# Fungrim entry: c6038c

Symbol: LogGamma $\log \Gamma\!\left(z\right)$ Logarithmic gamma function
The logarithmic gamma function $\log \Gamma\!\left(z\right)$ is a function of one complex variable $z$. It satisfies $\log \Gamma\!\left(x\right) = \log\!\left(\Gamma\!\left(x\right)\right)$ for real $x > 0$ and is defined on the complex plane through analytic continuation, with branch cuts on $\left(-\infty, 0\right]$. An explicit construction uses 37a95a combined with 774d37 for analytic continuation. In general, $\log \Gamma\!\left(z\right) \ne \log\!\left(\Gamma\!\left(z\right)\right)$ 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
Numbers
$z \in \left(0, \infty\right)$ $\log \Gamma\!\left(z\right) \in \left(-0.1215, \infty\right)$
$z \in \mathbb{C} \setminus \{0, -1, \ldots\}$ $\log \Gamma\!\left(z\right) \in \mathbb{C}$
Infinities
$z \in \left\{\infty\right\}$ $\log \Gamma\!\left(z\right) \in \left\{\infty\right\}$
Formal power series
$z \in \mathbb{R}[[x]] \,\mathbin{\operatorname{and}}\, [{x}^{0}] z > 0$ $\log \Gamma\!\left(z\right) \in \mathbb{R}[[x]]$
$z \in \mathbb{C}[[x]] \,\mathbin{\operatorname{and}}\, [{x}^{0}] z \notin \{0, -1, \ldots\}$ $\log \Gamma\!\left(z\right) \in \mathbb{C}[[x]]$
Table data: $\left(P, Q\right)$ such that $\left(P\right) \implies \left(Q\right)$
Definitions:
Fungrim symbol Notation Short description
LogGamma$\log \Gamma\!\left(z\right)$ Logarithmic gamma function
Log$\log\!\left(z\right)$ Natural logarithm
GammaFunction$\Gamma\!\left(z\right)$ Gamma function
OpenClosedInterval$\left(a, b\right]$ Open-closed interval
Infinity$\infty$ Positive infinity
OpenInterval$\left(a, b\right)$ Open interval
CC$\mathbb{C}$ Complex numbers
ZZLessEqual$\mathbb{Z}_{\le n}$ Integers less than or equal to n
FormalPowerSeries$K[[x]]$ Formal power series
RR$\mathbb{R}$ Real numbers
Source code for this entry:
Entry(ID("c6038c"),
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(GammaFunction(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,", Unequal(LogGamma(z), Log(GammaFunction(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, FormalPowerSeries(RR, x)), Greater(SeriesCoefficient(z, x, 0), 0)), And(Element(LogGamma(z), FormalPowerSeries(RR, x)))), Tuple(And(Element(z, FormalPowerSeries(CC, x)), NotElement(SeriesCoefficient(z, x, 0), ZZLessEqual(0))), And(Element(LogGamma(z), FormalPowerSeries(CC, x)))))))

## Topics using this entry

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

2019-08-21 11:44:15.926409 UTC