Exponential function

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))))))

f758c6

Symbol: ConstE —

e

— The constant e (2.718...)

The real number giving the base of the natural logarithm, also known as Euler's number.

Definitions:

Fungrim symbol	Notation	Short description
`ConstE`	$e$	The constant e (2.718...)

Source code for this entry:

Entry(ID("f758c6"),
    SymbolDefinition(ConstE, ConstE, "The constant e (2.718...)"),
    Description("The real number giving the base of the natural logarithm, also known as Euler's number."))

6ef3d1

Image: X-ray of

{e}^{z}

on

z \in \left[-5, 5\right] + \left[-5, 5\right] i

Download: png (small) — png (medium) — png (large) — pdf (small) — pdf (medium/large) — svg (small) — svg (medium/large)

An X-ray plot illustrates the geometry of a complex analytic function

f\!\left(z\right)

. Thick black curves show where

\operatorname{Im}\!\left(f\!\left(z\right)\right) = 0

(the function is pure real). Thick red curves show where

\operatorname{Re}\!\left(f\!\left(z\right)\right) = 0

(the function is pure imaginary). Points where black and red curves intersect are zeros or poles. Magnitude level curves

\left|f\!\left(z\right)\right| = C

are rendered as thin gray curves, with brighter shades corresponding to larger

C

. Blue lines show branch cuts. The value of the function is continuous with the branch cut on the side indicated with a solid line, and discontinuous on the side indicated with a dashed line. Yellow is used to highlight important regions.

Definitions:

Fungrim symbol	Notation	Short description
`Exp`	${e}^{z}$	Exponential function
`ClosedInterval`	$\left[a, b\right]$	Closed interval
`ConstI`	$i$	Imaginary unit
`Im`	$\operatorname{Im}\!\left(z\right)$	Imaginary part
`Re`	$\operatorname{Re}\!\left(z\right)$	Real part
`Abs`	$\left\|z\right\|$	Absolute value

Source code for this entry:

Entry(ID("6ef3d1"),
    Image(Description("X-ray of", Exp(z), "on", Element(z, Add(ClosedInterval(-5, 5), Mul(ClosedInterval(-5, 5), ConstI)))), ImageSource("xray_exp")),
    Description("An X-ray plot illustrates the geometry of a complex analytic function", f(z), ".", "Thick black curves show where", Equal(Im(f(z)), 0), "(the function is pure real).", "Thick red curves show where", Equal(Re(f(z)), 0), "(the function is pure imaginary).", "Points where black and red curves intersect are zeros or poles.", "Magnitude level curves", Equal(Abs(f(z)), C), "are rendered as thin gray curves, with brighter shades corresponding to larger", C, ".", "Blue lines show branch cuts.", "The value of the function is continuous with the branch cut on the side indicated with a solid line, and discontinuous on the side indicated with a dashed line.", "Yellow is used to highlight important regions."))

27ca8d

{e}^{0} = 1

TeX:

{e}^{0} = 1

Definitions:

Fungrim symbol	Notation	Short description
`Exp`	${e}^{z}$	Exponential function

Source code for this entry:

Entry(ID("27ca8d"),
    Formula(Equal(Exp(0), 1)))

9a944c

{e}^{1} = e

TeX:

{e}^{1} = e

Definitions:

Fungrim symbol	Notation	Short description
`Exp`	${e}^{z}$	Exponential function
`ConstE`	$e$	The constant e (2.718...)

Source code for this entry:

Entry(ID("9a944c"),
    Formula(Equal(Exp(1), ConstE)))

54aaf1

{e}^{\pi i} = -1

TeX:

{e}^{\pi i} = -1

Definitions:

Fungrim symbol	Notation	Short description
`Exp`	${e}^{z}$	Exponential function
`ConstPi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit

Source code for this entry:

Entry(ID("54aaf1"),
    Formula(Equal(Exp(Mul(ConstPi, ConstI)), -1)))

a90f35

{e}^{\pi i / 2} = i

TeX:

{e}^{\pi i / 2} = i

Definitions:

Fungrim symbol	Notation	Short description
`Exp`	${e}^{z}$	Exponential function
`ConstPi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit

Source code for this entry:

Entry(ID("a90f35"),
    Formula(Equal(Exp(Div(Mul(ConstPi, ConstI), 2)), ConstI)))

812707

{e}^{a + b} = {e}^{a} {e}^{b}

Assumptions:

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C}

TeX:

