\infty} \frac{1}{\sin^{2}\!\left(\pi m \tau\right)}

Assumptions:

\tau \in \mathbb{H}

TeX:

E_{2}\!\left(\tau\right) = 1 + 6 \sum_{m=1}^{\infty} \frac{1}{\sin^{2}\!\left(\pi m \tau\right)}

\tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`Sum`	$\sum_{n} f(n)$	Sum
`Pow`	${a}^{b}$	Power
`Sin`	$\sin(z)$	Sine
`Pi`	$\pi$	The constant pi (3.14...)
`Infinity`	$\infty$	Positive infinity
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("18a4d1"),
    Formula(Equal(EisensteinE(2, tau), Add(1, Mul(6, Sum(Div(1, Pow(Sin(Mul(Mul(Pi, m), tau)), 2)), For(m, 1, Infinity)))))),
    Variables(tau),
    Assumptions(And(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