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Fungrim entry: c6be24

G2k ⁣(i)=limτi[G2k ⁣(τ)]=2ζ ⁣(2k)G_{2 k}\!\left(i \infty\right) = \lim_{\tau \to i \infty} \left[ G_{2 k}\!\left(\tau\right) \right] = 2 \zeta\!\left(2 k\right)
Assumptions:kZ1k \in \mathbb{Z}_{\ge 1}
TeX:
G_{2 k}\!\left(i \infty\right) = \lim_{\tau \to i \infty} \left[ G_{2 k}\!\left(\tau\right) \right] = 2 \zeta\!\left(2 k\right)

k \in \mathbb{Z}_{\ge 1}
Definitions:
Fungrim symbol Notation Short description
EisensteinGGk ⁣(τ)G_{k}\!\left(\tau\right) Eisenstein series
ConstIii Imaginary unit
Infinity\infty Positive infinity
ComplexLimitlimzaf ⁣(z)\lim_{z \to a} f\!\left(z\right) Limiting value, complex variable
RiemannZetaζ ⁣(s)\zeta\!\left(s\right) Riemann zeta function
ZZGreaterEqualZn\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), tau, Mul(ConstI, Infinity)), Mul(2, RiemannZeta(Mul(2, k))))),
    Variables(k),
    Assumptions(And(Element(k, ZZGreaterEqual(1)))))

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

2019-08-21 11:44:15.926409 UTC