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http://functions.wolfram.com/09.14.20.0008.01
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D[WeierstrassPPrime[z, {Subscript[g, 2], Subscript[g, 3]}], {z, k}] ==
(-1)^(k - 1) (k + 2)!
Sum[1/(z - 2 m Subscript[\[Omega], 1] - 2 n Subscript[\[Omega], 3])^
(3 + k), {m, -Infinity, Infinity}, {n, -Infinity, Infinity}] /;
Element[k, Integers] && k > 0
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "k"]], "}"]]], RowBox[List["WeierstrassPPrime", "[", RowBox[List["z", ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]]]], "\[Equal]", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "-", "1"]]], RowBox[List[RowBox[List["(", RowBox[List["k", "+", "2"]], ")"]], "!"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", RowBox[List["-", "\[Infinity]"]]]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["n", "=", RowBox[List["-", "\[Infinity]"]]]], "\[Infinity]"], FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", RowBox[List["2", " ", "m", " ", SubscriptBox["\[Omega]", "1"]]], "-", RowBox[List["2", "n", " ", SubscriptBox["\[Omega]", "3"]]]]], ")"]], RowBox[List["3", "+", "k"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["k", "\[Element]", "Integers"]], "\[And]", RowBox[List["k", ">", "0"]]]]]]]]
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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> <mfrac> <mrow> <msup> <mo> ∂ </mo> <mi> k </mi> </msup> <mrow> <msup> <mi> ℘ </mi> <mo> ′ </mo> </msup> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> ; </mo> <msub> <mi> g </mi> <mn> 2 </mn> </msub> </mrow> <mo> , </mo> <msub> <mi> g </mi> <mn> 3 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> ∂ </mo> <msup> <mi> z </mi> <mi> k </mi> </msup> </mrow> </mfrac> <mo> ⩵ </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> m </mi> <mo> = </mo> <mrow> <mo> - </mo> <mi> ∞ </mi> </mrow> </mrow> <mi> ∞ </mi> </munderover> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> n </mi> <mo> = </mo> <mrow> <mo> - </mo> <mi> ∞ </mi> </mrow> </mrow> <mi> ∞ </mi> </munderover> <mfrac> <mn> 1 </mn> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> m </mi> <mo> ⁢ </mo> <msub> <mi> ω </mi> <mn> 1 </mn> </msub> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> <mo> ⁢ </mo> <msub> <mi> ω </mi> <mn> 3 </mn> </msub> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> k </mi> <mo> + </mo> <mn> 3 </mn> </mrow> </msup> </mfrac> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> k </mi> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> z </ci> <degree> <ci> k </ci> </degree> </bvar> <apply> <apply> <partialdiff /> <list> <cn type='integer'> 1 </cn> </list> <ci> ℘ </ci> </apply> <apply> <ci> CompoundExpression </ci> <ci> z </ci> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <factorial /> <apply> <plus /> <ci> k </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> n </ci> </bvar> <lowlimit> <apply> <times /> <cn type='integer'> -1 </cn> <infinity /> </apply> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <sum /> <bvar> <ci> m </ci> </bvar> <lowlimit> <apply> <times /> <cn type='integer'> -1 </cn> <infinity /> </apply> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> m </ci> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> </apply> <apply> <plus /> <ci> k </ci> <cn type='integer'> 3 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> k </ci> <apply> <ci> SuperPlus </ci> <ci> ℕ </ci> </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_", ",", "k"]], "}"]]]]], RowBox[List["WeierstrassPPrime", "[", RowBox[List["z_", ",", RowBox[List["{", RowBox[List[SubscriptBox["g_", "2"], ",", SubscriptBox["g_", "3"]]], "}"]]]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "-", "1"]]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "+", "2"]], ")"]], "!"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", RowBox[List["-", "\[Infinity]"]]]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["n", "=", RowBox[List["-", "\[Infinity]"]]]], "\[Infinity]"], FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", RowBox[List["2", " ", "m", " ", SubscriptBox["\[Omega]", "1"]]], "-", RowBox[List["2", " ", "n", " ", SubscriptBox["\[Omega]", "3"]]]]], ")"]], RowBox[List["3", "+", "k"]]]]]]]]]], "/;", RowBox[List[RowBox[List["k", "\[Element]", "Integers"]], "&&", RowBox[List["k", ">", "0"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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