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

Riemann zeta function