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variants of this functions
ProductLog




Mathematica Notation

Traditional Notation









Elementary Functions > ProductLog[z] > Limit representations





http://functions.wolfram.com/01.31.09.0001.01









  


  








Input Form





ProductLog[x] == Limit[Nest[x Exp[-#1] & , x, n], n -> Infinity] /; 0 < x < E








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ProductLog", "[", "x", "]"]], "\[Equal]", RowBox[List["Limit", "[", RowBox[List[RowBox[List["Nest", "[", RowBox[List[RowBox[List[RowBox[List["x", " ", RowBox[List["Exp", "[", RowBox[List["-", "#"]], "]"]]]], "&"]], ",", "x", ",", "n"]], "]"]], ",", RowBox[List["n", "\[Rule]", "\[Infinity]"]]]], "]"]]]], "/;", RowBox[List["0", "<", "x", "<", "\[ExponentialE]"]]]]]]








MathML Form







<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> &#8289; </mo> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mstyle scriptlevel='0'> <munder> <mi> lim </mi> <mrow> <mi> n </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mi> &#8734; </mi> </mrow> </munder> </mstyle> <mo> &#8290; </mo> <munder> <munder> <mrow> <mi> x </mi> <mo> &#8290; </mo> <mrow> <mi> exp </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> x </mi> </mrow> <mo> &#8290; </mo> <mrow> <mi> exp </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mo> &#8230; </mo> </mrow> <mo> &#8290; </mo> <mi> x </mi> <mo> &#8290; </mo> <mrow> <mi> exp </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mi> x </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#65080; </mo> </munder> <mrow> <mi> n </mi> <mo> - </mo> <mi> times </mi> </mrow> </munder> </mrow> </mrow> <mo> /; </mo> <mrow> <mn> 0 </mn> <mo> &lt; </mo> <mi> x </mi> <mo> &lt; </mo> <mi> &#8519; </mi> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> W </ci> <ci> x </ci> </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> <times /> <ci> x </ci> <apply> <exp /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> x </ci> </apply> <apply> <exp /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#8230; </ci> </apply> <ci> x </ci> <apply> <exp /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> x </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <ci> &#65080; </ci> </apply> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> times </ci> </apply> </apply> </apply> </apply> </apply> <apply> <lt /> <cn type='integer'> 0 </cn> <ci> x </ci> <exponentiale /> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ProductLog", "[", "x_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["Limit", "[", RowBox[List[RowBox[List["Nest", "[", RowBox[List[RowBox[List[RowBox[List["x", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", "#1"]]]]], "&"]], ",", "x", ",", "n"]], "]"]], ",", RowBox[List["n", "\[Rule]", "\[Infinity]"]]]], "]"]], "/;", RowBox[List["0", "<", "x", "<", "\[ExponentialE]"]]]]]]]]


Date Added to functions.wolfram.com (modification date)





2001-10-29