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http://functions.wolfram.com/01.31.07.0002.01
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ProductLog[z] == Integrate[(-(1/Pi)) Im[D[ProductLog[x], x]] Log[1 - z/x],
{x, -Infinity, -(1/E)}] /;
!IntervalMemberQ[Interval[{-Infinity, -(1/E)}], z]
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ProductLog", "[", "z", "]"]], "\[Equal]", RowBox[List[SubsuperscriptBox["\[Integral]", RowBox[List["-", "\[Infinity]"]], RowBox[List["-", FractionBox["1", "\[ExponentialE]"]]]], RowBox[List[RowBox[List["-", FractionBox["1", "\[Pi]"]]], RowBox[List["Im", "[", RowBox[List[SubscriptBox["\[PartialD]", "x"], RowBox[List["ProductLog", "[", "x", "]"]]]], "]"]], " ", RowBox[List["Log", "[", RowBox[List["1", "-", FractionBox["z", "x"]]], "]"]], RowBox[List["\[DifferentialD]", "x"]]]]]]]], "/;", RowBox[List["Not", "[", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "\[Infinity]"]], ",", RowBox[List["-", FractionBox["1", "\[ExponentialE]"]]]]], "}"]], "]"]], ",", "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> <mi> W </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <msubsup> <mo> ∫ </mo> <mrow> <mo> - </mo> <mi> ∞ </mi> </mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mi> ⅇ </mi> </mfrac> </mrow> </msubsup> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mi> π </mi> </mfrac> </mrow> <mo> ⁢ </mo> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mo> ∂ </mo> <mrow> <mi> W </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> ∂ </mo> <mi> x </mi> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mi> z </mi> <mi> x </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> x </mi> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> z </mi> <mo> ∉ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> ∞ </mi> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mi> ⅇ </mi> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> <mtext> ​ </mtext> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <mrow> <mi> W </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <msubsup> <mo> ∫ </mo> <mrow> <mo> - </mo> <mi> ∞ </mi> </mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mi> ⅇ </mi> </mfrac> </mrow> </msubsup> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mi> π </mi> </mfrac> </mrow> <mo> ⁢ </mo> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mo> ∂ </mo> <mrow> <mi> W </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> ∂ </mo> <mi> x </mi> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mi> z </mi> <mi> x </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> x </mi> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> z </mi> <mo> ∉ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> ∞ </mi> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mi> ⅇ </mi> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> <mtext> ​ </mtext> </mrow> </mrow> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ProductLog", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", RowBox[List["-", "\[Infinity]"]], RowBox[List["-", FractionBox["1", "\[ExponentialE]"]]]], RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["Im", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["x"]]], RowBox[List["ProductLog", "[", "x", "]"]]]], "]"]], " ", RowBox[List["Log", "[", RowBox[List["1", "-", FractionBox["z", "x"]]], "]"]]]], "\[Pi]"]]], RowBox[List["\[DifferentialD]", "x"]]]]]], "/;", RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "\[Infinity]"]], ",", RowBox[List["-", FractionBox["1", "\[ExponentialE]"]]]]], "}"]], "]"]], ",", "z"]], "]"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
