ζ(s)=2(2π)s−1sin(2πs)Γ(1−s)ζ(1−s)
Assumptions:s∈Cands∈/Z≥0
Alternative assumptions:s∈C[[x]]ands∈/Z≥0
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 0}
s \in \mathbb{C}[[x]] \;\mathbin{\operatorname{and}}\; s \notin \mathbb{Z}_{\ge 0}Definitions:
| Fungrim symbol | Notation | Short description |
|---|---|---|
| RiemannZeta | ζ(s) | Riemann zeta function |
| Pow | ab | Power |
| Pi | π | The constant pi (3.14...) |
| Sin | sin(z) | Sine |
| Gamma | Γ(z) | Gamma function |
| CC | C | Complex numbers |
| ZZGreaterEqual | Z≥n | Integers greater than or equal to n |
| PowerSeries | 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, Pi), Sub(s, 1))), Sin(Div(Mul(Pi, s), 2))), Gamma(Sub(1, s))), RiemannZeta(Sub(1, s))))),
Variables(s),
Assumptions(And(Element(s, CC), NotElement(s, ZZGreaterEqual(0))), And(Element(s, PowerSeries(CC, SerX)), NotElement(s, ZZGreaterEqual(0)))))