Lambert W-function

Symbol: LambertW —

W_{k}\!\left(z\right)

— Lambert W-function

Called with two arguments LambertW(k, z) (rendered

W_{k}\!\left(z\right)

) represents the

k

-th branch of the Lambert W-function.

Called with three arguments LambertW(k, z, r) (rendered

W^{(r)}_{k}\!\left(z\right)

) represents the

r

-th derivative of the

k

-th branch of the Lambert W-function, with inherited branch cuts.

LambertW(k, z) is equivalent to LambertW(k, z, 0).

The following table lists conditions such that LambertW(k, z, r) is defined in Fungrim.

Domain	Codomain
Numbers
$k \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{0\right\}$	$W_{k}\!\left(z\right) \in \mathbb{C}$
$k \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{0, -{e}^{-1}\right\} \,\mathbin{\operatorname{and}}\, r \in \mathbb{Z}_{\ge 0}$	$W^{(r)}_{k}\!\left(z\right) \in \mathbb{C}$
$r \in \mathbb{Z}_{\ge 0}$	$W^{(r)}_{0}\!\left(0\right) \in \mathbb{Q}$

Table data:

\left(P, Q\right)

such that

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

Definitions:

Fungrim symbol	Notation	Short description
`LambertW`	$W_{k}\!\left(z\right)$	Lambert W-function
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers
`Exp`	${e}^{z}$	Exponential function
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`QQ`	$\mathbb{Q}$	Rational numbers

Source code for this entry:

Entry(ID("6da738"),
    SymbolDefinition(LambertW, LambertW(k, z), "Lambert W-function"),
    Description("Called with two arguments", SourceForm(LambertW(k, z)), "(rendered", LambertW(k, z), ") represents the", k, "-th branch", "of the Lambert W-function."),
    Description("Called with three arguments", SourceForm(LambertW(k, z, r)), "(rendered", LambertW(k, z, r), ") represents the", r, "-th derivative of the", k, "-th branch of the Lambert W-function, with inherited branch cuts."),
    Description(SourceForm(LambertW(k, z)), "is equivalent to", SourceForm(LambertW(k, z, 0)), "."),
    Description("The following table lists conditions such that", SourceForm(LambertW(k, z, r)), "is defined in Fungrim."),
    Table(TableRelation(Tuple(P, Q), Implies(P, Q)), TableHeadings(Description("Domain"), Description("Codomain")), List(TableSection("Numbers"), Tuple(And(Element(k, ZZ), Element(z, SetMinus(CC, Set(0)))), Element(LambertW(k, z), CC)), Tuple(And(Element(k, ZZ), Element(z, SetMinus(CC, Set(0, Neg(Exp(-1))))), Element(r, ZZGreaterEqual(0))), Element(LambertW(k, z, r), CC)), Tuple(Element(r, ZZGreaterEqual(0)), Element(LambertW(0, 0, r), QQ)))))

cb0a9b

Image: X-ray of

W_{0}\!\left(z\right)

on

z \in \left[-3, 3\right] + \left[-3, 3\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
`LambertW`	$W_{k}\!\left(z\right)$	Lambert W-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("cb0a9b"),
    Image(Description("X-ray of", LambertW(0, z), "on", Element(z, Add(ClosedInterval(-3, 3), Mul(ClosedInterval(-3, 3), ConstI)))), ImageSource("xray_lambertw")),
    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."))

88168b

W_{k}\!\left(z\right) \exp\!\left(W_{k}\!\left(z\right)\right) = z

Assumptions:

\left(k \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{0\right\}\right) \,\mathbin{\operatorname{or}}\, \left(k = 0 \,\mathbin{\operatorname{and}}\, z = 0\right)

TeX:

W_{k}\!\left(z\right) \exp\!\left(W_{k}\!\left(z\right)\right) = z

\left(k \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{0\right\}\right) \,\mathbin{\operatorname{or}}\, \left(k = 0 \,\mathbin{\operatorname{and}}\, z = 0\right)

Definitions:

Fungrim symbol	Notation	Short description
`LambertW`	$W_{k}\!\left(z\right)$	Lambert W-function
`Exp`	${e}^{z}$	Exponential function
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("88168b"),
    Formula(Equal(Mul(LambertW(k, z), Exp(LambertW(k, z))), z)),
    Variables(k, z),
    Assumptions(Or(And(Element(k, ZZ), Element(z, SetMinus(CC, Set(0)))), And(Equal(k, 0), Equal(z, 0)))))

d7136f

\mathop{\operatorname{solutions}\,}\limits_{w \in \mathbb{C}} \left[w {e}^{w} = z\right] = \left\{ W_{k}\!\left(z\right) : k \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, \left(z \ne 0 \,\mathbin{\operatorname{or}}\, k = 0\right) \right\}

Assumptions:

z \in \mathbb{C}

TeX:

\mathop{\operatorname{solutions}\,}\limits_{w \in \mathbb{C}} \left[w {e}^{w} = z\right] = \left\{ W_{k}\!\left(z\right) : k \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, \left(z \ne 0 \,\mathbin{\operatorname{or}}\, k = 0\right) \right\}

z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`Exp`	${e}^{z}$	Exponential function
`CC`	$\mathbb{C}$	Complex numbers
`SetBuilder`	$\left\{ f\!\left(x\right) : P\!\left(x\right) \right\}$	Set comprehension
`LambertW`	$W_{k}\!\left(z\right)$	Lambert W-function
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("d7136f"),
    Formula(Equal(Solutions(Brackets(Equal(Mul(w, Exp(w)), z)), w, Element(w, CC)), SetBuilder(LambertW(k, z), k, And(Element(k, ZZ), Or(Unequal(z, 0), Equal(k, 0)))))),
    Variables(z),
    Assumptions(Element(z, CC)))

314807

\mathop{\operatorname{solution}\,}\limits_{w \in \left[-1, \infty\right)} \left[w {e}^{w} = x\right] = W_{0}\!\left(x\right)

Assumptions:

x \in \left[-\frac{1}{e}, \infty\right)

TeX:

\mathop{\operatorname{solution}\,}\limits_{w \in \left[-1, \infty\right)} \left[w {e}^{w} = x\right] = W_{0}\!\left(x\right)

x \in \left[-\frac{1}{e}, \infty\right)

Definitions:

Fungrim symbol	Notation	Short description
`Exp`	${e}^{z}$	Exponential function
`ClosedOpenInterval`	$\left[a, b\right)$	Closed-open interval
`Infinity`	$\infty$	Positive infinity
`LambertW`	$W_{k}\!\left(z\right)$	Lambert W-function
`ConstE`	$e$	The constant e (2.718...)

Source code for this entry:

Entry(ID("314807"),
    Formula(Equal(UniqueSolution(Brackets(Equal(Mul(w, Exp(w)), x)), w, Element(w, ClosedOpenInterval(-1, Infinity))), LambertW(0, x))),
    Variables(x),
    Assumptions(Element(x, ClosedOpenInterval(Neg(Div(1, ConstE)), Infinity))))

636929

\mathop{\operatorname{solution}\,}\limits_{w \in \left(-\infty, -1\right]} \left[w {e}^{w} = x\right] = W_{-1}\!\left(x\right)

Assumptions:

x \in \left[-\frac{1}{e}, 0\right)

TeX:

\mathop{\operatorname{solution}\,}\limits_{w \in \left(-\infty, -1\right]} \left[w {e}^{w} = x\right] = W_{-1}\!\left(x\right)

x \in \left[-\frac{1}{e}, 0\right)

Definitions:

Fungrim symbol	Notation	Short description
`Exp`	${e}^{z}$	Exponential function
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`Infinity`	$\infty$	Positive infinity
`LambertW`	$W_{k}\!\left(z\right)$	Lambert W-function
`ClosedOpenInterval`	$\left[a, b\right)$	Closed-open interval
`ConstE`	$e$	The constant e (2.718...)

Source code for this entry:

Entry(ID("636929"),
    Formula(Equal(UniqueSolution(Brackets(Equal(Mul(w, Exp(w)), x)), w, Element(w, OpenClosedInterval(Neg(Infinity), -1))), LambertW(-1, x))),
    Variables(x),
    Assumptions(Element(x, ClosedOpenInterval(Neg(Div(1, ConstE)), 0))))

8654a3

W_{0}\!\left(x {e}^{x}\right) = x

Assumptions:

x \in \left[-1, \infty\right)

TeX:

W_{0}\!\left(x {e}^{x}\right) = x

x \in \left[-1, \infty\right)

Definitions:

Fungrim symbol	Notation	Short description
`LambertW`	$W_{k}\!\left(z\right)$	Lambert W-function
`Exp`	${e}^{z}$	Exponential function
`ClosedOpenInterval`	$\left[a, b\right)$	Closed-open interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("8654a3"),
    Formula(Equal(LambertW(0, Mul(x, Exp(x))), x)),
    Variables(x),
    Assumptions(Element(x, ClosedOpenInterval(-1, Infinity))))

ed7dac

W_{-1}\!\left(x {e}^{x}\right) = x

Assumptions:

x \in \left(-\infty, -1\right]

TeX:

W_{-1}\!\left(x {e}^{x}\right) = x

x \in \left(-\infty, -1\right]

Definitions:

Fungrim symbol	Notation	Short description
`LambertW`	$W_{k}\!\left(z\right)$	Lambert W-function
`Exp`	${e}^{z}$	Exponential function
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("ed7dac"),
    Formula(Equal(LambertW(-1, Mul(x, Exp(x))), x)),
    Variables(x),
    Assumptions(Element(x, OpenClosedInterval(Neg(Infinity), -1))))

30bd5b

W_{0}\!\left(x \log\!\left(x\right)\right) = \log\!\left(x\right)

Assumptions:

x \in \left[-\frac{1}{e}, \infty\right)

TeX:

W_{0}\!\left(x \log\!\left(x\right)\right) = \log\!\left(x\right)

x \in \left[-\frac{1}{e}, \infty\right)

Definitions:

Fungrim symbol	Notation	Short description
`LambertW`	$W_{k}\!\left(z\right)$	Lambert W-function
`Log`	$\log\!\left(z\right)$	Natural logarithm
`ClosedOpenInterval`	$\left[a, b\right)$	Closed-open interval
`ConstE`	$e$	The constant e (2.718...)
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("30bd5b"),
    Formula(Equal(LambertW(0, Mul(x, Log(x))), Log(x))),
    Variables(x),
    Assumptions(Element(x, ClosedOpenInterval(Neg(Div(1, ConstE)), Infinity))))

a172c7

W_{-1}\!\left(x \log\!\left(x\right)\right) = \log\!\left(x\right)

Assumptions:

x \in \left(0, -\frac{1}{e}\right]

TeX:

W_{-1}\!\left(x \log\!\left(x\right)\right) = \log\!\left(x\right)

x \in \left(0, -\frac{1}{e}\right]

Definitions:

Fungrim symbol	Notation	Short description
`LambertW`	$W_{k}\!\left(z\right)$	Lambert W-function
`Log`	$\log\!\left(z\right)$	Natural logarithm
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`ConstE`	$e$	The constant e (2.718...)

