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WeierstrassInvariants






Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/09.19.20.0013.02









  


  










Input Form





D[Subscript[g, 2], Subscript[\[Omega], 3]] == (Subscript[\[Omega], 1]/(Pi Subscript[\[Omega], 3])) Sqrt[-(Subscript[\[Omega], 3]^2/Subscript[\[Omega], 1]^2)] (12 Subscript[g, 3] Subscript[\[Omega], 1] - 8 Subscript[g, 2] Subscript[\[Eta], 1]) /; {Subscript[g, 2], Subscript[g, 3]} == WeierstrassInvariants[{Subscript[\[Omega], 1], Subscript[\[Omega], 3]}] && Subscript[\[Eta], 1] == WeierstrassZeta[Subscript[\[Omega], 1], {Subscript[g, 2], Subscript[g, 3]}]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", SubscriptBox["\[Omega]", "3"]], SubscriptBox["g", "2"]]], "\[Equal]", RowBox[List[FractionBox[SubscriptBox["\[Omega]", "1"], RowBox[List["\[Pi]", " ", SubscriptBox["\[Omega]", "3"]]]], " ", SqrtBox[RowBox[List["-", FractionBox[SubsuperscriptBox["\[Omega]", "3", "2"], SubsuperscriptBox["\[Omega]", "1", "2"]]]]], RowBox[List["(", RowBox[List[RowBox[List["12", SubscriptBox["g", "3"], SubscriptBox["\[Omega]", "1"]]], "-", RowBox[List["8", SubscriptBox["g", "2"], SubscriptBox["\[Eta]", "1"]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]], "\[Equal]", RowBox[List["WeierstrassInvariants", "[", RowBox[List["{", RowBox[List[SubscriptBox["\[Omega]", "1"], ",", SubscriptBox["\[Omega]", "3"]]], "}"]], "]"]]]], "\[And]", RowBox[List[SubscriptBox["\[Eta]", "1"], "\[Equal]", RowBox[List["WeierstrassZeta", "[", RowBox[List[SubscriptBox["\[Omega]", "1"], ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]]]]]]]]]]










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> <mo> &#8706; </mo> <msub> <mi> g </mi> <mn> 2 </mn> </msub> </mrow> <mrow> <mo> &#8706; </mo> <msub> <mi> &#969; </mi> <mn> 3 </mn> </msub> </mrow> </mfrac> <mo> &#10869; </mo> <mrow> <mfrac> <msub> <mi> &#969; </mi> <mn> 1 </mn> </msub> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <msub> <mi> &#969; </mi> <mn> 3 </mn> </msub> </mrow> </mfrac> <mo> &#8290; </mo> <msqrt> <mrow> <mo> - </mo> <mfrac> <msubsup> <mi> &#969; </mi> <mn> 3 </mn> <mn> 2 </mn> </msubsup> <msubsup> <mi> &#969; </mi> <mn> 1 </mn> <mn> 2 </mn> </msubsup> </mfrac> </mrow> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 12 </mn> <mo> &#8290; </mo> <msub> <mi> g </mi> <mn> 3 </mn> </msub> <mo> &#8290; </mo> <msub> <mi> &#969; </mi> <mn> 1 </mn> </msub> </mrow> <mo> - </mo> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <msub> <mi> g </mi> <mn> 2 </mn> </msub> <mo> &#8290; </mo> <msub> <mi> &#951; </mi> <mn> 1 </mn> </msub> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mo> { </mo> <mrow> <msub> <mi> g </mi> <mn> 2 </mn> </msub> <mo> , </mo> <msub> <mi> g </mi> <mn> 3 </mn> </msub> </mrow> <mo> } </mo> </mrow> <mo> &#10869; </mo> <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> <mo> &#8743; </mo> <mrow> <msub> <mi> &#951; </mi> <mn> 1 </mn> </msub> <mo> &#10869; </mo> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <msub> <mi> &#969; </mi> <mn> 1 </mn> </msub> <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> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Zeta]&quot;, &quot;(&quot;, RowBox[List[RowBox[List[TagBox[SubscriptBox[&quot;\[Omega]&quot;, &quot;1&quot;], Rule[Editable, True]], &quot;;&quot;, TagBox[SubscriptBox[&quot;g&quot;, &quot;2&quot;], Rule[Editable, True]]]], &quot;,&quot;, TagBox[SubscriptBox[&quot;g&quot;, &quot;3&quot;], Rule[Editable, True]]]], &quot;)&quot;]], InterpretTemplate[Function[WeierstrassZeta[Slot[1], List[Slot[2], Slot[3]]]]]] </annotation> </semantics> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> D </ci> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <times /> <pi /> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <cn type='integer'> 3 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 12 </cn> <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> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> &#951; </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <eq /> <list> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 3 </cn> </apply> </list> <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> </apply> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> &#951; </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> WeierstrassZeta </ci> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <cn type='integer'> 1 </cn> </apply> <list> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 3 </cn> </apply> </list> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[SubscriptBox["\[Omega]_", "3"]]]], SubscriptBox["g_", "2"]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SubscriptBox["\[Omega]\[Omega]", "1"], " ", SqrtBox[RowBox[List["-", FractionBox[SubsuperscriptBox["\[Omega]\[Omega]", "3", "2"], SubsuperscriptBox["\[Omega]\[Omega]", "1", "2"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["12", " ", SubscriptBox["gg", "3"], " ", SubscriptBox["\[Omega]\[Omega]", "1"]]], "-", RowBox[List["8", " ", SubscriptBox["gg", "2"], " ", SubscriptBox["\[Eta]", "1"]]]]], ")"]]]], RowBox[List["\[Pi]", " ", SubscriptBox["\[Omega]\[Omega]", "3"]]]], "/;", RowBox[List[RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["gg", "2"], ",", SubscriptBox["gg", "3"]]], "}"]], "\[Equal]", RowBox[List["WeierstrassInvariants", "[", RowBox[List["{", RowBox[List[SubscriptBox["\[Omega]\[Omega]", "1"], ",", SubscriptBox["\[Omega]\[Omega]", "3"]]], "}"]], "]"]]]], "&&", RowBox[List[SubscriptBox["\[Eta]", "1"], "\[Equal]", RowBox[List["WeierstrassZeta", "[", RowBox[List[SubscriptBox["\[Omega]\[Omega]", "1"], ",", RowBox[List["{", RowBox[List[SubscriptBox["gg", "2"], ",", SubscriptBox["gg", "3"]]], "}"]]]], "]"]]]]]]]]]]]]










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





2002-12-18





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