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http://functions.wolfram.com/01.32.09.0001.01
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ProductLog[-1, -Log[x]] == (-Log[x]) Limit[Nest[Log[x, #1] & , E, n],
n -> Infinity] /; x > 1
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ProductLog", "[", RowBox[List[RowBox[List["-", "1"]], ",", " ", 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[RowBox[List["Log", "[", RowBox[List["x", ",", "#"]], "]"]], "&"]], ",", "E", ",", "n"]], "]"]], ",", RowBox[List["n", "\[Rule]", "\[Infinity]"]]]], "]"]]]]]], "/;", RowBox[List["x", ">", "1"]]]]]]
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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> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <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> <mstyle scriptlevel='0'> <munder> <mi> lim </mi> <mrow> <mi> n </mi> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mi> ∞ </mi> </mrow> </munder> </mstyle> <mo> ⁢ </mo> <munder> <munder> <mrow> <mtext> </mtext> <mrow> <msub> <mi> log </mi> <mi> x </mi> </msub> <mo> ( </mo> <mrow> <msub> <mi> log </mi> <mi> x </mi> </msub> <mo> ( </mo> <mrow> <msub> <mi> …log </mi> <mi> x </mi> </msub> <mo> ( </mo> <mi> 𝕖 </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ︸ </mo> </munder> <mrow> <mi> n </mi> <mo> - </mo> <mi> times </mi> </mrow> </munder> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> x </mi> <mo> > </mo> <mn> 1 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <apply> <ci> Subscript </ci> <ci> W </ci> <cn type='integer'> -1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <ci> x </ci> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <ci> x </ci> </apply> </apply> <apply> <limit /> <bvar> <ci> n </ci> </bvar> <condition> <apply> <tendsto /> <ci> n </ci> <infinity /> </apply> </condition> <apply> <ci> Underscript </ci> <apply> <ci> Underscript </ci> <apply> <log /> <logbase> <ci> x </ci> </logbase> <apply> <log /> <logbase> <ci> x </ci> </logbase> <apply> <apply> <ci> Subscript </ci> <ci> …log </ci> <ci> x </ci> </apply> <ci> 𝕖 </ci> </apply> </apply> </apply> <ci> ︸ </ci> </apply> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> times </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <gt /> <ci> x </ci> <cn type='integer'> 1 </cn> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ProductLog", "[", RowBox[List[RowBox[List["-", "1"]], ",", 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[RowBox[List["Log", "[", RowBox[List["x", ",", "#1"]], "]"]], "&"]], ",", "\[ExponentialE]", ",", "n"]], "]"]], ",", RowBox[List["n", "\[Rule]", "\[Infinity]"]]]], "]"]]]], "/;", RowBox[List["x", ">", "1"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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