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variants of this functions
ProductLog




Mathematica Notation

Traditional Notation









Elementary Functions > ProductLog[k,z] > General characteristics > Branch cuts > With respect to z





http://functions.wolfram.com/01.32.04.0018.01









  


  








Input Form





Limit[ProductLog[1, x - I \[Epsilon]], \[Epsilon] -> Plus[0]] == ProductLog[-1, x] /; -(1/E) < x < 0








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Limit", "[", RowBox[List[RowBox[List["ProductLog", "[", RowBox[List["1", ",", RowBox[List["x", "-", RowBox[List["\[ImaginaryI]", " ", "\[Epsilon]"]]]]]], "]"]], ",", RowBox[List["\[Epsilon]", "\[Rule]", RowBox[List["+", "0"]]]]]], "]"]], "\[Equal]", RowBox[List["ProductLog", "[", RowBox[List[RowBox[List["-", "1"]], ",", " ", "x"]], "]"]]]], "/;", " ", RowBox[List[RowBox[List["-", FractionBox["1", "\[ExponentialE]"]]], "<", "x", "<", "0"]]]]]]








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> <munder> <mi> lim </mi> <mrow> <mi> &#1013; </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mrow> <mo> + </mo> <mn> 0 </mn> </mrow> </mrow> </munder> <mo> &#8290; </mo> <mrow> <msub> <mi> W </mi> <mn> 1 </mn> </msub> <mo> ( </mo> <mrow> <mi> x </mi> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mtext> </mtext> <mi> &#1013; </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <msub> <mi> W </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mi> &#8519; </mi> </mfrac> </mrow> <mo> &lt; </mo> <mi> x </mi> <mo> &lt; </mo> <mn> 0 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <limit /> <bvar> <ci> &#1013; </ci> </bvar> <condition> <apply> <tendsto /> <ci> &#1013; </ci> <apply> <plus /> <cn type='integer'> 0 </cn> </apply> </apply> </condition> <apply> <apply> <ci> Subscript </ci> <ci> W </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> x </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> &#1013; </ci> </apply> </apply> </apply> </apply> </apply> <apply> <apply> <ci> Subscript </ci> <ci> W </ci> <cn type='integer'> -1 </cn> </apply> <ci> x </ci> </apply> </apply> <apply> <lt /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <exponentiale /> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <ci> x </ci> <cn type='integer'> 0 </cn> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Limit", "[", RowBox[List[RowBox[List["ProductLog", "[", RowBox[List["1", ",", RowBox[List["x_", "-", RowBox[List["\[ImaginaryI]", " ", "\[Epsilon]_"]]]]]], "]"]], ",", RowBox[List["\[Epsilon]_", "\[Rule]", RowBox[List["+", "0"]]]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["ProductLog", "[", RowBox[List[RowBox[List["-", "1"]], ",", "x"]], "]"]], "/;", RowBox[List[RowBox[List["-", FractionBox["1", "\[ExponentialE]"]]], "<", "x", "<", "0"]]]]]]]]


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





2001-10-29