Source code for this entry:

Entry(ID("a172c7"),
    Formula(Equal(LambertW(-1, Mul(x, Log(x))), Log(x))),
    Variables(x),
    Assumptions(Element(x, OpenClosedInterval(0, Neg(Div(1, ConstE))))))

0be17d

W_{0}\!\left(0\right) = 0

TeX:

W_{0}\!\left(0\right) = 0

Definitions:

Fungrim symbol	Notation	Short description
`LambertW`	$W_{k}\!\left(z\right)$	Lambert W-function

Source code for this entry:

Entry(ID("0be17d"),
    Formula(Equal(LambertW(0, 0), 0)))

c95c4f

W_{0}\!\left(e\right) = 1

TeX:

W_{0}\!\left(e\right) = 1

Definitions:

Fungrim symbol	Notation	Short description
`LambertW`	$W_{k}\!\left(z\right)$	Lambert W-function
`ConstE`	$e$	The constant e (2.718...)

Source code for this entry:

Entry(ID("c95c4f"),
    Formula(Equal(LambertW(0, ConstE), 1)))

b93d09

W_{0}\!\left(-\frac{1}{e}\right) = -1

TeX:

W_{0}\!\left(-\frac{1}{e}\right) = -1

Definitions:

Fungrim symbol	Notation	Short description
`LambertW`	$W_{k}\!\left(z\right)$	Lambert W-function
`ConstE`	$e$	The constant e (2.718...)

Source code for this entry:

Entry(ID("b93d09"),
    Formula(Equal(LambertW(0, Neg(Div(1, ConstE))), -1)))

d09380

W_{-1}\!\left(-\frac{1}{e}\right) = -1

TeX:

W_{-1}\!\left(-\frac{1}{e}\right) = -1

Definitions:

Fungrim symbol	Notation	Short description
`LambertW`	$W_{k}\!\left(z\right)$	Lambert W-function
`ConstE`	$e$	The constant e (2.718...)

Source code for this entry:

Entry(ID("d09380"),
    Formula(Equal(LambertW(-1, Neg(Div(1, ConstE))), -1)))

5d4cce

W_{0}\!\left(1\right) \in \left[0.56714329040978387299996866221035554975381578718651 \pm 2.51 \cdot 10^{-51}\right]

TeX:

W_{0}\!\left(1\right) \in \left[0.56714329040978387299996866221035554975381578718651 \pm 2.51 \cdot 10^{-51}\right]

Definitions:

Fungrim symbol	Notation	Short description
`LambertW`	$W_{k}\!\left(z\right)$	Lambert W-function

Source code for this entry:

Entry(ID("5d4cce"),
    Formula(Element(LambertW(0, 1), RealBall(Decimal("0.56714329040978387299996866221035554975381578718651"), Decimal("2.51e-51")))))

c87ff4

W'_{0}\!\left(0\right) = 1

TeX:

W'_{0}\!\left(0\right) = 1

Definitions:

Fungrim symbol	Notation	Short description
`LambertW`	$W_{k}\!\left(z\right)$	Lambert W-function

Source code for this entry:

Entry(ID("c87ff4"),
    Formula(Equal(LambertW(0, 0, 1), 1)))

8e8a59

W^{(r)}_{0}\!\left(0\right) = {\left(-r\right)}^{r - 1}

Assumptions:

r \in \mathbb{Z}_{\ge 1}

TeX:

W^{(r)}_{0}\!\left(0\right) = {\left(-r\right)}^{r - 1}

r \in \mathbb{Z}_{\ge 1}

Definitions:

Fungrim symbol	Notation	Short description
`LambertW`	$W_{k}\!\left(z\right)$	Lambert W-function
`Pow`	${a}^{b}$	Power
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("8e8a59"),
    Formula(Equal(LambertW(0, 0, r), Pow(Neg(r), Sub(r, 1)))),
    Variables(r),
    Assumptions(Element(r, ZZGreaterEqual(1))))

f372e9

W_{k}\!\left(0\right) = -\infty

Assumptions:

k \in \mathbb{Z} \setminus \left\{0\right\}

TeX:

W_{k}\!\left(0\right) = -\infty

k \in \mathbb{Z} \setminus \left\{0\right\}

Definitions:

Fungrim symbol	Notation	Short description
`LambertW`	$W_{k}\!\left(z\right)$	Lambert W-function
`Infinity`	$\infty$	Positive infinity
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("f372e9"),
    Formula(Equal(LambertW(k, 0), Neg(Infinity))),
    Variables(k),
    Assumptions(Element(k, SetMinus(ZZ, Set(0)))))

e1dd64

W_{0}\!\left(-\frac{\pi}{2}\right) = \frac{i \pi}{2}

TeX:

W_{0}\!\left(-\frac{\pi}{2}\right) = \frac{i \pi}{2}

Definitions:

Fungrim symbol	Notation	Short description
`LambertW`	$W_{k}\!\left(z\right)$	Lambert W-function
`ConstPi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit

Source code for this entry:

Entry(ID("e1dd64"),
    Formula(Equal(LambertW(0, Neg(Div(ConstPi, 2))), Div(Mul(ConstI, ConstPi), 2))))

6d936e

W_{k}\!\left(\overline{z}\right) = \overline{W_{-k}\!\left(z\right)}

Assumptions:

k \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left(\left(k = 0 \,\mathbin{\operatorname{and}}\, z \notin \left(-\infty, -{e}^{-1}\right)\right) \,\mathbin{\operatorname{or}}\, \left(k \ne 0 \,\mathbin{\operatorname{and}}\, z \notin \left(-\infty, 0\right]\right)\right)

TeX:

W_{k}\!\left(\overline{z}\right) = \overline{W_{-k}\!\left(z\right)}

k \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left(\left(k = 0 \,\mathbin{\operatorname{and}}\, z \notin \left(-\infty, -{e}^{-1}\right)\right) \,\mathbin{\operatorname{or}}\, \left(k \ne 0 \,\mathbin{\operatorname{and}}\, z \notin \left(-\infty, 0\right]\right)\right)

Definitions:

Fungrim symbol	Notation	Short description
`LambertW`	$W_{k}\!\left(z\right)$	Lambert W-function
`Conjugate`	$\overline{z}$	Complex conjugate
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers
`OpenInterval`	$\left(a, b\right)$	Open interval
`Infinity`	$\infty$	Positive infinity
`Exp`	${e}^{z}$	Exponential function
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval

Source code for this entry:

Entry(ID("6d936e"),
    Formula(Equal(LambertW(k, Conjugate(z)), Conjugate(LambertW(Neg(k), z)))),
    Variables(k, z),
    Assumptions(And(Element(k, ZZ), Element(z, CC), Or(And(Equal(k, 0), NotElement(z, OpenInterval(Neg(Infinity), Neg(Exp(-1))))), And(Unequal(k, 0), NotElement(z, OpenClosedInterval(Neg(Infinity), 0)))))))

0d3b91

\operatorname{HolomorphicDomain}\!\left(W_{0}\!\left(z\right), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \mathbb{C} \setminus \left(-\infty, -{e}^{-1}\right]

TeX:

\operatorname{HolomorphicDomain}\!\left(W_{0}\!\left(z\right), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \mathbb{C} \setminus \left(-\infty, -{e}^{-1}\right]

Definitions:

Fungrim symbol	Notation	Short description
`LambertW`	$W_{k}\!\left(z\right)$	Lambert W-function
`CC`	$\mathbb{C}$	Complex numbers
`UnsignedInfinity`	${\tilde \infty}$	Unsigned infinity
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`Infinity`	$\infty$	Positive infinity
`Exp`	${e}^{z}$	Exponential function

Source code for this entry:

Entry(ID("0d3b91"),
    Formula(Equal(HolomorphicDomain(LambertW(0, z), z, Union(CC, Set(UnsignedInfinity))), SetMinus(CC, OpenClosedInterval(Neg(Infinity), Neg(Exp(-1)))))))

2caf78

\operatorname{HolomorphicDomain}\!\left(W_{k}\!\left(z\right), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \mathbb{C} \setminus \left(-\infty, 0\right]

Assumptions:

k \in \mathbb{Z} \setminus \left\{0\right\}

TeX:

\operatorname{HolomorphicDomain}\!\left(W_{k}\!\left(z\right), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \mathbb{C} \setminus \left(-\infty, 0\right]

k \in \mathbb{Z} \setminus \left\{0\right\}

Definitions:

Fungrim symbol	Notation	Short description
`LambertW`	$W_{k}\!\left(z\right)$	Lambert W-function
`CC`	$\mathbb{C}$	Complex numbers
`UnsignedInfinity`	${\tilde \infty}$	Unsigned infinity
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`Infinity`	$\infty$	Positive infinity
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("2caf78"),
    Formula(Equal(HolomorphicDomain(LambertW(k, z), z, Union(CC, Set(UnsignedInfinity))), SetMinus(CC, OpenClosedInterval(Neg(Infinity), 0)))),
    Variables(k),
    Assumptions(Element(k, SetMinus(ZZ, Set(0)))))

aca420

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

Assumptions:

k \in \mathbb{Z}

TeX:

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

k \in \mathbb{Z}

Definitions:

Fungrim symbol	Notation	Short description
`LambertW`	$W_{k}\!\left(z\right)$	Lambert W-function
`CC`	$\mathbb{C}$	Complex numbers
`UnsignedInfinity`	${\tilde \infty}$	Unsigned infinity
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("aca420"),
    Formula(Equal(Poles(LambertW(k, z), z, Union(CC, Set(UnsignedInfinity))), Set())),
    Variables(k),
    Assumptions(Element(k, ZZ)))

41ece5

\operatorname{BranchPoints}\!\left(W_{0}\!\left(z\right), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{-{e}^{-1}, {\tilde \infty}\right\}

TeX:

\operatorname{BranchPoints}\!\left(W_{0}\!\left(z\right), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{-{e}^{-1}, {\tilde \infty}\right\}

Definitions:

Fungrim symbol	Notation	Short description
`LambertW`	$W_{k}\!\left(z\right)$	Lambert W-function
`CC`	$\mathbb{C}$	Complex numbers
`UnsignedInfinity`	${\tilde \infty}$	Unsigned infinity
`Exp`	${e}^{z}$	Exponential function

Source code for this entry:

Entry(ID("41ece5"),
    Formula(Equal(BranchPoints(LambertW(0, z), z, Union(CC, Set(UnsignedInfinity))), Set(Neg(Exp(-1)), UnsignedInfinity))))

17eaad

\operatorname{BranchPoints}\!\left(W_{k}\!\left(z\right), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{0, -{e}^{-1}, {\tilde \infty}\right\}

Assumptions:

k \in \left\{-1, 1\right\}

TeX:

\operatorname{BranchPoints}\!\left(W_{k}\!\left(z\right), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{0, -{e}^{-1}, {\tilde \infty}\right\}

k \in \left\{-1, 1\right\}

Definitions:

Fungrim symbol	Notation	Short description
`LambertW`	$W_{k}\!\left(z\right)$	Lambert W-function
`CC`	$\mathbb{C}$	Complex numbers
`UnsignedInfinity`	${\tilde \infty}$	Unsigned infinity
`Exp`	${e}^{z}$	Exponential function

Source code for this entry:

Entry(ID("17eaad"),
    Formula(Equal(BranchPoints(LambertW(k, z), z, Union(CC, Set(UnsignedInfinity))), Set(0, Neg(Exp(-1)), UnsignedInfinity))),
    Variables(k),
    Assumptions(Element(k, Set(-1, 1))))

e6e7a2

\operatorname{BranchPoints}\!\left(W_{k}\!\left(z\right), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{0, {\tilde \infty}\right\}

Assumptions:

k \in \mathbb{Z} \setminus \left\{-1, 0, 1\right\}

TeX:

\operatorname{BranchPoints}\!\left(W_{k}\!\left(z\right), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{0, {\tilde \infty}\right\}

k \in \mathbb{Z} \setminus \left\{-1, 0, 1\right\}

Definitions:

Fungrim symbol	Notation	Short description
`LambertW`	$W_{k}\!\left(z\right)$	Lambert W-function
`CC`	$\mathbb{C}$	Complex numbers
`UnsignedInfinity`	${\tilde \infty}$	Unsigned infinity
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("e6e7a2"),
    Formula(Equal(BranchPoints(LambertW(k, z), z, Union(CC, Set(UnsignedInfinity))), Set(0, UnsignedInfinity))),
    Variables(k),
    Assumptions(Element(k, SetMinus(ZZ, Set(-1, 0, 1)))))

fdfb16

\operatorname{BranchCuts}\!\left(W_{0}\!\left(z\right), z, \mathbb{C}\right) = \left\{\left(-\infty, -{e}^{-1}\right]\right\}

TeX:

\operatorname{BranchCuts}\!\left(W_{0}\!\left(z\right), z, \mathbb{C}\right) = \left\{\left(-\infty, -{e}^{-1}\right]\right\}

Definitions:

Fungrim symbol	Notation	Short description
`LambertW`	$W_{k}\!\left(z\right)$	Lambert W-function
`CC`	$\mathbb{C}$	Complex numbers
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`Infinity`	$\infty$	Positive infinity
`Exp`	${e}^{z}$	Exponential function

Source code for this entry:

Entry(ID("fdfb16"),
    Formula(Equal(BranchCuts(LambertW(0, z), z, CC), Set(OpenClosedInterval(Neg(Infinity), Neg(Exp(-1)))))))

276d78

\operatorname{BranchCuts}\!\left(W_{k}\!\left(z\right), z, \mathbb{C}\right) = \left\{\left(-\infty, -{e}^{-1}\right], \left[-{e}^{-1}, 0\right], \left(-\infty, 0\right]\right\}

Assumptions:

k \in \left\{-1, 1\right\}

TeX:

\operatorname{BranchCuts}\!\left(W_{k}\!\left(z\right), z, \mathbb{C}\right) = \left\{\left(-\infty, -{e}^{-1}\right], \left[-{e}^{-1}, 0\right], \left(-\infty, 0\right]\right\}

k \in \left\{-1, 1\right\}

Definitions:

Fungrim symbol	Notation	Short description
`LambertW`	$W_{k}\!\left(z\right)$	Lambert W-function
`CC`	$\mathbb{C}$	Complex numbers
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`Infinity`	$\infty$	Positive infinity
`Exp`	${e}^{z}$	Exponential function
`ClosedInterval`	$\left[a, b\right]$	Closed interval

Source code for this entry:

Entry(ID("276d78"),
    Formula(Equal(BranchCuts(LambertW(k, z), z, CC), Set(OpenClosedInterval(Neg(Infinity), Neg(Exp(-1))), ClosedInterval(Neg(Exp(-1)), 0), OpenClosedInterval(Neg(Infinity), 0)))),
    Variables(k),
    Assumptions(Element(k, Set(-1, 1))))

6191cd

\operatorname{BranchCuts}\!\left(W_{k}\!\left(z\right), z, \mathbb{C}\right) = \left\{\left(-\infty, 0\right]\right\}

Assumptions:

k \in \mathbb{Z} \setminus \left\{-1, 0, 1\right\}

TeX:

\operatorname{BranchCuts}\!\left(W_{k}\!\left(z\right), z, \mathbb{C}\right) = \left\{\left(-\infty, 0\right]\right\}

k \in \mathbb{Z} \setminus \left\{-1, 0, 1\right\}

Definitions:

Fungrim symbol	Notation	Short description
`LambertW`	$W_{k}\!\left(z\right)$	Lambert W-function
`CC`	$\mathbb{C}$	Complex numbers
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`Infinity`	$\infty$	Positive infinity
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("6191cd"),
    Formula(Equal(BranchCuts(LambertW(k, z), z, CC), Set(OpenClosedInterval(Neg(Infinity), 0)))),
    Variables(k),
    Assumptions(Element(k, SetMinus(ZZ, Set(-1, 0, 1)))))

f0f17c

\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} W_{0}\!\left(z\right) = \left\{0\right\}

TeX:

\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} W_{0}\!\left(z\right) = \left\{0\right\}

Definitions:

Fungrim symbol	Notation	Short description
`LambertW`	$W_{k}\!\left(z\right)$	Lambert W-function
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("f0f17c"),
    Formula(Equal(Zeros(LambertW(0, z), z, Element(z, CC)), Set(0))))

766302

\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} W_{k}\!\left(z\right) = \left\{\right\}

Assumptions:

k \in \mathbb{Z} \setminus \left\{0\right\}

TeX:

\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} W_{k}\!\left(z\right) = \left\{\right\}

k \in \mathbb{Z} \setminus \left\{0\right\}

Definitions:

Fungrim symbol	Notation	Short description
`LambertW`	$W_{k}\!\left(z\right)$	Lambert W-function
`CC`	$\mathbb{C}$	Complex numbers
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("766302"),
    Formula(Equal(Zeros(LambertW(k, z), z, Element(z, CC)), Set())),
    Variables(k),
    Assumptions(Element(k, SetMinus(ZZ, Set(0)))))

8d486c

W'_{k}\!\left(z\right) = \frac{1}{\left(1 + W_{k}\!\left(z\right)\right) \exp\!\left(W_{k}\!\left(z\right)\right)}

Assumptions:

\left(k = 0 \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{-{e}^{-1}\right\}\right) \,\mathbin{\operatorname{or}}\, \left(k = -1 \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{0, -{e}^{-1}\right\}\right) \,\mathbin{\operatorname{or}}\, \left(k \in \mathbb{Z} \setminus \left\{0, 1\right\} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{0\right\}\right)

TeX:

W'_{k}\!\left(z\right) = \frac{1}{\left(1 + W_{k}\!\left(z\right)\right) \exp\!\left(W_{k}\!\left(z\right)\right)}

\left(k = 0 \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{-{e}^{-1}\right\}\right) \,\mathbin{\operatorname{or}}\, \left(k = -1 \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{0, -{e}^{-1}\right\}\right) \,\mathbin{\operatorname{or}}\, \left(k \in \mathbb{Z} \setminus \left\{0, 1\right\} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{0\right\}\right)

Definitions:

Fungrim symbol	Notation	Short description
`LambertW`	$W_{k}\!\left(z\right)$	Lambert W-function
`Exp`	${e}^{z}$	Exponential function
`CC`	$\mathbb{C}$	Complex numbers
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("8d486c"),
    Formula(Equal(LambertW(k, z, 1), Div(1, Mul(Add(1, LambertW(k, z, 0)), Exp(LambertW(k, z, 0)))))),
    Variables(k, z),
    Assumptions(Or(And(Equal(k, 0), Element(z, SetMinus(CC, Set(Neg(Exp(-1)))))), And(Equal(k, -1), Element(z, SetMinus(CC, Set(0, Neg(Exp(-1)))))), And(Element(k, SetMinus(ZZ, Set(0, 1))), Element(z, SetMinus(CC, Set(0)))))))

72b6ca

W'_{k}\!\left(z\right) = \frac{W_{k}\!\left(z\right)}{z \left(1 + W_{k}\!\left(z\right)\right)}

Assumptions:

\left(k \in \left\{0, 1\right\} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{0, -{e}^{-1}\right\}\right) \,\mathbin{\operatorname{or}}\, \left(k \in \mathbb{Z} \setminus \left\{0, 1\right\} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{0\right\}\right)

TeX:

W'_{k}\!\left(z\right) = \frac{W_{k}\!\left(z\right)}{z \left(1 + W_{k}\!\left(z\right)\right)}

\left(k \in \left\{0, 1\right\} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{0, -{e}^{-1}\right\}\right) \,\mathbin{\operatorname{or}}\, \left(k \in \mathbb{Z} \setminus \left\{0, 1\right\} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{0\right\}\right)

Definitions:

Fungrim symbol	Notation	Short description
`LambertW`	$W_{k}\!\left(z\right)$	Lambert W-function
`CC`	$\mathbb{C}$	Complex numbers
`Exp`	${e}^{z}$	Exponential function
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("72b6ca"),
    Formula(Equal(LambertW(k, z, 1), Div(LambertW(k, z), Mul(z, Add(1, LambertW(k, z, 0)))))),
    Variables(k, z),
    Assumptions(Or(And(Element(k, Set(0, 1)), Element(z, SetMinus(CC, Set(0, Neg(Exp(-1)))))), And(Element(k, SetMinus(ZZ, Set(0, 1))), Element(z, SetMinus(CC, Set(0)))))))

58c19a

W_{0}\!\left(z\right) = \sum_{n=1}^{\infty} \frac{{\left(-n\right)}^{n - 1}}{n !} {z}^{n}

Assumptions:

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left|z\right| \lt \frac{1}{e}

TeX:

W_{0}\!\left(z\right) = \sum_{n=1}^{\infty} \frac{{\left(-n\right)}^{n - 1}}{n !} {z}^{n}

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left|z\right| \lt \frac{1}{e}

Definitions:

Fungrim symbol	Notation	Short description
`LambertW`	$W_{k}\!\left(z\right)$	Lambert W-function
`Pow`	${a}^{b}$	Power
`Factorial`	$n !$	Factorial
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`Abs`	$\left\|z\right\|$	Absolute value
`ConstE`	$e$	The constant e (2.718...)