{e}^{a + b} = {e}^{a} {e}^{b}

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`Exp`	${e}^{z}$	Exponential function
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("812707"),
    Formula(Equal(Exp(Add(a, b)), Mul(Exp(a), Exp(b)))),
    Variables(a, b),
    Assumptions(And(Element(a, CC), Element(b, CC))))

e51ec3

{\left({e}^{z}\right)}^{n} = {e}^{n z}

Assumptions:

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z}

TeX:

{\left({e}^{z}\right)}^{n} = {e}^{n z}

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z}

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`Exp`	${e}^{z}$	Exponential function
`CC`	$\mathbb{C}$	Complex numbers
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("e51ec3"),
    Formula(Equal(Pow(Exp(z), n), Exp(Mul(n, z)))),
    Variables(z, n),
    Assumptions(And(Element(z, CC), Element(n, ZZ))))

2f4f74

{e}^{-z} = \frac{1}{{e}^{z}}

Assumptions:

z \in \mathbb{C}

TeX:

{e}^{-z} = \frac{1}{{e}^{z}}

z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`Exp`	${e}^{z}$	Exponential function
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("2f4f74"),
    Formula(Equal(Exp(Neg(z)), Div(1, Exp(z)))),
    Variables(z),
    Assumptions(Element(z, CC)))

77d6bf

{e}^{a + b i} = {e}^{a} \left(\cos\!\left(b\right) + \sin\!\left(b\right) i\right)

Assumptions:

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C}

TeX:

{e}^{a + b i} = {e}^{a} \left(\cos\!\left(b\right) + \sin\!\left(b\right) i\right)

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`Exp`	${e}^{z}$	Exponential function
`ConstI`	$i$	Imaginary unit
`Sin`	$\sin\!\left(z\right)$	Sine
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("77d6bf"),
    Formula(Equal(Exp(Add(a, Mul(b, ConstI))), Mul(Exp(a), Add(Cos(b), Mul(Sin(b), ConstI))))),
    Variables(a, b),
    Assumptions(And(Element(a, CC), Element(b, CC))))

97ba8d

{e}^{z + n \pi i} = {\left(-1\right)}^{n} {e}^{z}

Assumptions:

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z}

TeX:

{e}^{z + n \pi i} = {\left(-1\right)}^{n} {e}^{z}

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z}

Definitions:

Fungrim symbol	Notation	Short description
`Exp`	${e}^{z}$	Exponential function
`ConstPi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("97ba8d"),
    Formula(Equal(Exp(Add(z, Mul(Mul(n, ConstPi), ConstI))), Mul(Pow(-1, n), Exp(z)))),
    Variables(z, n),
    Assumptions(And(Element(z, CC), Element(n, ZZ))))

1fa6b7

{e}^{z + 2 n \pi i} = {e}^{z}

Assumptions:

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z}

TeX:

{e}^{z + 2 n \pi i} = {e}^{z}

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z}

Definitions:

Fungrim symbol	Notation	Short description
`Exp`	${e}^{z}$	Exponential function
`ConstPi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`CC`	$\mathbb{C}$	Complex numbers
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("1fa6b7"),
    Formula(Equal(Exp(Add(z, Mul(Mul(Mul(2, n), ConstPi), ConstI))), Exp(z))),
    Variables(z, n),
    Assumptions(And(Element(z, CC), Element(n, ZZ))))

1568e1

{e}^{z} = \cosh\!\left(z\right) + \sinh\!\left(z\right)

Assumptions:

z \in \mathbb{C}

TeX:

{e}^{z} = \cosh\!\left(z\right) + \sinh\!\left(z\right)

z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`Exp`	${e}^{z}$	Exponential function
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("1568e1"),
    Formula(Equal(Exp(z), Add(Cosh(z), Sinh(z)))),
    Variables(z),
    Assumptions(Element(z, CC)))

e103e7

{e}^{i z} = \cos\!\left(z\right) + i \sin\!\left(z\right)

Assumptions:

z \in \mathbb{C}

TeX:

{e}^{i z} = \cos\!\left(z\right) + i \sin\!\left(z\right)

z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`Exp`	${e}^{z}$	Exponential function
`ConstI`	$i$	Imaginary unit
`Sin`	$\sin\!\left(z\right)$	Sine
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("e103e7"),
    Formula(Equal(Exp(Mul(ConstI, z)), Add(Cos(z), Mul(ConstI, Sin(z))))),
    Variables(z),
    Assumptions(Element(z, CC)))

28d158

\operatorname{HolomorphicDomain}\!\left({e}^{z}, z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \mathbb{C}

TeX:

\operatorname{HolomorphicDomain}\!\left({e}^{z}, z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`Exp`	${e}^{z}$	Exponential function
`CC`	$\mathbb{C}$	Complex numbers
`UnsignedInfinity`	${\tilde \infty}$	Unsigned infinity

