Fungrim entry: ef9f8a

Symbol: Pow —

{a}^{b}

— Power

The following table lists conditions such that Pow(a, b) is defined in Fungrim.

Domain	Codomain
Numbers
$a \in \mathbb{C} \setminus 0 \;\mathbin{\operatorname{and}}\; b \in \mathbb{C}$	${a}^{b} \in \mathbb{C}$
$a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \left\{0\right\}$	${a}^{b} \in \left\{1\right\}$
Infinities
$a \in \left\{\infty, -\infty, {\tilde \infty}\right\} \;\mathbin{\operatorname{and}}\; b \in \{-1, -2, \ldots\}$	${a}^{b} \in \left\{0\right\}$
General domains
$a \in R \;\mathbin{\operatorname{and}}\; R \in \operatorname{Rings} \;\mathbin{\operatorname{and}}\; b \in \mathbb{Z}_{\ge 0}$	${a}^{b} \in R$
$a \in K \setminus \left\{0\right\} \;\mathbin{\operatorname{and}}\; K \in \operatorname{Fields} \;\mathbin{\operatorname{and}}\; \mathbb{Q} \subseteq K \;\mathbin{\operatorname{and}}\; b \in \mathbb{Z}$	${a}^{b} \in R$

Table data:

\left(P, Q\right)

such that

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

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`Infinity`	$\infty$	Positive infinity
`UnsignedInfinity`	${\tilde \infty}$	Unsigned infinity
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`QQ`	$\mathbb{Q}$	Rational numbers
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("ef9f8a"),
    SymbolDefinition(Pow, Pow(a, b), "Power"),
    Description(""),
    Description("The following table lists conditions such that", SourceForm(Pow(a, b)), "is defined in Fungrim."),
    Table(TableRelation(Tuple(P, Q), Implies(P, Q)), TableHeadings(Description("Domain"), Description("Codomain")), List(TableSection("Numbers"), Tuple(And(Element(a, SetMinus(CC, 0)), Element(b, CC)), Element(Pow(a, b), CC)), Tuple(And(Element(a, CC), Element(b, Set(0))), Element(Pow(a, b), Set(1))), TableSection("Infinities"), Tuple(And(Element(a, Set(Infinity, Neg(Infinity), UnsignedInfinity)), Element(b, ZZLessEqual(-1))), Element(Pow(a, b), Set(0))), TableSection("General domains"), Tuple(And(Element(a, R), Element(R, Rings), Element(b, ZZGreaterEqual(0))), Element(Pow(a, b), R)), Tuple(And(Element(a, SetMinus(K, Set(0))), Element(K, Fields), SubsetEqual(QQ, K), Element(b, ZZ)), Element(Pow(a, b), R)))))

Topics using this entry

Powers