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http://functions.wolfram.com/01.31.07.0001.01
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ProductLog[z] == 1 + (Log[z] - 1)
Exp[(I/(2 Pi)) Integrate[(1/(1 + t)) Log[(Log[z] + t - Log[t] - I Pi)/
(Log[z] + t - Log[t] + I Pi)], {t, 0, Infinity}]] /;
!IntervalMemberQ[Interval[{-(1/E), 0}], z]
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ProductLog", "[", "z", "]"]], "\[Equal]", RowBox[List["1", "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "z", "]"]], "-", "1"]], ")"]], " ", RowBox[List["Exp", "[", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " "]], RowBox[List["2", " ", "\[Pi]"]]], RowBox[List[SubsuperscriptBox["\[Integral]", "0", InterpretationBox["\[Infinity]", DirectedInfinity[1]]], RowBox[List[FractionBox["1", RowBox[List["1", "+", "t"]]], RowBox[List["Log", "[", FractionBox[RowBox[List[RowBox[List["Log", "[", "z", "]"]], "+", "t", "-", RowBox[List["Log", "[", "t", "]"]], "-", RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]]]], RowBox[List[RowBox[List["Log", "[", "z", "]"]], "+", "t", "-", RowBox[List["Log", "[", "t", "]"]], "+", " ", RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]]]]], "]"]], RowBox[List["\[DifferentialD]", "t"]]]]]]]], "]"]]]]]]]], "/;", RowBox[List["Not", "[", 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> <mi> W </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> exp </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> ⅈ </mi> <mtext> </mtext> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <msubsup> <mo> ∫ </mo> <mn> 0 </mn> <semantics> <mi> ∞ </mi> <annotation-xml encoding='MathML-Content'> <infinity /> </annotation-xml> </semantics> </msubsup> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mi> t </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mi> t </mi> <mo> - </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mtext> </mtext> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mrow> <mrow> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mi> t </mi> <mo> - </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mtext> </mtext> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> t </mi> </mrow> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </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> <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> <mn> 1 </mn> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> exp </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> ⅈ </mi> <mtext> </mtext> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <msubsup> <mo> ∫ </mo> <mn> 0 </mn> <semantics> <mi> ∞ </mi> <annotation-xml encoding='MathML-Content'> <infinity /> </annotation-xml> </semantics> </msubsup> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mi> t </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mi> t </mi> <mo> - </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mtext> </mtext> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mrow> <mrow> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mi> t </mi> <mo> - </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mtext> </mtext> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> t </mi> </mrow> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </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> <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["1", "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "z", "]"]], "-", "1"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[FractionBox[RowBox[List["Log", "[", FractionBox[RowBox[List[RowBox[List["Log", "[", "z", "]"]], "+", "t", "-", RowBox[List["Log", "[", "t", "]"]], "-", RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]]]], RowBox[List[RowBox[List["Log", "[", "z", "]"]], "+", "t", "-", RowBox[List["Log", "[", "t", "]"]], "+", RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]]]]], "]"]], RowBox[List["1", "+", "t"]]], RowBox[List["\[DifferentialD]", "t"]]]]]]]], RowBox[List["2", " ", "\[Pi]"]]]]]]]], "/;", RowBox[List["!", 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}
