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Fungrim entry: 09e2ed

Symbol: GammaFunction Γ(z)\Gamma(z) Gamma function
The gamma function Γ(z)\Gamma(z) is a function of one complex variable zz. It is a meromorphic function with simple poles at the nonpositive integers and no zeros. It can be defined by the integral representation 4e4e0f in the right half-plane, together with the functional equation 78f1f4 for analytic continuation. The following table lists all conditions such that GammaFunction(z) is defined in Fungrim.
Domain Codomain
Numbers
zZ1z \in \mathbb{Z}_{\ge 1} Γ(z)Z1\Gamma(z) \in \mathbb{Z}_{\ge 1}
z(0,)z \in \left(0, \infty\right) Γ(z)(0.8856,)\Gamma(z) \in \left(0.8856, \infty\right)
zR{0,1,}z \in \mathbb{R} \setminus \{0, -1, \ldots\} Γ(z)R{0}\Gamma(z) \in \mathbb{R} \setminus \left\{0\right\}
zC{0,1,}z \in \mathbb{C} \setminus \{0, -1, \ldots\} Γ(z)C{0}\Gamma(z) \in \mathbb{C} \setminus \left\{0\right\}
Infinities
z{0,1,}z \in \{0, -1, \ldots\} Γ(z){~}\Gamma(z) \in \left\{{\tilde \infty}\right\}
z{}z \in \left\{\infty\right\} Γ(z){}\Gamma(z) \in \left\{\infty\right\}
z{i,i}z \in \left\{i \infty, -i \infty\right\} Γ(z){0}\Gamma(z) \in \left\{0\right\}
Formal power series
zR[[x]]and[x0]z{0,1,}z \in \mathbb{R}[[x]] \,\mathbin{\operatorname{and}}\, [{x}^{0}] z \notin \{0, -1, \ldots\} Γ(z)R[[x]]and[x0]Γ(z)0\Gamma(z) \in \mathbb{R}[[x]] \,\mathbin{\operatorname{and}}\, [{x}^{0}] \Gamma(z) \ne 0
zC[[x]]and[x0]z{0,1,}z \in \mathbb{C}[[x]] \,\mathbin{\operatorname{and}}\, [{x}^{0}] z \notin \{0, -1, \ldots\} Γ(z)C[[x]]and[x0]Γ(z)0\Gamma(z) \in \mathbb{C}[[x]] \,\mathbin{\operatorname{and}}\, [{x}^{0}] \Gamma(z) \ne 0
zR[[x]]andz{0,1,}z \in \mathbb{R}[[x]] \,\mathbin{\operatorname{and}}\, z \notin \{0, -1, \ldots\} Γ(z)R( ⁣(x) ⁣)\Gamma(z) \in \mathbb{R}(\!(x)\!)
zC[[x]]andz{0,1,}z \in \mathbb{C}[[x]] \,\mathbin{\operatorname{and}}\, z \notin \{0, -1, \ldots\} Γ(z)C( ⁣(x) ⁣)\Gamma(z) \in \mathbb{C}(\!(x)\!)
Table data: (P,Q)\left(P, Q\right) such that (P)    (Q)\left(P\right) \implies \left(Q\right)
Definitions:
Fungrim symbol Notation Short description
GammaFunctionΓ(z)\Gamma(z) Gamma function
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
OpenInterval(a,b)\left(a, b\right) Open interval
Infinity\infty Positive infinity
RRR\mathbb{R} Real numbers
ZZLessEqualZn\mathbb{Z}_{\le n} Integers less than or equal to n
CCC\mathbb{C} Complex numbers
UnsignedInfinity~{\tilde \infty} Unsigned infinity
ConstIii Imaginary unit
FormalPowerSeriesK[[x]]K[[x]] Formal power series
FormalLaurentSeriesK( ⁣(x) ⁣)K(\!(x)\!) Formal Laurent series
Source code for this entry:
Entry(ID("09e2ed"),
    SymbolDefinition(GammaFunction, GammaFunction(z), "Gamma function"),
    Description("The gamma function", GammaFunction(z), "is a function of one complex variable", z, ". It is a meromorphic function with simple poles at the nonpositive integers and no zeros.", "It can be defined by the integral representation", EntryReference("4e4e0f"), "in the right half-plane, together with the functional equation", EntryReference("78f1f4"), "for analytic continuation.", "The following table lists all conditions such that", SourceForm(GammaFunction(z)), "is defined in Fungrim."),
    Table(TableRelation(Tuple(P, Q), Implies(P, Q)), TableHeadings(Description("Domain"), Description("Codomain")), List(TableSection("Numbers"), Tuple(Element(z, ZZGreaterEqual(1)), Element(GammaFunction(z), ZZGreaterEqual(1))), Tuple(Element(z, OpenInterval(0, Infinity)), Element(GammaFunction(z), OpenInterval(Decimal("0.8856"), Infinity))), Tuple(Element(z, SetMinus(RR, ZZLessEqual(0))), Element(GammaFunction(z), SetMinus(RR, Set(0)))), Tuple(Element(z, SetMinus(CC, ZZLessEqual(0))), Element(GammaFunction(z), SetMinus(CC, Set(0)))), TableSection("Infinities"), Tuple(Element(z, ZZLessEqual(0)), Element(GammaFunction(z), Set(UnsignedInfinity))), Tuple(Element(z, Set(Infinity)), Element(GammaFunction(z), Set(Infinity))), Tuple(Element(z, Set(Mul(ConstI, Infinity), Neg(Mul(ConstI, Infinity)))), Element(GammaFunction(z), Set(0))), TableSection("Formal power series"), Tuple(And(Element(z, FormalPowerSeries(RR, x)), NotElement(SeriesCoefficient(z, x, 0), ZZLessEqual(0))), And(Element(GammaFunction(z), FormalPowerSeries(RR, x)), Unequal(SeriesCoefficient(GammaFunction(z), x, 0), 0))), Tuple(And(Element(z, FormalPowerSeries(CC, x)), NotElement(SeriesCoefficient(z, x, 0), ZZLessEqual(0))), And(Element(GammaFunction(z), FormalPowerSeries(CC, x)), Unequal(SeriesCoefficient(GammaFunction(z), x, 0), 0))), Tuple(And(Element(z, FormalPowerSeries(RR, x)), NotElement(z, ZZLessEqual(0))), Element(GammaFunction(z), FormalLaurentSeries(RR, x))), Tuple(And(Element(z, FormalPowerSeries(CC, x)), NotElement(z, ZZLessEqual(0))), Element(GammaFunction(z), FormalLaurentSeries(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.

2019-10-05 13:11:19.856591 UTC