Source code for this entry:

Entry(ID("58c19a"),
    Formula(Equal(LambertW(0, z), Sum(Mul(Div(Pow(Neg(n), Sub(n, 1)), Factorial(n)), Pow(z, n)), Tuple(n, 1, Infinity)))),
    Variables(z),
    Assumptions(And(Element(z, CC), Less(Abs(z), Div(1, ConstE)))))

c5a8c2

Symbol: LambertWPuiseuxCoefficient —

{\mu}_{k}

— Coefficient in scaled Puiseux expansion of Lambert W-function

Definitions:

Fungrim symbol	Notation	Short description
`LambertWPuiseuxCoefficient`	${\mu}_{k}$	Coefficient in scaled Puiseux expansion of Lambert W-function

Source code for this entry:

Entry(ID("c5a8c2"),
    SymbolDefinition(LambertWPuiseuxCoefficient, LambertWPuiseuxCoefficient(k), "Coefficient in scaled Puiseux expansion of Lambert W-function"))

0983d1

Table of

{\mu}_{k}

for

0 \le k \le 15

$k$	${\mu}_{k}$	$\operatorname{NearestDecimal}\!\left({\mu}_{k}, 30\right)$
0	-1	-1.00000000000000000000000000000
1	1	1.00000000000000000000000000000
2	-1/3	-0.333333333333333333333333333333
3	11/72	0.152777777777777777777777777778
4	-43/540	-0.0796296296296296296296296296296
5	769/17280	0.0445023148148148148148148148148
6	-221/8505	-0.0259847148736037624926513815403
7	680863/43545600	0.0156356325323339212228101116990
8	-1963/204120	-0.00961689202429943170683911424652
9	226287557/37623398400	0.00601454325295611786095325189975
10	-5776369/1515591000	-0.00381129803489199922670430215012
11	169709463197/69528040243200	0.00244087799114398266589685852864
12	-1118511313/709296588000	-0.00157693034468678425392340953993
13	667874164916771/650782456676352000	0.00102626332050760715443754815339
14	-500525573/744761417400	-0.000672061631156136204002020043419
15	103663334225097487/234281684403486720000	0.000442473061814620909930207608585

Table data:

\left(k, \mu, r\right)

such that

{\mu}_{k} = \mu \,\mathbin{\operatorname{and}}\, \operatorname{NearestDecimal}\!\left({\mu}_{k}, 30\right) = r

Definitions:

Fungrim symbol	Notation	Short description
`LambertWPuiseuxCoefficient`	${\mu}_{k}$	Coefficient in scaled Puiseux expansion of Lambert W-function

Source code for this entry:

Entry(ID("0983d1"),
    Description("Table of", LambertWPuiseuxCoefficient(k), "for", LessEqual(0, k, 15)),
    Table(TableRelation(Tuple(k, mu, r), And(Equal(LambertWPuiseuxCoefficient(k), mu), Equal(NearestDecimal(LambertWPuiseuxCoefficient(k), 30), r))), TableHeadings(k, LambertWPuiseuxCoefficient(k), NearestDecimal(LambertWPuiseuxCoefficient(k), 30)), TableSplit(1), List(Tuple(0, -1, Decimal("-1.00000000000000000000000000000")), Tuple(1, 1, Decimal("1.00000000000000000000000000000")), Tuple(2, Div(-1, 3), Decimal("-0.333333333333333333333333333333")), Tuple(3, Div(11, 72), Decimal("0.152777777777777777777777777778")), Tuple(4, Div(-43, 540), Decimal("-0.0796296296296296296296296296296")), Tuple(5, Div(769, 17280), Decimal("0.0445023148148148148148148148148")), Tuple(6, Div(-221, 8505), Decimal("-0.0259847148736037624926513815403")), Tuple(7, Div(680863, 43545600), Decimal("0.0156356325323339212228101116990")), Tuple(8, Div(-1963, 204120), Decimal("-0.00961689202429943170683911424652")), Tuple(9, Div(226287557, 37623398400), Decimal("0.00601454325295611786095325189975")), Tuple(10, Div(-5776369, 1515591000), Decimal("-0.00381129803489199922670430215012")), Tuple(11, Div(169709463197, 69528040243200), Decimal("0.00244087799114398266589685852864")), Tuple(12, Div(-1118511313, 709296588000), Decimal("-0.00157693034468678425392340953993")), Tuple(13, Div(667874164916771, 650782456676352000), Decimal("0.00102626332050760715443754815339")), Tuple(14, Div(-500525573, 744761417400), Decimal("-0.000672061631156136204002020043419")), Tuple(15, Div(103663334225097487, 234281684403486720000), Decimal("0.000442473061814620909930207608585")))))

d37d0f

{\mu}_{k} = \frac{k - 1}{k + 1} \left(\frac{{\mu}_{k - 2}}{2} + \frac{{\alpha}_{k - 2}}{4}\right) - \frac{{\alpha}_{k}}{2} - \frac{{\mu}_{k - 1}}{k + 1}\; \text{ where } {\alpha}_{0} = 2,\,{\alpha}_{1} = -1,\,{\alpha}_{k} = \sum_{j=2}^{k - 1} {\mu}_{j} {\mu}_{k + 1 - j}

Assumptions:

k \in \mathbb{Z}_{\ge 2}

TeX:

{\mu}_{k} = \frac{k - 1}{k + 1} \left(\frac{{\mu}_{k - 2}}{2} + \frac{{\alpha}_{k - 2}}{4}\right) - \frac{{\alpha}_{k}}{2} - \frac{{\mu}_{k - 1}}{k + 1}\; \text{ where } {\alpha}_{0} = 2,\,{\alpha}_{1} = -1,\,{\alpha}_{k} = \sum_{j=2}^{k - 1} {\mu}_{j} {\mu}_{k + 1 - j}

k \in \mathbb{Z}_{\ge 2}

Definitions:

Fungrim symbol	Notation	Short description
`LambertWPuiseuxCoefficient`	${\mu}_{k}$	Coefficient in scaled Puiseux expansion of Lambert W-function
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("d37d0f"),
    Formula(Where(Equal(LambertWPuiseuxCoefficient(k), Sub(Sub(Mul(Div(Sub(k, 1), Add(k, 1)), Add(Div(LambertWPuiseuxCoefficient(Sub(k, 2)), 2), Div(Subscript(alpha, Sub(k, 2)), 4))), Div(Subscript(alpha, k), 2)), Div(LambertWPuiseuxCoefficient(Sub(k, 1)), Add(k, 1)))), Equal(Subscript(alpha, 0), 2), Equal(Subscript(alpha, 1), -1), Equal(Subscript(alpha, k), Sum(Mul(LambertWPuiseuxCoefficient(j), LambertWPuiseuxCoefficient(Sub(Add(k, 1), j))), Tuple(j, 2, Sub(k, 1)))))),
    Variables(k),
    Assumptions(Element(k, ZZGreaterEqual(2))))

adf83a

\left|{\mu}_{k}\right| \lt 2 {\left(\frac{4}{5}\right)}^{k}

Assumptions:

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

TeX:

\left|{\mu}_{k}\right| \lt 2 {\left(\frac{4}{5}\right)}^{k}

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

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`LambertWPuiseuxCoefficient`	${\mu}_{k}$	Coefficient in scaled Puiseux expansion of Lambert W-function
`Pow`	${a}^{b}$	Power
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("adf83a"),
    Formula(Less(Abs(LambertWPuiseuxCoefficient(k)), Mul(2, Pow(Div(4, 5), k)))),
    Variables(k),
    Assumptions(Element(k, ZZGreaterEqual(0))))

e50532

W_{0}\!\left(z\right) = \sum_{n=0}^{\infty} {\mu}_{n} {v}^{n}\; \text{ where } v = \sqrt{2 \left(e z + 1\right)}

Assumptions:

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left|e z + 1\right| \lt 1

TeX:

W_{0}\!\left(z\right) = \sum_{n=0}^{\infty} {\mu}_{n} {v}^{n}\; \text{ where } v = \sqrt{2 \left(e z + 1\right)}

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left|e z + 1\right| \lt 1

Definitions:

Fungrim symbol	Notation	Short description
`LambertW`	$W_{k}\!\left(z\right)$	Lambert W-function
`LambertWPuiseuxCoefficient`	${\mu}_{k}$	Coefficient in scaled Puiseux expansion of Lambert W-function
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`Sqrt`	$\sqrt{z}$	Principal square root
`ConstE`	$e$	The constant e (2.718...)
`CC`	$\mathbb{C}$	Complex numbers
`Abs`	$\left\|z\right\|$	Absolute value

Source code for this entry:

Entry(ID("e50532"),
    Formula(Where(Equal(LambertW(0, z), Sum(Mul(LambertWPuiseuxCoefficient(n), Pow(v, n)), Tuple(n, 0, Infinity))), Equal(v, Sqrt(Mul(2, Add(Mul(ConstE, z), 1)))))),
    Variables(z),
    Assumptions(And(Element(z, CC), Less(Abs(Add(Mul(ConstE, z), 1)), 1))))

99ff4c

W_{k}\!\left(z\right) = \sum_{n=0}^{\infty} {\mu}_{n} {v}^{n}\; \text{ where } v = -\sqrt{2 \left(e z + 1\right)}

Assumptions:

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left|e z + 1\right| \lt 1 \,\mathbin{\operatorname{and}}\, \left(\left(k = -1 \,\mathbin{\operatorname{and}}\, \operatorname{Im}\!\left(z\right) \ge 0\right) \,\mathbin{\operatorname{or}}\, \left(k = 1 \,\mathbin{\operatorname{and}}\, \operatorname{Im}\!\left(z\right) \lt 0\right)\right)

TeX:

W_{k}\!\left(z\right) = \sum_{n=0}^{\infty} {\mu}_{n} {v}^{n}\; \text{ where } v = -\sqrt{2 \left(e z + 1\right)}

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left|e z + 1\right| \lt 1 \,\mathbin{\operatorname{and}}\, \left(\left(k = -1 \,\mathbin{\operatorname{and}}\, \operatorname{Im}\!\left(z\right) \ge 0\right) \,\mathbin{\operatorname{or}}\, \left(k = 1 \,\mathbin{\operatorname{and}}\, \operatorname{Im}\!\left(z\right) \lt 0\right)\right)

Definitions:

Fungrim symbol	Notation	Short description
`LambertW`	$W_{k}\!\left(z\right)$	Lambert W-function
`LambertWPuiseuxCoefficient`	${\mu}_{k}$	Coefficient in scaled Puiseux expansion of Lambert W-function
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`Sqrt`	$\sqrt{z}$	Principal square root
`ConstE`	$e$	The constant e (2.718...)
`CC`	$\mathbb{C}$	Complex numbers
`Abs`	$\left\|z\right\|$	Absolute value
`Im`	$\operatorname{Im}\!\left(z\right)$	Imaginary part

Source code for this entry:

Entry(ID("99ff4c"),
    Formula(Where(Equal(LambertW(k, z), Sum(Mul(LambertWPuiseuxCoefficient(n), Pow(v, n)), Tuple(n, 0, Infinity))), Equal(v, Neg(Sqrt(Mul(2, Add(Mul(ConstE, z), 1))))))),
    Variables(k, z),
    Assumptions(And(Element(z, CC), Less(Abs(Add(Mul(ConstE, z), 1)), 1), Or(And(Equal(k, -1), GreaterEqual(Im(z), 0)), And(Equal(k, 1), Less(Im(z), 0))))))

