Fungrim entry: 1a63af

\zeta\!\left(1 - s\right) = \frac{2 \cos\!\left(\frac{1}{2} \pi s\right)}{{\left(2 \pi\right)}^{s}} \Gamma(s) \zeta\!\left(s\right)

Assumptions:

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; s \notin \{1, 0, \ldots\}

Alternative assumptions:

s \in \mathbb{C}[[x]] \;\mathbin{\operatorname{and}}\; s \notin \{1, 0, \ldots\}

TeX:

\zeta\!\left(1 - s\right) = \frac{2 \cos\!\left(\frac{1}{2} \pi s\right)}{{\left(2 \pi\right)}^{s}} \Gamma(s) \zeta\!\left(s\right)

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; s \notin \{1, 0, \ldots\}

s \in \mathbb{C}[[x]] \;\mathbin{\operatorname{and}}\; s \notin \{1, 0, \ldots\}

Definitions:

Fungrim symbol	Notation	Short description
`RiemannZeta`	$\zeta\!\left(s\right)$	Riemann zeta function
`Cos`	$\cos(z)$	Cosine
`Pi`	$\pi$	The constant pi (3.14...)
`Pow`	${a}^{b}$	Power
`Gamma`	$\Gamma(z)$	Gamma function
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n
`PowerSeries`	$K[[x]]$	Formal power series

Source code for this entry:

Entry(ID("1a63af"),
    Formula(Equal(RiemannZeta(Sub(1, s)), Mul(Mul(Div(Mul(2, Cos(Mul(Mul(Div(1, 2), Pi), s))), Pow(Mul(2, Pi), s)), Gamma(s)), RiemannZeta(s)))),
    Variables(s),
    Assumptions(And(Element(s, CC), NotElement(s, ZZLessEqual(1))), And(Element(s, PowerSeries(CC, SerX)), NotElement(s, ZZLessEqual(1)))))

Topics using this entry

Riemann zeta function