Fungrim entry: 9ee8bc

$\zeta\!\left(s\right) = 2 {\left(2 \pi\right)}^{s - 1} \sin\!\left(\frac{\pi s}{2}\right) \Gamma\!\left(1 - s\right) \zeta\!\left(1 - s\right)$
Assumptions:$s \in \mathbb{C} \,\mathbin{\operatorname{and}}\, s \notin \mathbb{Z}_{\ge 1}$
Alternative assumptions:$s \in \mathbb{C}[[x]] \,\mathbin{\operatorname{and}}\, s \notin \mathbb{Z}_{\ge 1}$
TeX:
\zeta\!\left(s\right) = 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}
Definitions:
Fungrim symbol Notation Short description
RiemannZeta$\zeta\!\left(s\right)$ Riemann zeta function
Pow${a}^{b}$ Power
ConstPi$\pi$ The constant pi (3.14...)
Sin$\sin\!\left(z\right)$ Sine
GammaFunction$\Gamma\!\left(z\right)$ Gamma function
CC$\mathbb{C}$ Complex numbers
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
FormalPowerSeries$K[[x]]$ Formal power series
Source code for this entry:
Entry(ID("9ee8bc"),
Formula(Equal(RiemannZeta(s), Mul(Mul(Mul(Mul(2, Pow(Mul(2, ConstPi), Sub(s, 1))), Sin(Div(Mul(ConstPi, s), 2))), GammaFunction(Sub(1, s))), RiemannZeta(Sub(1, s))))),
Variables(s),
Assumptions(And(Element(s, CC), NotElement(s, ZZGreaterEqual(1))), And(Element(s, FormalPowerSeries(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.

2019-06-18 07:49:59.356594 UTC