# Fungrim entry: 09e2ed

Symbol: Gamma $\Gamma(z)$ Gamma function
The gamma function $\Gamma(z)$ 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 4e4e0f in the right half-plane, together with the functional equation 78f1f4 for analytic continuation. The following table lists all conditions such that Gamma(z) is defined in Fungrim.
Domain Codomain
Numbers
$z \in \mathbb{Z}_{\ge 1}$ $\Gamma(z) \in \mathbb{Z}_{\ge 1}$
$z \in \left(0, \infty\right)$ $\Gamma(z) \in \left(0.8856, \infty\right)$
$z \in \mathbb{R} \setminus \{0, -1, \ldots\}$ $\Gamma(z) \in \mathbb{R} \setminus \left\{0\right\}$
$z \in \mathbb{C} \setminus \{0, -1, \ldots\}$ $\Gamma(z) \in \mathbb{C} \setminus \left\{0\right\}$
Infinities
$z \in \{0, -1, \ldots\}$ $\Gamma(z) \in \left\{{\tilde \infty}\right\}$
$z \in \left\{\infty\right\}$ $\Gamma(z) \in \left\{\infty\right\}$
$z \in \left\{i \infty, -i \infty\right\}$ $\Gamma(z) \in \left\{0\right\}$
Formal power series
$z \in \mathbb{R}[[x]] \;\mathbin{\operatorname{and}}\; [{x}^{0}] z \notin \{0, -1, \ldots\}$ $\Gamma(z) \in \mathbb{R}[[x]] \;\mathbin{\operatorname{and}}\; [{x}^{0}] \Gamma(z) \ne 0$
$z \in \mathbb{C}[[x]] \;\mathbin{\operatorname{and}}\; [{x}^{0}] z \notin \{0, -1, \ldots\}$ $\Gamma(z) \in \mathbb{C}[[x]] \;\mathbin{\operatorname{and}}\; [{x}^{0}] \Gamma(z) \ne 0$
$z \in \mathbb{R}[[x]] \;\mathbin{\operatorname{and}}\; z \notin \{0, -1, \ldots\}$ $\Gamma(z) \in \mathbb{R}(\!(x)\!)$
$z \in \mathbb{C}[[x]] \;\mathbin{\operatorname{and}}\; z \notin \{0, -1, \ldots\}$ $\Gamma(z) \in \mathbb{C}(\!(x)\!)$
Table data: $\left(P, Q\right)$ such that $\left(P\right) \;\implies\; \left(Q\right)$
Definitions:
Fungrim symbol Notation Short description
Gamma$\Gamma(z)$ 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
PowerSeries$K[[x]]$ Formal power series
LaurentSeries$K(\!(x)\!)$ Formal Laurent series
Source code for this entry:
Entry(ID("09e2ed"),
SymbolDefinition(Gamma, Gamma(z), "Gamma function"),
Description("The gamma function", Gamma(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(Gamma(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(Gamma(z), ZZGreaterEqual(1))), Tuple(Element(z, OpenInterval(0, Infinity)), Element(Gamma(z), OpenInterval(Decimal("0.8856"), Infinity))), Tuple(Element(z, SetMinus(RR, ZZLessEqual(0))), Element(Gamma(z), SetMinus(RR, Set(0)))), Tuple(Element(z, SetMinus(CC, ZZLessEqual(0))), Element(Gamma(z), SetMinus(CC, Set(0)))), TableSection("Infinities"), Tuple(Element(z, ZZLessEqual(0)), Element(Gamma(z), Set(UnsignedInfinity))), Tuple(Element(z, Set(Infinity)), Element(Gamma(z), Set(Infinity))), Tuple(Element(z, Set(Mul(ConstI, Infinity), Neg(Mul(ConstI, Infinity)))), Element(Gamma(z), Set(0))), TableSection("Formal power series"), Tuple(And(Element(z, PowerSeries(RR, x)), NotElement(SeriesCoefficient(z, x, 0), ZZLessEqual(0))), And(Element(Gamma(z), PowerSeries(RR, x)), NotEqual(SeriesCoefficient(Gamma(z), x, 0), 0))), Tuple(And(Element(z, PowerSeries(CC, x)), NotElement(SeriesCoefficient(z, x, 0), ZZLessEqual(0))), And(Element(Gamma(z), PowerSeries(CC, x)), NotEqual(SeriesCoefficient(Gamma(z), x, 0), 0))), Tuple(And(Element(z, PowerSeries(RR, x)), NotElement(z, ZZLessEqual(0))), Element(Gamma(z), LaurentSeries(RR, x))), Tuple(And(Element(z, PowerSeries(CC, x)), NotElement(z, ZZLessEqual(0))), Element(Gamma(z), LaurentSeries(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.

2021-03-15 19:12:00.328586 UTC