1fc63b

W_{k}\!\left(z\right) = {L}_{1} - {L}_{2} + \sum_{n=0}^{\infty} \sum_{m=1}^{\infty} \frac{{\left(-1\right)}^{n}}{m !} \left[{n + m \atop n + 1}\right] {\sigma}^{n} {\tau}^{m}\; \text{ where } {L}_{1} = \log\!\left(z\right) + 2 \pi i k,\,{L}_{2} = \log\!\left({L}_{1}\right),\,\sigma = \frac{1}{{L}_{1}},\,\tau = \frac{{L}_{2}}{{L}_{1}}

Assumptions:

k \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{0\right\} \,\mathbin{\operatorname{and}}\, \left|\sigma\right| \lt \frac{1}{4} \,\mathbin{\operatorname{and}}\, \left|\tau\right| \lt \frac{1}{4} \,\mathbin{\operatorname{and}}\, \left(k \ne 0 \,\mathbin{\operatorname{or}}\, \left|z\right| \gt 1\right)\; \text{ where } {L}_{1} = \log\!\left(z\right) + 2 \pi i k,\,{L}_{2} = \log\!\left({L}_{1}\right),\,\sigma = \frac{1}{{L}_{1}},\,\tau = \frac{{L}_{2}}{{L}_{1}}

TeX:

W_{k}\!\left(z\right) = {L}_{1} - {L}_{2} + \sum_{n=0}^{\infty} \sum_{m=1}^{\infty} \frac{{\left(-1\right)}^{n}}{m !} \left[{n + m \atop n + 1}\right] {\sigma}^{n} {\tau}^{m}\; \text{ where } {L}_{1} = \log\!\left(z\right) + 2 \pi i k,\,{L}_{2} = \log\!\left({L}_{1}\right),\,\sigma = \frac{1}{{L}_{1}},\,\tau = \frac{{L}_{2}}{{L}_{1}}

k \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{0\right\} \,\mathbin{\operatorname{and}}\, \left|\sigma\right| \lt \frac{1}{4} \,\mathbin{\operatorname{and}}\, \left|\tau\right| \lt \frac{1}{4} \,\mathbin{\operatorname{and}}\, \left(k \ne 0 \,\mathbin{\operatorname{or}}\, \left|z\right| \gt 1\right)\; \text{ where } {L}_{1} = \log\!\left(z\right) + 2 \pi i k,\,{L}_{2} = \log\!\left({L}_{1}\right),\,\sigma = \frac{1}{{L}_{1}},\,\tau = \frac{{L}_{2}}{{L}_{1}}

Definitions:

Fungrim symbol	Notation	Short description
`LambertW`	$W_{k}\!\left(z\right)$	Lambert W-function
`Pow`	${a}^{b}$	Power
`Factorial`	$n !$	Factorial
`StirlingCycle`	$\left[{n \atop k}\right]$	Unsigned Stirling number of the first kind
`Infinity`	$\infty$	Positive infinity
`Log`	$\log\!\left(z\right)$	Natural logarithm
`ConstPi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers
`Abs`	$\left\|z\right\|$	Absolute value

Source code for this entry:

Entry(ID("1fc63b"),
    Formula(Equal(LambertW(k, z), Where(Add(Sub(Subscript(L, 1), Subscript(L, 2)), Sum(Sum(Mul(Mul(Mul(Div(Pow(-1, n), Factorial(m)), StirlingCycle(Add(n, m), Add(n, 1))), Pow(sigma, n)), Pow(tau, m)), Tuple(m, 1, Infinity)), Tuple(n, 0, Infinity))), Equal(Subscript(L, 1), Add(Log(z), Mul(Mul(Mul(2, ConstPi), ConstI), k))), Equal(Subscript(L, 2), Log(Subscript(L, 1))), Equal(sigma, Div(1, Subscript(L, 1))), Equal(tau, Div(Subscript(L, 2), Subscript(L, 1)))))),
    Variables(k, z),
    Assumptions(Where(And(Element(k, ZZ), Element(z, SetMinus(CC, Set(0))), Less(Abs(sigma), Div(1, 4)), Less(Abs(tau), Div(1, 4)), Or(Unequal(k, 0), Greater(Abs(z), 1))), Equal(Subscript(L, 1), Add(Log(z), Mul(Mul(Mul(2, ConstPi), ConstI), k))), Equal(Subscript(L, 2), Log(Subscript(L, 1))), Equal(sigma, Div(1, Subscript(L, 1))), Equal(tau, Div(Subscript(L, 2), Subscript(L, 1))))))

da0f15

\left|W_{k}\!\left(z\right) - \left({L}_{1} - {L}_{2} + \sum_{n=0}^{N - 1} \sum_{m=1}^{M - 1} \frac{{\left(-1\right)}^{n}}{m !} \left[{n + m \atop n + 1}\right] {\sigma}^{n} {\tau}^{m}\right)\right| \le \frac{4 \left|\tau\right| {\left(4 \left|\sigma\right|\right)}^{N} + {\left(4 \left|\tau\right|\right)}^{M}}{\left(1 - 4 \left|\sigma\right|\right) \left(1 - 4 \left|\tau\right|\right)}\; \text{ where } {L}_{1} = \log\!\left(z\right) + 2 \pi i k,\,{L}_{2} = \log\!\left({L}_{1}\right),\,\sigma = \frac{1}{{L}_{1}},\,\tau = \frac{{L}_{2}}{{L}_{1}}

Assumptions:

N \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, M \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, k \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{0\right\} \,\mathbin{\operatorname{and}}\, \left|\sigma\right| \lt \frac{1}{4} \,\mathbin{\operatorname{and}}\, \left|\tau\right| \lt \frac{1}{4} \,\mathbin{\operatorname{and}}\, \left(k \ne 0 \,\mathbin{\operatorname{or}}\, \left|z\right| \gt 1\right)\; \text{ where } {L}_{1} = \log\!\left(z\right) + 2 \pi i k,\,{L}_{2} = \log\!\left({L}_{1}\right),\,\sigma = \frac{1}{{L}_{1}},\,\tau = \frac{{L}_{2}}{{L}_{1}}

TeX:

\left|W_{k}\!\left(z\right) - \left({L}_{1} - {L}_{2} + \sum_{n=0}^{N - 1} \sum_{m=1}^{M - 1} \frac{{\left(-1\right)}^{n}}{m !} \left[{n + m \atop n + 1}\right] {\sigma}^{n} {\tau}^{m}\right)\right| \le \frac{4 \left|\tau\right| {\left(4 \left|\sigma\right|\right)}^{N} + {\left(4 \left|\tau\right|\right)}^{M}}{\left(1 - 4 \left|\sigma\right|\right) \left(1 - 4 \left|\tau\right|\right)}\; \text{ where } {L}_{1} = \log\!\left(z\right) + 2 \pi i k,\,{L}_{2} = \log\!\left({L}_{1}\right),\,\sigma = \frac{1}{{L}_{1}},\,\tau = \frac{{L}_{2}}{{L}_{1}}

N \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, M \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, k \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{0\right\} \,\mathbin{\operatorname{and}}\, \left|\sigma\right| \lt \frac{1}{4} \,\mathbin{\operatorname{and}}\, \left|\tau\right| \lt \frac{1}{4} \,\mathbin{\operatorname{and}}\, \left(k \ne 0 \,\mathbin{\operatorname{or}}\, \left|z\right| \gt 1\right)\; \text{ where } {L}_{1} = \log\!\left(z\right) + 2 \pi i k,\,{L}_{2} = \log\!\left({L}_{1}\right),\,\sigma = \frac{1}{{L}_{1}},\,\tau = \frac{{L}_{2}}{{L}_{1}}

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`LambertW`	$W_{k}\!\left(z\right)$	Lambert W-function
`Pow`	${a}^{b}$	Power
`Factorial`	$n !$	Factorial
`StirlingCycle`	$\left[{n \atop k}\right]$	Unsigned Stirling number of the first kind
`Log`	$\log\!\left(z\right)$	Natural logarithm
`ConstPi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("da0f15"),
    Formula(Where(LessEqual(Abs(Sub(LambertW(k, z), Parentheses(Add(Sub(Subscript(L, 1), Subscript(L, 2)), Sum(Sum(Mul(Mul(Mul(Div(Pow(-1, n), Factorial(m)), StirlingCycle(Add(n, m), Add(n, 1))), Pow(sigma, n)), Pow(tau, m)), Tuple(m, 1, Sub(M, 1))), Tuple(n, 0, Sub(N, 1))))))), Div(Add(Mul(Mul(4, Abs(tau)), Pow(Mul(4, Abs(sigma)), N)), Pow(Mul(4, Abs(tau)), M)), Mul(Sub(1, Mul(4, Abs(sigma))), Sub(1, Mul(4, Abs(tau)))))), Equal(Subscript(L, 1), Add(Log(z), Mul(Mul(Mul(2, ConstPi), ConstI), k))), Equal(Subscript(L, 2), Log(Subscript(L, 1))), Equal(sigma, Div(1, Subscript(L, 1))), Equal(tau, Div(Subscript(L, 2), Subscript(L, 1))))),
    Variables(k, z, N, M),
    Assumptions(Where(And(Element(N, ZZGreaterEqual(0)), Element(M, ZZGreaterEqual(0)), Element(k, ZZ), Element(z, SetMinus(CC, Set(0))), Less(Abs(sigma), Div(1, 4)), Less(Abs(tau), Div(1, 4)), Or(Unequal(k, 0), Greater(Abs(z), 1))), Equal(Subscript(L, 1), Add(Log(z), Mul(Mul(Mul(2, ConstPi), ConstI), k))), Equal(Subscript(L, 2), Log(Subscript(L, 1))), Equal(sigma, Div(1, Subscript(L, 1))), Equal(tau, Div(Subscript(L, 2), Subscript(L, 1))))))

c0ae5b

\left\{ W_{k}\!\left(z\right) : k \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left(z \ne 0 \,\mathbin{\operatorname{or}}\, k = 0\right) \right\} = \mathbb{C}

TeX:

\left\{ W_{k}\!\left(z\right) : k \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left(z \ne 0 \,\mathbin{\operatorname{or}}\, k = 0\right) \right\} = \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`SetBuilder`	$\left\{ f\!\left(x\right) : P\!\left(x\right) \right\}$	Set comprehension
`LambertW`	$W_{k}\!\left(z\right)$	Lambert W-function
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("c0ae5b"),
    Formula(Equal(SetBuilder(LambertW(k, z), Tuple(k, z), And(Element(k, ZZ), Element(z, CC), Or(Unequal(z, 0), Equal(k, 0)))), CC)))

