Fungrim entry: c6be24

G_{2 k}\!\left(i \infty\right) = \lim_{\tau \to i \infty} G_{2 k}\!\left(\tau\right) = 2 \zeta\!\left(2 k\right)

Assumptions:

k \in \mathbb{Z}_{\ge 1}

TeX:

G_{2 k}\!\left(i \infty\right) = \lim_{\tau \to i \infty} G_{2 k}\!\left(\tau\right) = 2 \zeta\!\left(2 k\right)

k \in \mathbb{Z}_{\ge 1}

Definitions:

Fungrim symbol	Notation	Short description
`EisensteinG`	$G_{k}\!\left(\tau\right)$	Eisenstein series
`ConstI`	$i$	Imaginary unit
`Infinity`	$\infty$	Positive infinity
`ComplexLimit`	$\lim_{z \to a} f(z)$	Limiting value, complex variable
`RiemannZeta`	$\zeta\!\left(s\right)$	Riemann zeta function
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("c6be24"),
    Formula(Equal(EisensteinG(Mul(2, k), Mul(ConstI, Infinity)), ComplexLimit(EisensteinG(Mul(2, k), tau), For(tau, Mul(ConstI, Infinity))), Mul(2, RiemannZeta(Mul(2, k))))),
    Variables(k),
    Assumptions(And(Element(k, ZZGreaterEqual(1)))))

Topics using this entry

Eisenstein series