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http://functions.wolfram.com/01.31.09.0002.01
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ProductLog[-Log[x]] == (-Log[x]) Limit[Nest[x^#1 & , x, n], n -> Infinity] /;
E^(-E) < x < E^(1/E)
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ProductLog", "[", RowBox[List["-", RowBox[List["Log", "[", "x", "]"]]]], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", RowBox[List["Log", "[", "x", "]"]]]], " ", RowBox[List["Limit", "[", RowBox[List[RowBox[List["Nest", "[", RowBox[List[RowBox[List[SuperscriptBox["x", "#1"], "&"]], ",", "x", ",", "n"]], "]"]], ",", RowBox[List["n", "\[Rule]", "\[Infinity]"]]]], "]"]]]]]], "/;", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", "\[ExponentialE]"]]], "<", "x", "<", SuperscriptBox["\[ExponentialE]", RowBox[List["1", "/", "\[ExponentialE]"]]]]]]]]]
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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> <mrow> <mo> - </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mo> - </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> <mo> ⁢ </mo> <mrow> <munder> <mi> lim </mi> <mrow> <mi> n </mi> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mi> ∞ </mi> </mrow> </munder> <mo> ⁢ </mo> <mtext>   </mtext> <msup> <mi> x </mi> <msup> <mi> x </mi> <mrow> <mrow> <mstyle minsize='4'> <mrow> <msup> <mo> . </mo> <msup> <mo> . </mo> <msup> <mo> . </mo> <mstyle minsize='3'> <mi> x </mi> </mstyle> </msup> </msup> </msup> <mo> } </mo> </mrow> </mstyle> <mo> ⁢ </mo> <mstyle minsize='3'> <mi> n </mi> </mstyle> </mrow> <mo minsize='3'> - </mo> <mstyle minsize='3'> <mi> times </mi> </mstyle> </mrow> </msup> </msup> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mo> - </mo> <mi> ⅇ </mi> </mrow> </msup> <mo> < </mo> <mi> x </mi> <mo> < </mo> <msup> <mi> ⅇ </mi> <mrow> <mn> 1 </mn> <mo> / </mo> <mi> ⅇ </mi> </mrow> </msup> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <mrow> <mi> W </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mo> - </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> <mo> ⁢ </mo> <mrow> <munder> <mi> lim </mi> <mrow> <mi> n </mi> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mi> ∞ </mi> </mrow> </munder> <mo> ⁢ </mo> <mtext>   </mtext> <msup> <mi> x </mi> <msup> <mi> x </mi> <mrow> <mrow> <mstyle minsize='4'> <mrow> <msup> <mo> . </mo> <msup> <mo> . </mo> <msup> <mo> . </mo> <mstyle minsize='3'> <mi> x </mi> </mstyle> </msup> </msup> </msup> <mo> } </mo> </mrow> </mstyle> <mo> ⁢ </mo> <mstyle minsize='3'> <mi> n </mi> </mstyle> </mrow> <mo minsize='3'> - </mo> <mstyle minsize='3'> <mi> times </mi> </mstyle> </mrow> </msup> </msup> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mo> - </mo> <mi> ⅇ </mi> </mrow> </msup> <mo> < </mo> <mi> x </mi> <mo> < </mo> <msup> <mi> ⅇ </mi> <mrow> <mn> 1 </mn> <mo> / </mo> <mi> ⅇ </mi> </mrow> </msup> </mrow> </mrow> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ProductLog", "[", RowBox[List["-", RowBox[List["Log", "[", "x_", "]"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["Log", "[", "x", "]"]]]], " ", RowBox[List["Limit", "[", RowBox[List[RowBox[List["Nest", "[", RowBox[List[RowBox[List[SuperscriptBox["x", "#1"], "&"]], ",", "x", ",", "n"]], "]"]], ",", RowBox[List["n", "\[Rule]", "\[Infinity]"]]]], "]"]]]], "/;", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", "\[ExponentialE]"]]], "<", "x", "<", SuperscriptBox["\[ExponentialE]", RowBox[List["1", "/", "\[ExponentialE]"]]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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