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http://functions.wolfram.com/01.32.06.0002.01
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ProductLog[k, z] == w - \[Zeta] -
Sum[Sum[(((-1)^n StirlingS1[n, 1 - m + n])/m!) \[Zeta]^m, {m, 1, n}]/w^n,
{n, 1, Infinity}] /; (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])
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ProductLog", "[", RowBox[List["k", ",", "z"]], "]"]], "\[Equal]", RowBox[List["w", "-", "\[Zeta]", "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["n", "=", "1"]], InterpretationBox["\[Infinity]", DirectedInfinity[1]]], RowBox[List[SuperscriptBox["w", RowBox[List["-", "n"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "1"]], "n"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", " ", RowBox[List["StirlingS1", "[", RowBox[List["n", ",", RowBox[List["1", "-", "m", "+", "n"]]]], "]"]]]], RowBox[List["m", "!"]]], SuperscriptBox["\[Zeta]", "m"]]]]]]]]]]]]], "/;", " ", RowBox[List[RowBox[List["(", RowBox[List["z", "\[Rule]", RowBox[List["0", "|", "\[Infinity]"]]]], ")"]], "\[And]", RowBox[List["w", "\[Equal]", RowBox[List[RowBox[List["Log", "[", "z", "]"]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "k", " ", "\[Pi]"]]]]]], "\[And]", RowBox[List["\[Zeta]", "\[Equal]", RowBox[List["Log", "[", "w", "]"]]]], "\[And]", RowBox[List["Not", "[", RowBox[List[RowBox[List["k", "\[Equal]", "0"]], "\[And]", " ", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "<", SuperscriptBox["\[ExponentialE]", "2"]]]]], "]"]], "\[And]", RowBox[List["Not", "[", RowBox[List[RowBox[List["k", "\[Equal]", RowBox[List["-", "1"]]]], "\[And]", " ", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["1", "\[ExponentialE]"]]], ",", "0"]], "}"]], "]"]], ",", "z"]], "]"]]]], "]"]]]]]]]]
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<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> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> n </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <semantics> <mi> ∞ </mi> <annotation-xml encoding='MathML-Content'> <infinity /> </annotation-xml> </semantics> </munderover> <mrow> <msup> <mi> w </mi> <mrow> <mo> - </mo> <mi> n </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> m </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msup> <mrow> <mi> m </mi> <mo> ! </mo> </mrow> </mfrac> <mo> ⁢ </mo> <msubsup> <semantics> <mi> S </mi> <annotation encoding='Mathematica'> TagBox["S", StirlingS1] </annotation> </semantics> <mi> n </mi> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> m </mi> </mrow> <mo> + </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </msubsup> <mo> ⁢ </mo> <msup> <mi> ζ </mi> <mi> m </mi> </msup> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mrow> <mn> 0 </mn> <mo> | </mo> <mi> ∞ </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ∧ </mo> <mrow> <mi> w </mi> <mo> ⩵ </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> + </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mi> ζ </mi> <mo> ⩵ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> w </mi> <mo> ) </mo> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mo> ¬ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> k </mi> <mo> ⩵ </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mi> z </mi> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> < </mo> <msup> <mi> ⅇ </mi> <mn> 2 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mo> ¬ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> k </mi> <mo> ⩵ </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mi> z </mi> <mo> ∈ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mi> ⅇ </mi> </mfrac> </mrow> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <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> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> n </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <semantics> <mi> ∞ </mi> <annotation-xml encoding='MathML-Content'> <infinity /> </annotation-xml> </semantics> </munderover> <mrow> <msup> <mi> w </mi> <mrow> <mo> - </mo> <mi> n </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> m </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msup> <mrow> <mi> m </mi> <mo> ! </mo> </mrow> </mfrac> <mo> ⁢ </mo> <msubsup> <semantics> <mi> S </mi> <annotation encoding='Mathematica'> TagBox["S", StirlingS1] </annotation> </semantics> <mi> n </mi> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> m </mi> </mrow> <mo> + </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </msubsup> <mo> ⁢ </mo> <msup> <mi> ζ </mi> <mi> m </mi> </msup> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mrow> <mn> 0 </mn> <mo> | </mo> <mi> ∞ </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ∧ </mo> <mrow> <mi> w </mi> <mo> ⩵ </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> + </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mi> ζ </mi> <mo> ⩵ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> w </mi> <mo> ) </mo> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mo> ¬ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> k </mi> <mo> ⩵ </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mi> z </mi> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> < </mo> <msup> <mi> ⅇ </mi> <mn> 2 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mo> ¬ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> k </mi> <mo> ⩵ </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mi> z </mi> <mo> ∈ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mi> ⅇ </mi> </mfrac> </mrow> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ProductLog", "[", RowBox[List["k_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["w", "-", "\[Zeta]", "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["n", "=", "1"]], "\[Infinity]"], RowBox[List[SuperscriptBox["w", RowBox[List["-", "n"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "1"]], "n"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["StirlingS1", "[", RowBox[List["n", ",", RowBox[List["1", "-", "m", "+", "n"]]]], "]"]]]], ")"]], " ", SuperscriptBox["\[Zeta]", "m"]]], RowBox[List["m", "!"]]]]]]]]]]], "/;", 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"]], "]"]]]], ")"]]]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
