# Fungrim entry: dfbcd9

Symbol: Exp ${e}^{z}$ Exponential function
The exponential function ${e}^{z}$ is a function of one complex variable $z$. It can be defined by the Taylor series 1635f5.
In rendered formulas, Exp(z) is shown as ${e}^{z}$ or as $\exp\!\left(z\right)$ depending on the typographical requirements; no semantic difference is implied.
The following table lists all conditions such that Exp(z) is defined in Fungrim.
Domain Codomain
Numbers
$z \in \left\{0\right\}$ ${e}^{z} \in \left\{1\right\}$
$z \in \mathbb{R}$ ${e}^{z} \in \left(0, \infty\right)$
$z \in \mathbb{C}$ ${e}^{z} \in \mathbb{C} \setminus \left\{0\right\}$
Infinities
$z \in \left\{\infty\right\}$ ${e}^{z} \in \left\{\infty\right\}$
$z \in \left\{-\infty\right\}$ ${e}^{z} \in \left\{0\right\}$
Formal power series
$z \in \mathbb{Q}[[x]] \,\mathbin{\operatorname{and}}\, [{x}^{0}] z = 0$ ${e}^{z} \in \mathbb{Q}[[x]] \,\mathbin{\operatorname{and}}\, [{x}^{0}] {e}^{z} = 1$
$z \in \mathbb{R}[[x]]$ ${e}^{z} \in \mathbb{R}[[x]] \,\mathbin{\operatorname{and}}\, [{x}^{0}] {e}^{z} \ne 0$
$z \in \mathbb{C}[[x]]$ ${e}^{z} \in \mathbb{C}[[x]] \,\mathbin{\operatorname{and}}\, [{x}^{0}] {e}^{z} \ne 0$
Table data: $\left(P, Q\right)$ such that $\left(P\right) \implies \left(Q\right)$
Definitions:
Fungrim symbol Notation Short description
Exp${e}^{z}$ Exponential function
RR$\mathbb{R}$ Real numbers
OpenInterval$\left(a, b\right)$ Open interval
Infinity$\infty$ Positive infinity
CC$\mathbb{C}$ Complex numbers
FormalPowerSeries$K[[x]]$ Formal power series
QQ$\mathbb{Q}$ Rational numbers
Source code for this entry:
Entry(ID("dfbcd9"),
SymbolDefinition(Exp, Exp(z), "Exponential function"),
Description("The exponential function", Exp(z), "is a function of one complex variable", z, ".", "It can be defined by the Taylor series", EntryReference("1635f5"), "."),
Description("In rendered formulas,", SourceForm(Exp(z)), "is shown as", Exp(z), "or as", Call(Exp, z), "depending on the typographical requirements; no semantic difference is implied."),
Description("The following table lists all conditions such that", SourceForm(Exp(z)), "is defined in Fungrim."),
Table(TableRelation(Tuple(P, Q), Implies(P, Q)), TableHeadings(Description("Domain"), Description("Codomain")), List(TableSection("Numbers"), Tuple(Element(z, Set(0)), Element(Exp(z), Set(1))), Tuple(Element(z, RR), Element(Exp(z), OpenInterval(0, Infinity))), Tuple(Element(z, CC), Element(Exp(z), SetMinus(CC, Set(0)))), TableSection("Infinities"), Tuple(Element(z, Set(Infinity)), Element(Exp(z), Set(Infinity))), Tuple(Element(z, Set(Neg(Infinity))), Element(Exp(z), Set(0))), TableSection("Formal power series"), Tuple(And(Element(z, FormalPowerSeries(QQ, x)), Equal(SeriesCoefficient(z, x, 0), 0)), And(Element(Exp(z), FormalPowerSeries(QQ, x)), Equal(SeriesCoefficient(Exp(z), x, 0), 1))), Tuple(Element(z, FormalPowerSeries(RR, x)), And(Element(Exp(z), FormalPowerSeries(RR, x)), Unequal(SeriesCoefficient(Exp(z), x, 0), 0))), Tuple(Element(z, FormalPowerSeries(CC, x)), And(Element(Exp(z), FormalPowerSeries(CC, x)), Unequal(SeriesCoefficient(Exp(z), x, 0), 0))))))

## Topics using this entry

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-10-05 13:11:19.856591 UTC