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http://functions.wolfram.com/01.32.20.0002.01
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D[ProductLog[k, z], {z, 2}] == -((ProductLog[k, z]^2 (2 + ProductLog[k, z]))/
(z^2 (1 + ProductLog[k, z])^3))
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Cell[BoxData[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "2"]], "}"]]], RowBox[List["ProductLog", "[", RowBox[List["k", ",", "z"]], "]"]]]], "\[Equal]", RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["ProductLog", "[", RowBox[List["k", ",", "z"]], "]"]], "2"], " ", RowBox[List["(", RowBox[List["2", "+", RowBox[List["ProductLog", "[", RowBox[List["k", ",", "z"]], "]"]]]], ")"]]]], RowBox[List[SuperscriptBox["z", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["ProductLog", "[", RowBox[List["k", ",", "z"]], "]"]]]], ")"]], "3"]]]]]]]]]]
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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> <mfrac> <mrow> <msup> <mo> ∂ </mo> <mn> 2 </mn> </msup> <mrow> <msub> <mi> W </mi> <mi> k </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> ∂ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> ⩵ </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <msup> <mrow> <msub> <mi> W </mi> <mi> k </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msub> <mi> W </mi> <mi> k </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <msub> <mi> W </mi> <mi> k </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mrow> </mfrac> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> z </ci> <degree> <cn type='integer'> 2 </cn> </degree> </bvar> <apply> <apply> <ci> Subscript </ci> <ci> W </ci> <ci> k </ci> </apply> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <apply> <ci> Subscript </ci> <ci> W </ci> <ci> k </ci> </apply> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <apply> <ci> Subscript </ci> <ci> W </ci> <ci> k </ci> </apply> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <apply> <ci> Subscript </ci> <ci> W </ci> <ci> k </ci> </apply> <ci> z </ci> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 3 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z_", ",", "2"]], "}"]]]]], RowBox[List["ProductLog", "[", RowBox[List["k_", ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["ProductLog", "[", RowBox[List["k", ",", "z"]], "]"]], "2"], " ", RowBox[List["(", RowBox[List["2", "+", RowBox[List["ProductLog", "[", RowBox[List["k", ",", "z"]], "]"]]]], ")"]]]], RowBox[List[SuperscriptBox["z", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["ProductLog", "[", RowBox[List["k", ",", "z"]], "]"]]]], ")"]], "3"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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