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 \gt 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
$z \in \mathbb{R}[[x]] \,\mathbin{\operatorname{and}}\, [{x}^{0}] z \gt 0$	$\log \Gamma\!\left(z\right) \in \mathbb{R}[[x]]$
$z \in \mathbb{C}[[x]] \,\mathbin{\operatorname{and}}\, [{x}^{0}] z \notin \{0, -1, \ldots\}$	$\log \Gamma\!\left(z\right) \in \mathbb{C}[[x]]$

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

Gamma function