Fungrim home page

Fungrim entry: 9ee8bc

ζ(s)=2(2π)s1sin ⁣(πs2)Γ ⁣(1s)ζ ⁣(1s)\zeta(s) = 2 {\left(2 \pi\right)}^{s - 1} \sin\!\left(\frac{\pi s}{2}\right) \Gamma\!\left(1 - s\right) \zeta\!\left(1 - s\right)
Assumptions:sCandsZ1s \in \mathbb{C} \,\mathbin{\operatorname{and}}\, s \notin \mathbb{Z}_{\ge 1}
Alternative assumptions:sC[[x]]andsZ1s \in \mathbb{C}[[x]] \,\mathbin{\operatorname{and}}\, s \notin \mathbb{Z}_{\ge 1}
\zeta(s) = 2 {\left(2 \pi\right)}^{s - 1} \sin\!\left(\frac{\pi s}{2}\right) \Gamma\!\left(1 - s\right) \zeta\!\left(1 - s\right)

s \in \mathbb{C} \,\mathbin{\operatorname{and}}\, s \notin \mathbb{Z}_{\ge 1}

s \in \mathbb{C}[[x]] \,\mathbin{\operatorname{and}}\, s \notin \mathbb{Z}_{\ge 1}
Fungrim symbol Notation Short description
RiemannZetaζ(s)\zeta(s) Riemann zeta function
Powab{a}^{b} Power
Piπ\pi The constant pi (3.14...)
Sinsin(z)\sin(z) Sine
GammaΓ(z)\Gamma(z) Gamma function
CCC\mathbb{C} Complex numbers
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
PowerSeriesK[[x]]K[[x]] Formal power series
Source code for this entry:
    Formula(Equal(RiemannZeta(s), Mul(Mul(Mul(Mul(2, Pow(Mul(2, Pi), Sub(s, 1))), Sin(Div(Mul(Pi, s), 2))), Gamma(Sub(1, s))), RiemannZeta(Sub(1, s))))),
    Assumptions(And(Element(s, CC), NotElement(s, ZZGreaterEqual(1))), And(Element(s, PowerSeries(CC, x)), NotElement(s, ZZGreaterEqual(1)))))

Topics using this entry

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

2020-01-31 18:09:28.494564 UTC