Fungrim entry: 09e2ed

Symbol: GammaFunction —

\Gamma\!\left(z\right)

— Gamma function

The gamma function

\Gamma\!\left(z\right)

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 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
$z \in \mathbb{Z}_{\ge 1}$	$\Gamma\!\left(z\right) \in \mathbb{Z}_{\ge 1}$
$z \in \left(0, \infty\right)$	$\Gamma\!\left(z\right) \in \left(0.8856, \infty\right)$
$z \in \mathbb{R} \setminus \{0, -1, \ldots\}$	$\Gamma\!\left(z\right) \in \mathbb{R} \setminus \left\{0\right\}$
$z \in \mathbb{C} \setminus \{0, -1, \ldots\}$	$\Gamma\!\left(z\right) \in \mathbb{C} \setminus \left\{0\right\}$
Infinities
$z \in \{0, -1, \ldots\}$	$\Gamma\!\left(z\right) \in \left\{{\tilde \infty}\right\}$
$z \in \left\{\infty\right\}$	$\Gamma\!\left(z\right) \in \left\{\infty\right\}$
$z \in \left\{i \infty, -i \infty\right\}$	$\Gamma\!\left(z\right) \in \left\{0\right\}$
Formal power series
$z \in \mathbb{R}[[x]] \,\mathbin{\operatorname{and}}\, [{x}^{0}] z \notin \{0, -1, \ldots\}$	$\Gamma\!\left(z\right) \in \mathbb{R}[[x]] \,\mathbin{\operatorname{and}}\, [{x}^{0}] \Gamma\!\left(z\right) \ne 0$
$z \in \mathbb{C}[[x]] \,\mathbin{\operatorname{and}}\, [{x}^{0}] z \notin \{0, -1, \ldots\}$	$\Gamma\!\left(z\right) \in \mathbb{C}[[x]] \,\mathbin{\operatorname{and}}\, [{x}^{0}] \Gamma\!\left(z\right) \ne 0$
$z \in \mathbb{R}[[x]] \,\mathbin{\operatorname{and}}\, z \notin \{0, -1, \ldots\}$	$\Gamma\!\left(z\right) \in \mathbb{R}(\!(x)\!)$
$z \in \mathbb{C}[[x]] \,\mathbin{\operatorname{and}}\, z \notin \{0, -1, \ldots\}$	$\Gamma\!\left(z\right) \in \mathbb{C}(\!(x)\!)$

Table data:

\left(P, Q\right)

such that

\left(P\right) \implies \left(Q\right)

Definitions:

Fungrim symbol	Notation	Short description
`GammaFunction`	$\Gamma\!\left(z\right)$	Gamma function
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`OpenInterval`	$\left(a, b\right)$	Open interval
`Infinity`	$\infty$	Positive infinity
`RR`	$\mathbb{R}$	Real numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n
`CC`	$\mathbb{C}$	Complex numbers
`UnsignedInfinity`	${\tilde \infty}$	Unsigned infinity
`ConstI`	$i$	Imaginary unit
`FormalPowerSeries`	$K[[x]]$	Formal power series
`FormalLaurentSeries`	$K(\!(x)\!)$	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))))))

Topics using this entry

Gamma function