Source code for this entry:

Entry(ID("28d158"),
    Formula(Equal(HolomorphicDomain(Exp(z), z, Union(CC, Set(UnsignedInfinity))), CC)))

0901a1

\operatorname{Poles}\!\left({e}^{z}, z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{\right\}

TeX:

\operatorname{Poles}\!\left({e}^{z}, z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{\right\}

Definitions:

Fungrim symbol	Notation	Short description
`Exp`	${e}^{z}$	Exponential function
`CC`	$\mathbb{C}$	Complex numbers
`UnsignedInfinity`	${\tilde \infty}$	Unsigned infinity

Source code for this entry:

Entry(ID("0901a1"),
    Formula(Equal(Poles(Exp(z), z, Union(CC, Set(UnsignedInfinity))), Set())))

be4b28

\operatorname{EssentialSingularities}\!\left({e}^{z}, z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{{\tilde \infty}\right\}

TeX:

\operatorname{EssentialSingularities}\!\left({e}^{z}, z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{{\tilde \infty}\right\}

Definitions:

Fungrim symbol	Notation	Short description
`Exp`	${e}^{z}$	Exponential function
`CC`	$\mathbb{C}$	Complex numbers
`UnsignedInfinity`	${\tilde \infty}$	Unsigned infinity

Source code for this entry:

Entry(ID("be4b28"),
    Formula(Equal(EssentialSingularities(Exp(z), z, Union(CC, Set(UnsignedInfinity))), Set(UnsignedInfinity))))

184c11

\operatorname{BranchPoints}\!\left({e}^{z}, z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{\right\}

TeX:

\operatorname{BranchPoints}\!\left({e}^{z}, z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{\right\}

Definitions:

Fungrim symbol	Notation	Short description
`Exp`	${e}^{z}$	Exponential function
`CC`	$\mathbb{C}$	Complex numbers
`UnsignedInfinity`	${\tilde \infty}$	Unsigned infinity

Source code for this entry:

Entry(ID("184c11"),
    Formula(Equal(BranchPoints(Exp(z), z, Union(CC, Set(UnsignedInfinity))), Set())))

b62d05

\operatorname{BranchCuts}\!\left({e}^{z}, z, \mathbb{C}\right) = \left\{\right\}

TeX:

\operatorname{BranchCuts}\!\left({e}^{z}, z, \mathbb{C}\right) = \left\{\right\}

Definitions:

Fungrim symbol	Notation	Short description
`Exp`	${e}^{z}$	Exponential function
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("b62d05"),
    Formula(Equal(BranchCuts(Exp(z), z, CC), Set())))

bceb84

\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} {e}^{z} = \left\{\right\}

TeX:

\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} {e}^{z} = \left\{\right\}

Definitions:

Fungrim symbol	Notation	Short description
`Exp`	${e}^{z}$	Exponential function
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("bceb84"),
    Formula(Equal(Zeros(Exp(z), z, Element(z, CC)), Set())))

1b3014

\left|{e}^{z}\right| = \exp\!\left(\operatorname{Re}\!\left(z\right)\right)

Assumptions:

z \in \mathbb{C}

TeX:

\left|{e}^{z}\right| = \exp\!\left(\operatorname{Re}\!\left(z\right)\right)

z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`Exp`	${e}^{z}$	Exponential function
`Re`	$\operatorname{Re}\!\left(z\right)$	Real part
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("1b3014"),
    Formula(Equal(Abs(Exp(z)), Exp(Re(z)))),
    Variables(z),
    Assumptions(Element(z, CC)))

caf706

\operatorname{sgn}\!\left({e}^{z}\right) = \exp\!\left(\operatorname{Im}\!\left(z\right) i\right)

Assumptions:

z \in \mathbb{C}

TeX:

\operatorname{sgn}\!\left({e}^{z}\right) = \exp\!\left(\operatorname{Im}\!\left(z\right) i\right)

z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`Sign`	$\operatorname{sgn}\!\left(z\right)$	Sign function
`Exp`	${e}^{z}$	Exponential function
`Im`	$\operatorname{Im}\!\left(z\right)$	Imaginary part
`ConstI`	$i$	Imaginary unit
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("caf706"),
    Formula(Equal(Sign(Exp(z)), Exp(Mul(Im(z), ConstI)))),
    Variables(z),
    Assumptions(Element(z, CC)))

b7d62b

\operatorname{Re}\!\left({e}^{z}\right) = \exp\!\left(\operatorname{Re}\!\left(z\right)\right) \cos\!\left(\operatorname{Im}\!\left(z\right)\right)

Assumptions:

z \in \mathbb{C}

TeX:

