Fungrim entry: 7b62e4

E_{2}\!\left(\tau\right) = 1 - 12 \sum_{m=1}^{\infty} \frac{1}{\cos\!\left(2 \pi m \tau\right) - 1}

Assumptions:

\tau \in \mathbb{H}

TeX:

E_{2}\!\left(\tau\right) = 1 - 12 \sum_{m=1}^{\infty} \frac{1}{\cos\!\left(2 \pi m \tau\right) - 1}

\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
`Cos`	$\cos(z)$	Cosine
`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("7b62e4"),
    Formula(Equal(EisensteinE(2, tau), Sub(1, Mul(12, Sum(Div(1, Sub(Cos(Mul(Mul(Mul(2, Pi), m), tau)), 1)), 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