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Fungrim entry: e0a6a2

Symbol: RiemannZeta ζ ⁣(s)\zeta\!\left(s\right) Riemann zeta function
The Riemann zeta function ζ ⁣(s)\zeta\!\left(s\right) is a function of one complex variable ss. It is a meromorphic function with a pole at s=1s = 1. The following table lists all conditions such that RiemannZeta(s) is defined in Fungrim.
Domain Codomain
s(1,)s \in \left(1, \infty\right) ζ ⁣(s)(1,)\zeta\!\left(s\right) \in \left(1, \infty\right)
sR{1}s \in \mathbb{R} \setminus \left\{1\right\} ζ ⁣(s)R\zeta\!\left(s\right) \in \mathbb{R}
sC{1}s \in \mathbb{C} \setminus \left\{1\right\} ζ ⁣(s)C\zeta\!\left(s\right) \in \mathbb{C}
s{1}s \in \left\{1\right\} ζ ⁣(s){~}\zeta\!\left(s\right) \in \left\{{\tilde \infty}\right\}
s{}s \in \left\{\infty\right\} ζ ⁣(s){1}\zeta\!\left(s\right) \in \left\{1\right\}
Formal power series
sR[[x]]  and  [x0]s1s \in \mathbb{R}[[x]] \;\mathbin{\operatorname{and}}\; [{x}^{0}] s \ne 1 ζ ⁣(s)R[[x]]\zeta\!\left(s\right) \in \mathbb{R}[[x]]
sC[[x]]  and  [x0]s1s \in \mathbb{C}[[x]] \;\mathbin{\operatorname{and}}\; [{x}^{0}] s \ne 1 ζ ⁣(s)C[[x]]\zeta\!\left(s\right) \in \mathbb{C}[[x]]
sR[[x]]  and  s1s \in \mathbb{R}[[x]] \;\mathbin{\operatorname{and}}\; s \ne 1 ζ ⁣(s)R( ⁣(x) ⁣)\zeta\!\left(s\right) \in \mathbb{R}(\!(x)\!)
sC[[x]]  and  s1s \in \mathbb{C}[[x]] \;\mathbin{\operatorname{and}}\; s \ne 1 ζ ⁣(s)C( ⁣(x) ⁣)\zeta\!\left(s\right) \in \mathbb{C}(\!(x)\!)
Table data: (P,Q)\left(P, Q\right) such that (P)        (Q)\left(P\right) \;\implies\; \left(Q\right)
Fungrim symbol Notation Short description
RiemannZetaζ ⁣(s)\zeta\!\left(s\right) Riemann zeta function
OpenInterval(a,b)\left(a, b\right) Open interval
Infinity\infty Positive infinity
RRR\mathbb{R} Real numbers
CCC\mathbb{C} Complex numbers
UnsignedInfinity~{\tilde \infty} Unsigned infinity
PowerSeriesK[[x]]K[[x]] Formal power series
LaurentSeriesK( ⁣(x) ⁣)K(\!(x)\!) Formal Laurent series
Source code for this entry:
    SymbolDefinition(RiemannZeta, RiemannZeta(s), "Riemann zeta function"),
    Description("The Riemann zeta function", RiemannZeta(s), "is a function of one complex variable", s, ". It is a meromorphic function with a pole at", Equal(s, 1), ".", "The following table lists all conditions such that", SourceForm(RiemannZeta(s)), "is defined in Fungrim."),
    Table(TableRelation(Tuple(P, Q), Implies(P, Q)), TableHeadings(Description("Domain"), Description("Codomain")), List(TableSection("Numbers"), Tuple(Element(s, OpenInterval(1, Infinity)), Element(RiemannZeta(s), OpenInterval(1, Infinity))), Tuple(Element(s, SetMinus(RR, Set(1))), Element(RiemannZeta(s), RR)), Tuple(Element(s, SetMinus(CC, Set(1))), Element(RiemannZeta(s), CC)), TableSection("Infinities"), Tuple(Element(s, Set(1)), Element(RiemannZeta(s), Set(UnsignedInfinity))), Tuple(Element(s, Set(Infinity)), Element(RiemannZeta(s), Set(1))), TableSection("Formal power series"), Tuple(And(Element(s, PowerSeries(RR, x)), NotEqual(SeriesCoefficient(s, x, 0), 1)), Element(RiemannZeta(s), PowerSeries(RR, x))), Tuple(And(Element(s, PowerSeries(CC, x)), NotEqual(SeriesCoefficient(s, x, 0), 1)), Element(RiemannZeta(s), PowerSeries(CC, x))), Tuple(And(Element(s, PowerSeries(RR, x)), NotEqual(s, 1)), Element(RiemannZeta(s), LaurentSeries(RR, x))), Tuple(And(Element(s, PowerSeries(CC, x)), NotEqual(s, 1)), Element(RiemannZeta(s), LaurentSeries(CC, x))))))

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

2021-03-15 19:12:00.328586 UTC