\operatorname{Re}\!\left({e}^{z}\right) = \exp\!\left(\operatorname{Re}\!\left(z\right)\right) \cos\!\left(\operatorname{Im}\!\left(z\right)\right)

z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`Re`	$\operatorname{Re}\!\left(z\right)$	Real part
`Exp`	${e}^{z}$	Exponential function
`Im`	$\operatorname{Im}\!\left(z\right)$	Imaginary part
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("b7d62b"),
    Formula(Equal(Re(Exp(z)), Mul(Exp(Re(z)), Cos(Im(z))))),
    Variables(z),
    Assumptions(Element(z, CC)))

e2fac7

\operatorname{Im}\!\left({e}^{z}\right) = \exp\!\left(\operatorname{Re}\!\left(z\right)\right) \sin\!\left(\operatorname{Im}\!\left(z\right)\right)

Assumptions:

z \in \mathbb{C}

TeX:

\operatorname{Im}\!\left({e}^{z}\right) = \exp\!\left(\operatorname{Re}\!\left(z\right)\right) \sin\!\left(\operatorname{Im}\!\left(z\right)\right)

z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`Im`	$\operatorname{Im}\!\left(z\right)$	Imaginary part
`Exp`	${e}^{z}$	Exponential function
`Re`	$\operatorname{Re}\!\left(z\right)$	Real part
`Sin`	$\sin\!\left(z\right)$	Sine
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("e2fac7"),
    Formula(Equal(Im(Exp(z)), Mul(Exp(Re(z)), Sin(Im(z))))),
    Variables(z),
    Assumptions(Element(z, CC)))

a0d93c

\arg\!\left({e}^{z}\right) = \operatorname{Im}\!\left(z\right)

Assumptions:

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \operatorname{Im}\!\left(z\right) \in \left(-\pi, \pi\right]

TeX:

\arg\!\left({e}^{z}\right) = \operatorname{Im}\!\left(z\right)

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \operatorname{Im}\!\left(z\right) \in \left(-\pi, \pi\right]

Definitions:

Fungrim symbol	Notation	Short description
`Arg`	$\arg\!\left(z\right)$	Complex argument
`Exp`	${e}^{z}$	Exponential function
`Im`	$\operatorname{Im}\!\left(z\right)$	Imaginary part
`CC`	$\mathbb{C}$	Complex numbers
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`ConstPi`	$\pi$	The constant pi (3.14...)

Source code for this entry:

Entry(ID("a0d93c"),
    Formula(Equal(Arg(Exp(z)), Im(z))),
    Variables(z),
    Assumptions(And(Element(z, CC), Element(Im(z), OpenClosedInterval(Neg(ConstPi), ConstPi)))))

52d827

\exp\!\left(\overline{z}\right) = \overline{{e}^{z}}

Assumptions:

z \in \mathbb{C}

TeX:

\exp\!\left(\overline{z}\right) = \overline{{e}^{z}}

z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`Exp`	${e}^{z}$	Exponential function
`Conjugate`	$\overline{z}$	Complex conjugate
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("52d827"),
    Formula(Equal(Exp(Conjugate(z)), Conjugate(Exp(z)))),
    Variables(z),
    Assumptions(Element(z, CC)))

1635f5

{e}^{z} = \sum_{k=0}^{\infty} \frac{{z}^{k}}{k !}

Assumptions:

z \in \mathbb{C}

TeX:

{e}^{z} = \sum_{k=0}^{\infty} \frac{{z}^{k}}{k !}

z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`Exp`	${e}^{z}$	Exponential function
`Pow`	${a}^{b}$	Power
`Factorial`	$n !$	Factorial
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("1635f5"),
    Formula(Equal(Exp(z), Sum(Div(Pow(z, k), Factorial(k)), Tuple(k, 0, Infinity)))),
    Variables(z),
    Assumptions(Element(z, CC)))

bad502

{e}^{c + z} = {e}^{c} \sum_{k=0}^{\infty} \frac{{z}^{k}}{k !}

Assumptions:

c \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C}

TeX:

{e}^{c + z} = {e}^{c} \sum_{k=0}^{\infty} \frac{{z}^{k}}{k !}

c \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`Exp`	${e}^{z}$	Exponential function
`Pow`	${a}^{b}$	Power
`Factorial`	$n !$	Factorial
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("bad502"),
    Formula(Equal(Exp(Add(c, z)), Mul(Exp(c), Sum(Div(Pow(z, k), Factorial(k)), Tuple(k, 0, Infinity))))),
    Variables(c, z),
    Assumptions(And(Element(c, CC), Element(z, CC))))