6e05c9

W_{{k}_{1}}\!\left({z}_{1}\right) \ne W_{{k}_{2}}\!\left({z}_{2}\right)

Assumptions:

{k}_{1} \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, {k}_{2} \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, {z}_{1} \in \mathbb{C} \,\mathbin{\operatorname{and}}\, {z}_{2} \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left({k}_{1} \ne {k}_{2} \,\mathbin{\operatorname{or}}\, {z}_{1} \ne {z}_{2}\right) \,\mathbin{\operatorname{and}}\, W_{{k}_{1}}\!\left({z}_{1}\right) \notin \left\{-1, -\infty\right\}

TeX:

W_{{k}_{1}}\!\left({z}_{1}\right) \ne W_{{k}_{2}}\!\left({z}_{2}\right)

{k}_{1} \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, {k}_{2} \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, {z}_{1} \in \mathbb{C} \,\mathbin{\operatorname{and}}\, {z}_{2} \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left({k}_{1} \ne {k}_{2} \,\mathbin{\operatorname{or}}\, {z}_{1} \ne {z}_{2}\right) \,\mathbin{\operatorname{and}}\, W_{{k}_{1}}\!\left({z}_{1}\right) \notin \left\{-1, -\infty\right\}

Definitions:

Fungrim symbol	Notation	Short description
`LambertW`	$W_{k}\!\left(z\right)$	Lambert W-function
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("6e05c9"),
    Formula(Unequal(LambertW(Subscript(k, 1), Subscript(z, 1)), LambertW(Subscript(k, 2), Subscript(z, 2)))),
    Variables(Subscript(k, 1), Subscript(z, 1), Subscript(k, 2), Subscript(z, 2)),
    Assumptions(And(Element(Subscript(k, 1), ZZ), Element(Subscript(k, 2), ZZ), Element(Subscript(z, 1), CC), Element(Subscript(z, 2), CC), Or(Unequal(Subscript(k, 1), Subscript(k, 2)), Unequal(Subscript(z, 1), Subscript(z, 2))), NotElement(LambertW(Subscript(k, 1), Subscript(z, 1)), Set(-1, Neg(Infinity))))))

ee86fb

\left\{ W_{0}\!\left(x\right) : x \in \left(-{e}^{-1}, \infty\right) \right\} = \left(-1, \infty\right)

TeX:

\left\{ W_{0}\!\left(x\right) : x \in \left(-{e}^{-1}, \infty\right) \right\} = \left(-1, \infty\right)

Definitions:

Fungrim symbol	Notation	Short description
`SetBuilder`	$\left\{ f\!\left(x\right) : P\!\left(x\right) \right\}$	Set comprehension
`LambertW`	$W_{k}\!\left(z\right)$	Lambert W-function
`OpenInterval`	$\left(a, b\right)$	Open interval
`Exp`	${e}^{z}$	Exponential function
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("ee86fb"),
    Formula(Equal(SetBuilder(LambertW(0, x), x, Element(x, OpenInterval(Neg(Exp(-1)), Infinity))), OpenInterval(-1, Infinity))))

55498b

\left\{ W_{0}\!\left(x\right) : x \in \left\{-{e}^{-1}\right\} \right\} = \left\{-1\right\}

TeX:

\left\{ W_{0}\!\left(x\right) : x \in \left\{-{e}^{-1}\right\} \right\} = \left\{-1\right\}

Definitions:

Fungrim symbol	Notation	Short description
`SetBuilder`	$\left\{ f\!\left(x\right) : P\!\left(x\right) \right\}$	Set comprehension
`LambertW`	$W_{k}\!\left(z\right)$	Lambert W-function
`Exp`	${e}^{z}$	Exponential function

Source code for this entry:

Entry(ID("55498b"),
    Formula(Equal(SetBuilder(LambertW(0, x), x, Element(x, Set(Neg(Exp(-1))))), Set(-1))))

44ad09

\left\{ W_{0}\!\left(x\right) : x \in \left(-\infty, -{e}^{-1}\right) \right\} = \left\{ -y \cot\!\left(y\right) + y i : y \in \left(0, \pi\right) \right\}

TeX:

\left\{ W_{0}\!\left(x\right) : x \in \left(-\infty, -{e}^{-1}\right) \right\} = \left\{ -y \cot\!\left(y\right) + y i : y \in \left(0, \pi\right) \right\}

Definitions:

Fungrim symbol	Notation	Short description
`SetBuilder`	$\left\{ f\!\left(x\right) : P\!\left(x\right) \right\}$	Set comprehension
`LambertW`	$W_{k}\!\left(z\right)$	Lambert W-function
`OpenInterval`	$\left(a, b\right)$	Open interval
`Infinity`	$\infty$	Positive infinity
`Exp`	${e}^{z}$	Exponential function
`ConstI`	$i$	Imaginary unit
`ConstPi`	$\pi$	The constant pi (3.14...)

Source code for this entry:

Entry(ID("44ad09"),
    Formula(Equal(SetBuilder(LambertW(0, x), x, Element(x, OpenInterval(Neg(Infinity), Neg(Exp(-1))))), SetBuilder(Add(Mul(Neg(y), Cot(y)), Mul(y, ConstI)), y, Element(y, OpenInterval(0, ConstPi))))))

2d3356

\left\{ W_{0}\!\left(z\right) : z \in \mathbb{C} \setminus \mathbb{R} \right\} = \left\{ x + y i : y \in \left(-\pi, \pi\right) \setminus \left\{0\right\} \,\mathbin{\operatorname{and}}\, x \in \left(-y \cot\!\left(y\right), \infty\right) \right\}

TeX:

\left\{ W_{0}\!\left(z\right) : z \in \mathbb{C} \setminus \mathbb{R} \right\} = \left\{ x + y i : y \in \left(-\pi, \pi\right) \setminus \left\{0\right\} \,\mathbin{\operatorname{and}}\, x \in \left(-y \cot\!\left(y\right), \infty\right) \right\}

Definitions:

Fungrim symbol	Notation	Short description
`SetBuilder`	$\left\{ f\!\left(x\right) : P\!\left(x\right) \right\}$	Set comprehension
`LambertW`	$W_{k}\!\left(z\right)$	Lambert W-function
`CC`	$\mathbb{C}$	Complex numbers
`RR`	$\mathbb{R}$	Real numbers
`ConstI`	$i$	Imaginary unit
`OpenInterval`	$\left(a, b\right)$	Open interval
`ConstPi`	$\pi$	The constant pi (3.14...)
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("2d3356"),
    Formula(Equal(SetBuilder(LambertW(0, z), z, Element(z, SetMinus(CC, RR))), SetBuilder(Add(x, Mul(y, ConstI)), Tuple(x, y), And(Element(y, SetMinus(OpenInterval(Neg(ConstPi), ConstPi), Set(0))), Element(x, OpenInterval(Mul(Neg(y), Cot(y)), Infinity)))))))

21d9a0

\left\{ W_{-1}\!\left(z\right) : z \in \mathbb{C} \setminus \left\{0\right\} \right\} = \left(-\infty, -1\right] \cup \left\{ x + y i : x \in \mathbb{R} \,\mathbin{\operatorname{and}}\, y \in \mathbb{R} \,\mathbin{\operatorname{and}}\, \left(\left(0 \lt u \lt 2 \,\mathbin{\operatorname{and}}\, t \le v\right) \,\mathbin{\operatorname{or}}\, \left(1 \le u \le 2\right) \,\mathbin{\operatorname{or}}\, \left(1 \lt u \lt 3 \,\mathbin{\operatorname{and}}\, t \gt v\right)\right)\; \text{ where } t = x \operatorname{sinc}\!\left(y\right),\,v = -\cos\!\left(y\right),\,u = -\frac{y}{\pi} \right\}

TeX:

\left\{ W_{-1}\!\left(z\right) : z \in \mathbb{C} \setminus \left\{0\right\} \right\} = \left(-\infty, -1\right] \cup \left\{ x + y i : x \in \mathbb{R} \,\mathbin{\operatorname{and}}\, y \in \mathbb{R} \,\mathbin{\operatorname{and}}\, \left(\left(0 \lt u \lt 2 \,\mathbin{\operatorname{and}}\, t \le v\right) \,\mathbin{\operatorname{or}}\, \left(1 \le u \le 2\right) \,\mathbin{\operatorname{or}}\, \left(1 \lt u \lt 3 \,\mathbin{\operatorname{and}}\, t \gt v\right)\right)\; \text{ where } t = x \operatorname{sinc}\!\left(y\right),\,v = -\cos\!\left(y\right),\,u = -\frac{y}{\pi} \right\}

Definitions:

Fungrim symbol	Notation	Short description
`SetBuilder`	$\left\{ f\!\left(x\right) : P\!\left(x\right) \right\}$	Set comprehension
`LambertW`	$W_{k}\!\left(z\right)$	Lambert W-function
`CC`	$\mathbb{C}$	Complex numbers
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`Infinity`	$\infty$	Positive infinity
`ConstI`	$i$	Imaginary unit
`RR`	$\mathbb{R}$	Real numbers
`ConstPi`	$\pi$	The constant pi (3.14...)

Source code for this entry:

Entry(ID("21d9a0"),
    Formula(Equal(SetBuilder(LambertW(-1, z), z, Element(z, SetMinus(CC, Set(0)))), Union(OpenClosedInterval(Neg(Infinity), -1), SetBuilder(Add(x, Mul(y, ConstI)), Tuple(x, y), Where(And(Element(x, RR), Element(y, RR), Or(And(Less(0, u, 2), LessEqual(t, v)), Parentheses(LessEqual(1, u, 2)), And(Less(1, u, 3), Greater(t, v)))), Equal(t, Mul(x, Sinc(y))), Equal(v, Neg(Cos(y))), Equal(u, Neg(Div(y, ConstPi)))))))))

d5917b

\left\{ W_{k}\!\left(z\right) : z \in \mathbb{C} \setminus \left\{0\right\} \right\} = \left\{ x + y i : x \in \mathbb{R} \,\mathbin{\operatorname{and}}\, y \in \mathbb{R} \,\mathbin{\operatorname{and}}\, \left(\left(2 k - 2 \lt u \lt 2 k \,\mathbin{\operatorname{and}}\, t \lt v\right) \,\mathbin{\operatorname{or}}\, \left(2 k - 1 \le u \le 2 k\right) \,\mathbin{\operatorname{or}}\, \left(2 k - 1 \lt u \lt 2 k + 1 \,\mathbin{\operatorname{and}}\, t \ge v\right)\right)\; \text{ where } t = x \operatorname{sinc}\!\left(y\right),\,v = -\cos\!\left(y\right),\,u = \frac{y}{\pi} \right\}

Assumptions:

k \in \mathbb{Z}_{\ge 1}

TeX:

\left\{ W_{k}\!\left(z\right) : z \in \mathbb{C} \setminus \left\{0\right\} \right\} = \left\{ x + y i : x \in \mathbb{R} \,\mathbin{\operatorname{and}}\, y \in \mathbb{R} \,\mathbin{\operatorname{and}}\, \left(\left(2 k - 2 \lt u \lt 2 k \,\mathbin{\operatorname{and}}\, t \lt v\right) \,\mathbin{\operatorname{or}}\, \left(2 k - 1 \le u \le 2 k\right) \,\mathbin{\operatorname{or}}\, \left(2 k - 1 \lt u \lt 2 k + 1 \,\mathbin{\operatorname{and}}\, t \ge v\right)\right)\; \text{ where } t = x \operatorname{sinc}\!\left(y\right),\,v = -\cos\!\left(y\right),\,u = \frac{y}{\pi} \right\}

k \in \mathbb{Z}_{\ge 1}

Definitions:

Fungrim symbol	Notation	Short description
`SetBuilder`	$\left\{ f\!\left(x\right) : P\!\left(x\right) \right\}$	Set comprehension
`LambertW`	$W_{k}\!\left(z\right)$	Lambert W-function
`CC`	$\mathbb{C}$	Complex numbers
`ConstI`	$i$	Imaginary unit
`RR`	$\mathbb{R}$	Real numbers
`ConstPi`	$\pi$	The constant pi (3.14...)
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("d5917b"),
    Formula(Equal(SetBuilder(LambertW(k, z), z, Element(z, SetMinus(CC, Set(0)))), SetBuilder(Add(x, Mul(y, ConstI)), Tuple(x, y), Where(And(Element(x, RR), Element(y, RR), Or(And(Less(Sub(Mul(2, k), 2), u, Mul(2, k)), Less(t, v)), Parentheses(LessEqual(Sub(Mul(2, k), 1), u, Mul(2, k))), And(Less(Sub(Mul(2, k), 1), u, Add(Mul(2, k), 1)), GreaterEqual(t, v)))), Equal(t, Mul(x, Sinc(y))), Equal(v, Neg(Cos(y))), Equal(u, Div(y, ConstPi)))))),
    Variables(k),
    Assumptions(Element(k, ZZGreaterEqual(1))))

bf3e29

\left\{ W_{-k}\!\left(z\right) : z \in \mathbb{C} \setminus \left\{0\right\} \right\} = \left\{ x + y i : x \in \mathbb{R} \,\mathbin{\operatorname{and}}\, y \in \mathbb{R} \,\mathbin{\operatorname{and}}\, \left(\left(2 k - 2 \lt u \lt 2 k \,\mathbin{\operatorname{and}}\, t \le v\right) \,\mathbin{\operatorname{or}}\, \left(2 k - 1 \le u \le 2 k\right) \,\mathbin{\operatorname{or}}\, \left(2 k - 1 \lt u \lt 2 k + 1 \,\mathbin{\operatorname{and}}\, t \gt v\right)\right)\; \text{ where } t = x \operatorname{sinc}\!\left(y\right),\,v = -\cos\!\left(y\right),\,u = -\frac{y}{\pi} \right\}

Assumptions:

k \in \mathbb{Z}_{\ge 2}

TeX:

\left\{ W_{-k}\!\left(z\right) : z \in \mathbb{C} \setminus \left\{0\right\} \right\} = \left\{ x + y i : x \in \mathbb{R} \,\mathbin{\operatorname{and}}\, y \in \mathbb{R} \,\mathbin{\operatorname{and}}\, \left(\left(2 k - 2 \lt u \lt 2 k \,\mathbin{\operatorname{and}}\, t \le v\right) \,\mathbin{\operatorname{or}}\, \left(2 k - 1 \le u \le 2 k\right) \,\mathbin{\operatorname{or}}\, \left(2 k - 1 \lt u \lt 2 k + 1 \,\mathbin{\operatorname{and}}\, t \gt v\right)\right)\; \text{ where } t = x \operatorname{sinc}\!\left(y\right),\,v = -\cos\!\left(y\right),\,u = -\frac{y}{\pi} \right\}

k \in \mathbb{Z}_{\ge 2}

Definitions:

Fungrim symbol	Notation	Short description
`SetBuilder`	$\left\{ f\!\left(x\right) : P\!\left(x\right) \right\}$	Set comprehension
`LambertW`	$W_{k}\!\left(z\right)$	Lambert W-function
`CC`	$\mathbb{C}$	Complex numbers
`ConstI`	$i$	Imaginary unit
`RR`	$\mathbb{R}$	Real numbers
`ConstPi`	$\pi$	The constant pi (3.14...)
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("bf3e29"),
    Formula(Equal(SetBuilder(LambertW(Neg(k), z), z, Element(z, SetMinus(CC, Set(0)))), SetBuilder(Add(x, Mul(y, ConstI)), Tuple(x, y), Where(And(Element(x, RR), Element(y, RR), Or(And(Less(Sub(Mul(2, k), 2), u, Mul(2, k)), LessEqual(t, v)), Parentheses(LessEqual(Sub(Mul(2, k), 1), u, Mul(2, k))), And(Less(Sub(Mul(2, k), 1), u, Add(Mul(2, k), 1)), Greater(t, v)))), Equal(t, Mul(x, Sinc(y))), Equal(v, Neg(Cos(y))), Equal(u, Neg(Div(y, ConstPi))))))),
    Variables(k),
    Assumptions(Element(k, ZZGreaterEqual(2))))

4257f4

\left|\operatorname{Im}\!\left(W_{0}\!\left(z\right)\right)\right| \lt \pi

Assumptions:

z \in \mathbb{C}

TeX:

\left|\operatorname{Im}\!\left(W_{0}\!\left(z\right)\right)\right| \lt \pi

z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`Im`	$\operatorname{Im}\!\left(z\right)$	Imaginary part
`LambertW`	$W_{k}\!\left(z\right)$	Lambert W-function
`ConstPi`	$\pi$	The constant pi (3.14...)
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("4257f4"),
    Formula(Less(Abs(Im(LambertW(0, z))), ConstPi)),
    Variables(z),
    Assumptions(Element(z, CC)))

82926c

\operatorname{Im}\!\left(W_{1}\!\left(z\right)\right) \in \left(0, 3 \pi\right)

Assumptions:

z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

\operatorname{Im}\!\left(W_{1}\!\left(z\right)\right) \in \left(0, 3 \pi\right)

z \in \mathbb{C} \setminus \left\{0\right\}

Definitions:

Fungrim symbol	Notation	Short description
`Im`	$\operatorname{Im}\!\left(z\right)$	Imaginary part
`LambertW`	$W_{k}\!\left(z\right)$	Lambert W-function
`OpenInterval`	$\left(a, b\right)$	Open interval
`ConstPi`	$\pi$	The constant pi (3.14...)
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("82926c"),
    Formula(Element(Im(LambertW(1, z)), OpenInterval(0, Mul(3, ConstPi)))),
    Variables(z),
    Assumptions(Element(z, SetMinus(CC, Set(0)))))

e5bba3

\operatorname{Im}\!\left(W_{-1}\!\left(z\right)\right) \in \left(-3 \pi, 0\right]

Assumptions:

z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

\operatorname{Im}\!\left(W_{-1}\!\left(z\right)\right) \in \left(-3 \pi, 0\right]

z \in \mathbb{C} \setminus \left\{0\right\}

Definitions:

Fungrim symbol	Notation	Short description
`Im`	$\operatorname{Im}\!\left(z\right)$	Imaginary part
`LambertW`	$W_{k}\!\left(z\right)$	Lambert W-function
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`ConstPi`	$\pi$	The constant pi (3.14...)
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("e5bba3"),
    Formula(Element(Im(LambertW(-1, z)), OpenClosedInterval(Neg(Mul(3, ConstPi)), 0))),
    Variables(z),
    Assumptions(Element(z, SetMinus(CC, Set(0)))))

a68e0e

\operatorname{Im}\!\left(W_{k}\!\left(z\right)\right) \in \left(\operatorname{sgn}\!\left(k\right) \left(2 \left|k\right| - 2\right) \pi, \operatorname{sgn}\!\left(k\right) \left(2 \left|k\right| + 1\right) \pi\right)

Assumptions:

z \in \mathbb{C} \setminus \left\{0\right\} \,\mathbin{\operatorname{and}}\, k \in \mathbb{Z} \setminus \left\{-1, 0, 1\right\}

TeX:

\operatorname{Im}\!\left(W_{k}\!\left(z\right)\right) \in \left(\operatorname{sgn}\!\left(k\right) \left(2 \left|k\right| - 2\right) \pi, \operatorname{sgn}\!\left(k\right) \left(2 \left|k\right| + 1\right) \pi\right)

z \in \mathbb{C} \setminus \left\{0\right\} \,\mathbin{\operatorname{and}}\, k \in \mathbb{Z} \setminus \left\{-1, 0, 1\right\}

Definitions:

Fungrim symbol	Notation	Short description
`Im`	$\operatorname{Im}\!\left(z\right)$	Imaginary part
`LambertW`	$W_{k}\!\left(z\right)$	Lambert W-function
`OpenInterval`	$\left(a, b\right)$	Open interval
`Sign`	$\operatorname{sgn}\!\left(z\right)$	Sign function
`Abs`	$\left\|z\right\|$	Absolute value
`ConstPi`	$\pi$	The constant pi (3.14...)
`CC`	$\mathbb{C}$	Complex numbers
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("a68e0e"),
    Formula(Element(Im(LambertW(k, z)), OpenInterval(Mul(Mul(Sign(k), Sub(Mul(2, Abs(k)), 2)), ConstPi), Mul(Mul(Sign(k), Add(Mul(2, Abs(k)), 1)), ConstPi)))),
    Variables(z, k),
    Assumptions(And(Element(z, SetMinus(CC, Set(0))), Element(k, SetMinus(ZZ, Set(-1, 0, 1))))))

f171a6

W'_{0}\!\left(x\right) \le \frac{1}{x + 1}

Assumptions:

x \in \left[0, \infty\right)

TeX:

W'_{0}\!\left(x\right) \le \frac{1}{x + 1}

x \in \left[0, \infty\right)

Definitions:

Fungrim symbol	Notation	Short description
`LambertW`	$W_{k}\!\left(z\right)$	Lambert W-function
`ClosedOpenInterval`	$\left[a, b\right)$	Closed-open interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("f171a6"),
    Formula(LessEqual(LambertW(0, x, 1), Div(1, Add(x, 1)))),
    Variables(x),
    Assumptions(Element(x, ClosedOpenInterval(0, Infinity))))

a34260

W'_{0}\!\left(x\right) \lt \frac{2}{\sqrt{1 + e x}}

Assumptions:

x \in \left(-{e}^{-1}, \infty\right)

TeX:

W'_{0}\!\left(x\right) \lt \frac{2}{\sqrt{1 + e x}}

x \in \left(-{e}^{-1}, \infty\right)

Definitions:

