Generalized Lambert function: Series representations (formula 01.32.06.0001)

ProductLog

$Mathematica Notation$

Elementary Functions

ProductLog[k,z]

Series representations

Asymptotic series expansions

http://functions.wolfram.com/01.32.06.0001.02

Input Form

ProductLog[k, z] \[Proportional] w - \[Zeta] + \[Zeta]/w - (2 \[Zeta] - \[Zeta]^2)/(2 w^2) + (\[Zeta] (6 - 9 \[Zeta] + 2 \[Zeta]^2))/ (6 w^3) + \[Ellipsis] /; (z -> 0 | Infinity) && w == Log[z] + 2 I k Pi && \[Zeta] == Log[w] && !(k == 0 && Abs[z] < E^2) && !(k == -1 && IntervalMemberQ[Interval[{-(1/E), 0}], z])

Standard Form

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MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msub> <mi> W </mi> <mi> k </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ∝ </mo> <mrow> <mi> w </mi> <mo> - </mo> <mi> ζ </mi> <mo> + </mo> <mfrac> <mi> ζ </mi> <mi> w </mi> </mfrac> <mo> - </mo> <mfrac> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ζ </mi> </mrow> <mo> - </mo> <msup> <mi> ζ </mi> <mn> 2 </mn> </msup> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mi> w </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> + </mo> <mfrac> <mrow> <mi> ζ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 6 </mn> <mo> - </mo> <mrow> <mn> 9 </mn> <mo> ⁢ </mo> <mi> ζ </mi> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mi> ζ </mi> <mn> 2 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 6 </mn> <mo> ⁢ </mo> <msup> <mi> w </mi> <mn> 3 </mn> </msup> </mrow> </mfrac> <mo> + </mo> <mo> … </mo> </mrow> </mrow> <mo> /; 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Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ProductLog", "[", RowBox[List["k_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["w", "-", "\[Zeta]", "+", FractionBox["\[Zeta]", "w"], "-", FractionBox[RowBox[List[RowBox[List["2", " ", "\[Zeta]"]], "-", SuperscriptBox["\[Zeta]", "2"]]], RowBox[List["2", " ", SuperscriptBox["w", "2"]]]], "+", FractionBox[RowBox[List["\[Zeta]", " ", RowBox[List["(", RowBox[List["6", "-", RowBox[List["9", " ", "\[Zeta]"]], "+", RowBox[List["2", " ", SuperscriptBox["\[Zeta]", "2"]]]]], ")"]]]], RowBox[List["6", " ", SuperscriptBox["w", "3"]]]], "+", "\[Ellipsis]"]], "/;", RowBox[List[RowBox[List["(", RowBox[List["z", "\[Rule]", RowBox[List["0", "|", "\[Infinity]"]]]], ")"]], "&&", RowBox[List["w", "\[Equal]", RowBox[List[RowBox[List["Log", "[", "z", "]"]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "k", " ", "\[Pi]"]]]]]], "&&", RowBox[List["\[Zeta]", "\[Equal]", RowBox[List["Log", "[", "w", "]"]]]], "&&", RowBox[List["!", RowBox[List["(", RowBox[List[RowBox[List["k", "\[Equal]", "0"]], "&&", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "<", SuperscriptBox["\[ExponentialE]", "2"]]]]], ")"]]]], "&&", RowBox[List["!", RowBox[List["(", RowBox[List[RowBox[List["k", "\[Equal]", RowBox[List["-", "1"]]]], "&&", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["1", "\[ExponentialE]"]]], ",", "0"]], "}"]], "]"]], ",", "z"]], "]"]]]], ")"]]]]]]]]]]]]

Date Added to functions.wolfram.com (modification date)

2001-10-29

ProductLog[z]