935b2f

\int_{a}^{b} {e}^{z} \, dz = {e}^{b} - {e}^{a}

Assumptions:

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C}

TeX:

\int_{a}^{b} {e}^{z} \, dz = {e}^{b} - {e}^{a}

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`Exp`	${e}^{z}$	Exponential function
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("935b2f"),
    Formula(Equal(Integral(Exp(z), Tuple(z, a, b)), Sub(Exp(b), Exp(a)))),
    Variables(a, b),
    Assumptions(And(Element(a, CC), Element(b, CC))))

96af56

\frac{d}{d z}\, {e}^{z} = {e}^{z}

Assumptions:

z \in \mathbb{C}

TeX:

\frac{d}{d z}\, {e}^{z} = {e}^{z}

z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`Derivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Derivative
`Exp`	${e}^{z}$	Exponential function
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("96af56"),
    Formula(Equal(Derivative(Exp(z), Tuple(z, z, 1)), Exp(z))),
    Variables(z),
    Assumptions(Element(z, CC)))

4491b8

\frac{d^{n}}{{d z}^{n}} {e}^{z} = {e}^{z}

Assumptions:

z \in \mathbb{C}

Alternative assumptions:

n \in \mathbb{Z}_{\ge 0}

TeX:

\frac{d^{n}}{{d z}^{n}} {e}^{z} = {e}^{z}

z \in \mathbb{C}

n \in \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`Derivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Derivative
`Exp`	${e}^{z}$	Exponential function
`CC`	$\mathbb{C}$	Complex numbers
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("4491b8"),
    Formula(Equal(Derivative(Exp(z), Tuple(z, z, n)), Exp(z))),
    Variables(z, n),
    Assumptions(And(Element(z, CC)), Element(n, ZZGreaterEqual(0))))

3c4480

\left|{e}^{z} - \sum_{k=0}^{N - 1} \frac{{z}^{k}}{k !}\right| \le \frac{{\left|z\right|}^{N}}{N ! \left(1 - \frac{\left|z\right|}{N}\right)}

Assumptions:

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, N \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, N \gt \left|z\right|

TeX:

\left|{e}^{z} - \sum_{k=0}^{N - 1} \frac{{z}^{k}}{k !}\right| \le \frac{{\left|z\right|}^{N}}{N ! \left(1 - \frac{\left|z\right|}{N}\right)}

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, N \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, N \gt \left|z\right|

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`Exp`	${e}^{z}$	Exponential function
`Pow`	${a}^{b}$	Power
`Factorial`	$n !$	Factorial
`CC`	$\mathbb{C}$	Complex numbers
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("3c4480"),
    Formula(LessEqual(Abs(Sub(Exp(z), Sum(Div(Pow(z, k), Factorial(k)), Tuple(k, 0, Sub(N, 1))))), Div(Pow(Abs(z), N), Mul(Factorial(N), Sub(1, Div(Abs(z), N)))))),
    Variables(z, N),
    Assumptions(And(Element(z, CC), Element(N, ZZ), Greater(N, Abs(z)))))

2f57ad

\left|{e}^{z}\right| \le {e}^{\left|z\right|}

Assumptions:

z \in \mathbb{C}

TeX:

\left|{e}^{z}\right| \le {e}^{\left|z\right|}

z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`Exp`	${e}^{z}$	Exponential function
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("2f57ad"),
    Formula(LessEqual(Abs(Exp(z)), Exp(Abs(z)))),
    Variables(z),
    Assumptions(Element(z, CC)))

3ac8a5

\left|{e}^{x + a} - {e}^{x}\right| \le {e}^{x} \left({e}^{\left|a\right|} - 1\right)

Assumptions:

x \in \mathbb{R} \,\mathbin{\operatorname{and}}\, a \in \mathbb{R}

TeX:

\left|{e}^{x + a} - {e}^{x}\right| \le {e}^{x} \left({e}^{\left|a\right|} - 1\right)

x \in \mathbb{R} \,\mathbin{\operatorname{and}}\, a \in \mathbb{R}

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`Exp`	${e}^{z}$	Exponential function
`RR`	$\mathbb{R}$	Real numbers

Source code for this entry:

Entry(ID("3ac8a5"),
    Formula(LessEqual(Abs(Sub(Exp(Add(x, a)), Exp(x))), Mul(Exp(x), Sub(Exp(Abs(a)), 1)))),
    Variables(x, a),
    Assumptions(And(Element(x, RR), Element(a, RR))))

Definitions

Illustrations

Particular values

Functional equations and connection formulas

Analytic properties

Complex parts

Taylor series

Derivatives and integrals

Approximations

Bounds and inequalities