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WeierstrassInvariants






Mathematica Notation

Traditional Notation









Elliptic Functions > WeierstrassInvariants[{w1,w3}] > Differentiation > Symbolic differentiation > With respect to omega3





http://functions.wolfram.com/09.19.20.0008.01









  


  










Input Form





D[WeierstrassInvariants[{Subscript[\[Omega], 1], Subscript[\[Omega], 3]}], {Subscript[\[Omega], 3], k}] == {(5/4) Sum[((-1)^k (k + 3)! n^k)/(m Subscript[\[Omega], 1] + n Subscript[\[Omega], 3])^(4 + k), {m, -Infinity, Infinity}, {n, 1, Infinity}], (7/192) Sum[((-1)^k (k + 5)! n^k)/ (m Subscript[\[Omega], 1] + n Subscript[\[Omega], 3])^(6 + k), {m, -Infinity, Infinity}, {n, 1, Infinity}]} /; Element[k, Integers] && k > 0










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List[SubscriptBox["\[Omega]", "3"], ",", "k"]], "}"]]], RowBox[List["WeierstrassInvariants", "[", RowBox[List["{", RowBox[List[SubscriptBox["\[Omega]", "1"], ",", SubscriptBox["\[Omega]", "3"]]], "}"]], "]"]]]], "\[Equal]", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["5", "4"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", RowBox[List["-", "\[Infinity]"]]]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["n", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], RowBox[List[RowBox[List["(", RowBox[List["k", "+", "3"]], ")"]], "!"]], SuperscriptBox["n", RowBox[List["k", " "]]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["m", " ", SubscriptBox["\[Omega]", "1"]]], "+", RowBox[List["n", " ", SubscriptBox["\[Omega]", "3"]]]]], ")"]], RowBox[List["4", "+", "k"]]]]]]]]]], ",", RowBox[List[FractionBox["7", "192"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", RowBox[List["-", "\[Infinity]"]]]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["n", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], RowBox[List[RowBox[List["(", RowBox[List["k", "+", "5"]], ")"]], "!"]], SuperscriptBox["n", RowBox[List["k", " "]]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["m", " ", SubscriptBox["\[Omega]", "1"]]], "+", RowBox[List["n", " ", SubscriptBox["\[Omega]", "3"]]]]], ")"]], RowBox[List["6", "+", "k"]]]]]]]]]]]], "}"]]]], "/;", RowBox[List[RowBox[List["k", "\[Element]", "Integers"]], "\[And]", RowBox[List["k", ">", "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> <mfrac> <mrow> <msup> <mo> &#8706; </mo> <mi> k </mi> </msup> <mrow> <mo> { </mo> <mrow> <mrow> <msub> <mi> g </mi> <mn> 2 </mn> </msub> <mo> ( </mo> <mrow> <msub> <mi> &#969; </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> &#969; </mi> <mn> 3 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> , </mo> <mrow> <msub> <mi> g </mi> <mn> 3 </mn> </msub> <mo> ( </mo> <mrow> <msub> <mi> &#969; </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> &#969; </mi> <mn> 3 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> } </mo> </mrow> </mrow> <mrow> <mo> &#8706; </mo> <msubsup> <mi> &#969; </mi> <mn> 3 </mn> <mi> k </mi> </msubsup> </mrow> </mfrac> <mo> &#10869; </mo> <mrow> <mo> { </mo> <mrow> <mrow> <mfrac> <mn> 5 </mn> <mn> 4 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> m </mi> <mo> = </mo> <mrow> <mo> - </mo> <mi> &#8734; </mi> </mrow> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> n </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> n </mi> <mi> k </mi> </msup> </mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> m </mi> <mo> &#8290; </mo> <msub> <mi> &#969; </mi> <mn> 1 </mn> </msub> </mrow> <mo> + </mo> <mrow> <mi> n </mi> <mo> &#8290; </mo> <msub> <mi> &#969; </mi> <mn> 3 </mn> </msub> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> k </mi> <mo> + </mo> <mn> 4 </mn> </mrow> </msup> </mfrac> </mrow> </mrow> </mrow> <mo> , </mo> <mrow> <mfrac> <mn> 7 </mn> <mn> 192 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> m </mi> <mo> = </mo> <mrow> <mo> - </mo> <mi> &#8734; </mi> </mrow> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> n </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mn> 5 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> n </mi> <mi> k </mi> </msup> </mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> m </mi> <mo> &#8290; </mo> <msub> <mi> &#969; </mi> <mn> 1 </mn> </msub> </mrow> <mo> + </mo> <mrow> <mi> n </mi> <mo> &#8290; </mo> <msub> <mi> &#969; </mi> <mn> 3 </mn> </msub> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> k </mi> <mo> + </mo> <mn> 6 </mn> </mrow> </msup> </mfrac> </mrow> </mrow> </mrow> </mrow> <mo> } </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> k </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> D </ci> <list> <apply> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 3 </cn> </apply> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <cn type='integer'> 3 </cn> </apply> </apply> </list> <list> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <cn type='integer'> 3 </cn> </apply> <ci> k </ci> </list> </apply> <list> <apply> <times /> <cn type='rational'> 5 <sep /> 4 </cn> <apply> <sum /> <bvar> <ci> n </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </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 /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <factorial /> <apply> <plus /> <ci> k </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <power /> <ci> n </ci> <ci> k </ci> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <ci> m </ci> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <ci> n </ci> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <plus /> <ci> k </ci> <cn type='integer'> 4 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 7 <sep /> 192 </cn> <apply> <sum /> <bvar> <ci> n </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </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 /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <factorial /> <apply> <plus /> <ci> k </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <power /> <ci> n </ci> <ci> k </ci> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <ci> m </ci> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <ci> n </ci> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <plus /> <ci> k </ci> <cn type='integer'> 6 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </list> </apply> <apply> <in /> <ci> k </ci> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["\[Omega]_", "3"], ",", "k"]], "}"]]]]], RowBox[List["WeierstrassInvariants", "[", RowBox[List["{", RowBox[List[SubscriptBox["\[Omega]_", "1"], ",", SubscriptBox["\[Omega]_", "3"]]], "}"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["5", "4"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", RowBox[List["-", "\[Infinity]"]]]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["n", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "+", "3"]], ")"]], "!"]], " ", SuperscriptBox["n", "k"]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["m", " ", SubscriptBox["\[Omega]\[Omega]", "1"]]], "+", RowBox[List["n", " ", SubscriptBox["\[Omega]\[Omega]", "3"]]]]], ")"]], RowBox[List["4", "+", "k"]]]]]]]]]], ",", RowBox[List[FractionBox["7", "192"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", RowBox[List["-", "\[Infinity]"]]]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["n", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "+", "5"]], ")"]], "!"]], " ", SuperscriptBox["n", "k"]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["m", " ", SubscriptBox["\[Omega]\[Omega]", "1"]]], "+", RowBox[List["n", " ", SubscriptBox["\[Omega]\[Omega]", "3"]]]]], ")"]], RowBox[List["6", "+", "k"]]]]]]]]]]]], "}"]], "/;", RowBox[List[RowBox[List["k", "\[Element]", "Integers"]], "&&", RowBox[List["k", ">", "0"]]]]]]]]]]










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





2001-10-29





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