Fungrim entry: 171724

E_{6}\!\left(\tau\right) = 1 + 63 \sum_{m=1}^{\infty} \frac{2 \cos^{4}\!\left(\pi m \tau\right) + 11 \cos^{2}\!\left(\pi m \tau\right) + 2}{\sin^{6}\!\left(\pi m \tau\right)}

Assumptions:

\tau \in \mathbb{H}

TeX:

E_{6}\!\left(\tau\right) = 1 + 63 \sum_{m=1}^{\infty} \frac{2 \cos^{4}\!\left(\pi m \tau\right) + 11 \cos^{2}\!\left(\pi m \tau\right) + 2}{\sin^{6}\!\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
`Cos`	$\cos(z)$	Cosine
`Pi`	$\pi$	The constant pi (3.14...)
`Sin`	$\sin(z)$	Sine
`Infinity`	$\infty$	Positive infinity
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("171724"),
    Formula(Equal(EisensteinE(6, tau), Add(1, Mul(63, Sum(Div(Add(Add(Mul(2, Pow(Cos(Mul(Mul(Pi, m), tau)), 4)), Mul(11, Pow(Cos(Mul(Mul(Pi, m), tau)), 2))), 2), Pow(Sin(Mul(Mul(Pi, m), tau)), 6)), For(m, 1, Infinity)))))),
    Variables(tau),
    Assumptions(And(Element(tau, HH))))

Topics using this entry

Eisenstein series