E^(n ProductLog[k, z]) == z^n/ProductLog[k, z]^n /; Element[n, Integers]

Cell[BoxData[RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["n", " ", RowBox[List["ProductLog", "[", RowBox[List["k", ",", "z"]], "]"]]]]], "\[Equal]", RowBox[List[SuperscriptBox["z", "n"], " ", SuperscriptBox[RowBox[List["ProductLog", "[", RowBox[List["k", ",", "z"]], "]"]], RowBox[List["-", "n"]]]]]]], "/;", RowBox[List["Element", "[", RowBox[List["n", ",", "Integers"]], "]"]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mi> n </mi> <mo> &#8290; </mo> <mrow> <mstyle fontweight='normal' fontstyle='normal'> <msub> <mi> W </mi> <mi> k </mi> </msub> </mstyle> <mo> ( </mo> <mi fontweight='normal' fontstyle='italic'> z </mi> <mo fontweight='normal' fontstyle='normal'> ) </mo> </mrow> </mrow> </msup> <mo> &#10869; </mo> <mrow> <msup> <mi> z </mi> <mi> n </mi> </msup> <mo> &#8290; </mo> <msup> <mrow> <mstyle fontweight='normal' fontstyle='normal'> <msub> <mi> W </mi> <mi> k </mi> </msub> </mstyle> <mo> ( </mo> <mi fontweight='normal' fontstyle='italic'> z </mi> <mo fontweight='normal' fontstyle='normal'> ) </mo> </mrow> <mrow> <mo> - </mo> <mi> n </mi> </mrow> </msup> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <power /> <exponentiale /> <apply> <times /> <ci> n </ci> <apply> <apply> <ci> Subscript </ci> <ci> W </ci> <ci> k </ci> </apply> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <ci> z </ci> <ci> n </ci> </apply> <apply> <power /> <apply> <apply> <ci> Subscript </ci> <ci> W </ci> <ci> k </ci> </apply> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <integers /> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", SuperscriptBox["\[ExponentialE]", RowBox[List["n_", " ", RowBox[List["ProductLog", "[", RowBox[List["k_", ",", "z_"]], "]"]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["z", "n"], " ", SuperscriptBox[RowBox[List["ProductLog", "[", RowBox[List["k", ",", "z"]], "]"]], RowBox[List["-", "n"]]]]], "/;", RowBox[List["n", "\[Element]", "Integers"]]]]]]]]

2001-10-29