# 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

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