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Fungrim entry: c6038c

Symbol: LogGamma logΓ(z)\log \Gamma(z) Logarithmic gamma function
The logarithmic gamma function logΓ(z)\log \Gamma(z) is a function of one complex variable zz. It satisfies logΓ(x)=log ⁣(Γ(x))\log \Gamma(x) = \log\!\left(\Gamma(x)\right) for real x>0x > 0 and is defined on the complex plane through analytic continuation, with branch cuts on (,0]\left(-\infty, 0\right]. An explicit construction uses 37a95a combined with 774d37 for analytic continuation. In general, logΓ(z)log ⁣(Γ(z))\log \Gamma(z) \ne \log\!\left(\Gamma(z)\right) as the latter has an infinite set of branch cuts off the real line. The following table lists all conditions such that LogGamma(z) is defined in Fungrim.
Domain Codomain
z(0,)z \in \left(0, \infty\right) logΓ(z)(0.1215,)\log \Gamma(z) \in \left(-0.1215, \infty\right)
zC{0,1,}z \in \mathbb{C} \setminus \{0, -1, \ldots\} logΓ(z)C\log \Gamma(z) \in \mathbb{C}
z{}z \in \left\{\infty\right\} logΓ(z){}\log \Gamma(z) \in \left\{\infty\right\}
Formal power series
zR[[x]]  and  [x0]z>0z \in \mathbb{R}[[x]] \;\mathbin{\operatorname{and}}\; [{x}^{0}] z > 0 logΓ(z)R[[x]]\log \Gamma(z) \in \mathbb{R}[[x]]
zC[[x]]  and  [x0]z{0,1,}z \in \mathbb{C}[[x]] \;\mathbin{\operatorname{and}}\; [{x}^{0}] z \notin \{0, -1, \ldots\} logΓ(z)C[[x]]\log \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)
Fungrim symbol Notation Short description
LogGammalogΓ(z)\log \Gamma(z) Logarithmic gamma function
Loglog(z)\log(z) Natural logarithm
GammaΓ(z)\Gamma(z) Gamma function
OpenClosedInterval(a,b]\left(a, b\right] Open-closed interval
Infinity\infty Positive infinity
OpenInterval(a,b)\left(a, b\right) Open interval
CCC\mathbb{C} Complex numbers
ZZLessEqualZn\mathbb{Z}_{\le n} Integers less than or equal to n
PowerSeriesK[[x]]K[[x]] Formal power series
RRR\mathbb{R} Real numbers
Source code for this entry:
    SymbolDefinition(LogGamma, LogGamma(z), "Logarithmic gamma function"),
    Description("The logarithmic gamma function", LogGamma(z), "is a function of one complex variable", z, ".", "It satisfies", Equal(LogGamma(x), Log(Gamma(x))), "for real", Greater(x, 0), "and is defined on the complex plane", "through analytic continuation, with branch cuts on", OpenClosedInterval(Neg(Infinity), 0), ".", "An explicit construction uses", EntryReference("37a95a"), "combined with", EntryReference("774d37"), "for analytic continuation.", "In general,", NotEqual(LogGamma(z), Log(Gamma(z))), " as the latter has an infinite set of branch cuts off the real line.", "The following table lists all conditions such that", SourceForm(LogGamma(z)), "is defined in Fungrim."),
    Table(TableRelation(Tuple(P, Q), Implies(P, Q)), TableHeadings(Description("Domain"), Description("Codomain")), List(TableSection("Numbers"), Tuple(Element(z, OpenInterval(0, Infinity)), Element(LogGamma(z), OpenInterval(Decimal("-0.1215"), Infinity))), Tuple(Element(z, SetMinus(CC, ZZLessEqual(0))), Element(LogGamma(z), CC)), TableSection("Infinities"), Tuple(Element(z, Set(Infinity)), Element(LogGamma(z), Set(Infinity))), TableSection("Formal power series"), Tuple(And(Element(z, PowerSeries(RR, x)), Greater(SeriesCoefficient(z, x, 0), 0)), And(Element(LogGamma(z), PowerSeries(RR, x)))), Tuple(And(Element(z, PowerSeries(CC, x)), NotElement(SeriesCoefficient(z, x, 0), ZZLessEqual(0))), And(Element(LogGamma(z), PowerSeries(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