Fungrim symbol	Notation	Short description
`LambertW`	$W_{k}\!\left(z\right)$	Lambert W-function
`Sqrt`	$\sqrt{z}$	Principal square root
`ConstE`	$e$	The constant e (2.718...)
`OpenInterval`	$\left(a, b\right)$	Open interval
`Exp`	${e}^{z}$	Exponential function
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("a34260"),
    Formula(Less(LambertW(0, x, 1), Div(2, Sqrt(Add(1, Mul(ConstE, x)))))),
    Variables(x),
    Assumptions(Element(x, OpenInterval(Neg(Exp(-1)), Infinity))))

9be916

\left|W'_{-1}\!\left(x\right)\right| \lt \frac{2}{\sqrt{1 + e x}} + \frac{2}{\left|x\right|}

Assumptions:

x \in \left(-{e}^{-1}, 0\right)

TeX:

\left|W'_{-1}\!\left(x\right)\right| \lt \frac{2}{\sqrt{1 + e x}} + \frac{2}{\left|x\right|}

x \in \left(-{e}^{-1}, 0\right)

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`LambertW`	$W_{k}\!\left(z\right)$	Lambert W-function
`Sqrt`	$\sqrt{z}$	Principal square root
`ConstE`	$e$	The constant e (2.718...)
`OpenInterval`	$\left(a, b\right)$	Open interval
`Exp`	${e}^{z}$	Exponential function

Source code for this entry:

Entry(ID("9be916"),
    Formula(Less(Abs(LambertW(-1, x, 1)), Add(Div(2, Sqrt(Add(1, Mul(ConstE, x)))), Div(2, Abs(x))))),
    Variables(x),
    Assumptions(Element(x, OpenInterval(Neg(Exp(-1)), 0))))

b3d435

\left|W'_{k}\!\left(z\right)\right| \le \left|\frac{1.2}{z}\right|

Assumptions:

k \in \mathbb{Z} \setminus \left\{-1, 0, 1\right\} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C}

TeX:

\left|W'_{k}\!\left(z\right)\right| \le \left|\frac{1.2}{z}\right|

k \in \mathbb{Z} \setminus \left\{-1, 0, 1\right\} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`LambertW`	$W_{k}\!\left(z\right)$	Lambert W-function
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("b3d435"),
    Formula(LessEqual(Abs(LambertW(k, z, 1)), Abs(Div(Decimal("1.2"), z)))),
    Variables(k, z),
    Assumptions(And(Element(k, SetMinus(ZZ, Set(-1, 0, 1))), Element(z, CC))))

8e06be

\left|W'_{k}\!\left(z\right)\right| \le \left|\frac{1.5}{z}\right|

Assumptions:

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left(\left(k = 1 \,\mathbin{\operatorname{and}}\, \operatorname{Im}\!\left(z\right) \ge 0\right) \,\mathbin{\operatorname{or}}\, \left(k = -1 \,\mathbin{\operatorname{and}}\, \operatorname{Im}\!\left(z\right) \lt 0\right)\right)

TeX:

\left|W'_{k}\!\left(z\right)\right| \le \left|\frac{1.5}{z}\right|

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left(\left(k = 1 \,\mathbin{\operatorname{and}}\, \operatorname{Im}\!\left(z\right) \ge 0\right) \,\mathbin{\operatorname{or}}\, \left(k = -1 \,\mathbin{\operatorname{and}}\, \operatorname{Im}\!\left(z\right) \lt 0\right)\right)

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`LambertW`	$W_{k}\!\left(z\right)$	Lambert W-function
`CC`	$\mathbb{C}$	Complex numbers
`Im`	$\operatorname{Im}\!\left(z\right)$	Imaginary part

Source code for this entry:

Entry(ID("8e06be"),
    Formula(LessEqual(Abs(LambertW(k, z, 1)), Abs(Div(Decimal("1.5"), z)))),
    Variables(k, z),
    Assumptions(And(Element(z, CC), Or(And(Equal(k, 1), GreaterEqual(Im(z), 0)), And(Equal(k, -1), Less(Im(z), 0))))))

72712c

\left|W'_{k}\!\left(z\right)\right| \le \frac{1}{\left|z\right|}

Assumptions:

\left(k = 0 \,\mathbin{\operatorname{and}}\, \left|z\right| \ge 1\right) \,\mathbin{\operatorname{or}}\, \left(k \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left|z\right| \ge 4 \left(\left|k\right| + 1\right)\right)

TeX:

\left|W'_{k}\!\left(z\right)\right| \le \frac{1}{\left|z\right|}

\left(k = 0 \,\mathbin{\operatorname{and}}\, \left|z\right| \ge 1\right) \,\mathbin{\operatorname{or}}\, \left(k \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left|z\right| \ge 4 \left(\left|k\right| + 1\right)\right)

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`LambertW`	$W_{k}\!\left(z\right)$	Lambert W-function
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("72712c"),
    Formula(LessEqual(Abs(LambertW(k, z, 1)), Div(1, Abs(z)))),
    Variables(k, z),
    Assumptions(Or(And(Equal(k, 0), GreaterEqual(Abs(z), 1)), And(Element(k, ZZ), Element(z, CC), GreaterEqual(Abs(z), Mul(4, Add(Abs(k), 1)))))))

9136b9

\left|W'_{k}\!\left(z\right)\right| \le \left|\frac{1}{z}\right| \max\!\left(3, \left|\frac{1.5}{\sqrt{\left|e z + 1\right|}}\right|\right)

Assumptions:

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

TeX:

\left|W'_{k}\!\left(z\right)\right| \le \left|\frac{1}{z}\right| \max\!\left(3, \left|\frac{1.5}{\sqrt{\left|e z + 1\right|}}\right|\right)

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

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`LambertW`	$W_{k}\!\left(z\right)$	Lambert W-function
`Sqrt`	$\sqrt{z}$	Principal square root
`ConstE`	$e$	The constant e (2.718...)
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("9136b9"),
    Formula(LessEqual(Abs(LambertW(k, z, 1)), Mul(Abs(Div(1, z)), Max(3, Abs(Div(Decimal("1.5"), Sqrt(Abs(Add(Mul(ConstE, z), 1))))))))),
    Variables(k, z),
    Assumptions(And(Element(k, ZZ), Element(z, CC))))

0eb699

\left|W'_{0}\!\left(z\right)\right| \le \left|\frac{2.25}{\sqrt{t \left(1 + t\right)}}\right|\; \text{ where } t = \left|e z + 1\right|

Assumptions:

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left|z\right| \le 64

TeX:

\left|W'_{0}\!\left(z\right)\right| \le \left|\frac{2.25}{\sqrt{t \left(1 + t\right)}}\right|\; \text{ where } t = \left|e z + 1\right|

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left|z\right| \le 64

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`LambertW`	$W_{k}\!\left(z\right)$	Lambert W-function
`Sqrt`	$\sqrt{z}$	Principal square root
`ConstE`	$e$	The constant e (2.718...)
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("0eb699"),
    Formula(Where(LessEqual(Abs(LambertW(0, z, 1)), Abs(Div(Decimal("2.25"), Sqrt(Mul(t, Add(1, t)))))), Equal(t, Abs(Add(Mul(ConstE, z), 1))))),
    Variables(z),
    Assumptions(And(Element(z, CC), LessEqual(Abs(z), 64))))

214b1c

\left|W'_{k}\!\left(z\right)\right| \le \left|\frac{1}{\left|z\right|} \left(1 + \frac{1}{4 + {\left|z\right|}^{2}}\right)\right|

Assumptions:

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left(\left(k \in \left\{1, -1\right\} \,\mathbin{\operatorname{and}}\, \operatorname{Re}\!\left(z\right) \ge 0\right) \,\mathbin{\operatorname{or}}\, \left(k = -1 \,\mathbin{\operatorname{and}}\, \operatorname{Im}\!\left(z\right) \lt 0\right) \,\mathbin{\operatorname{or}}\, \left(k = 1 \,\mathbin{\operatorname{and}}\, \operatorname{Im}\!\left(z\right) \ge 0\right)\right)

TeX:

\left|W'_{k}\!\left(z\right)\right| \le \left|\frac{1}{\left|z\right|} \left(1 + \frac{1}{4 + {\left|z\right|}^{2}}\right)\right|

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left(\left(k \in \left\{1, -1\right\} \,\mathbin{\operatorname{and}}\, \operatorname{Re}\!\left(z\right) \ge 0\right) \,\mathbin{\operatorname{or}}\, \left(k = -1 \,\mathbin{\operatorname{and}}\, \operatorname{Im}\!\left(z\right) \lt 0\right) \,\mathbin{\operatorname{or}}\, \left(k = 1 \,\mathbin{\operatorname{and}}\, \operatorname{Im}\!\left(z\right) \ge 0\right)\right)

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`LambertW`	$W_{k}\!\left(z\right)$	Lambert W-function
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`Re`	$\operatorname{Re}\!\left(z\right)$	Real part
`Im`	$\operatorname{Im}\!\left(z\right)$	Imaginary part

Source code for this entry:

Entry(ID("214b1c"),
    Formula(LessEqual(Abs(LambertW(k, z, 1)), Abs(Mul(Div(1, Abs(z)), Add(1, Div(1, Add(4, Pow(Abs(z), 2)))))))),
    Variables(k, z),
    Assumptions(And(Element(z, CC), Or(And(Element(k, Set(1, -1)), GreaterEqual(Re(z), 0)), And(Equal(k, -1), Less(Im(z), 0)), And(Equal(k, 1), GreaterEqual(Im(z), 0))))))

a1e634

\left|W'_{k}\!\left(z\right)\right| \le \left|\frac{1}{\left|z\right|} \left(1 + \frac{23}{32} \frac{1}{\sqrt{\left|e z + 1\right|}}\right)\right|

Assumptions:

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, k \in \left\{-1, 1\right\}

TeX:

\left|W'_{k}\!\left(z\right)\right| \le \left|\frac{1}{\left|z\right|} \left(1 + \frac{23}{32} \frac{1}{\sqrt{\left|e z + 1\right|}}\right)\right|

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, k \in \left\{-1, 1\right\}

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`LambertW`	$W_{k}\!\left(z\right)$	Lambert W-function
`Sqrt`	$\sqrt{z}$	Principal square root
`ConstE`	$e$	The constant e (2.718...)
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("a1e634"),
    Formula(LessEqual(Abs(LambertW(k, z, 1)), Abs(Mul(Div(1, Abs(z)), Add(1, Mul(Div(23, 32), Div(1, Sqrt(Abs(Add(Mul(ConstE, z), 1)))))))))),
    Variables(k, z),
    Assumptions(And(Element(z, CC), Element(k, Set(-1, 1)))))

Definition

Illustrations

Transcendental equations

Specific values

Symmetry

Analytic properties

Derivatives and integrals

Series expansions

Taylor series

Puiseux series

Logarithmic expansion

Range

Tiling of the plane

Image of the principal branch

Image of the non-principal branches

Bounds and inequalities

Complex parts

Derivative bounds