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WeierstrassZeta






Mathematica Notation

Traditional Notation









Elliptic Functions > WeierstrassZeta[z,{g2,g3}] > Differentiation > Low-order differentiation > With respect to omega3





http://functions.wolfram.com/09.17.20.0012.02









  


  










Input Form





D[WeierstrassZeta[z, {Subscript[g, 2], Subscript[g, 3]}], Subscript[\[Omega], 3]] == (-(Subscript[\[Omega], 3]/(6 Pi Sqrt[-Subscript[\[Omega], 3]^2]))) (WeierstrassInvariants[{1, Subscript[\[Omega], 3]}][[1]] - 6 WeierstrassPPrime[1, WeierstrassInvariants[ {1, Subscript[\[Omega], 3]}]] - 12 WeierstrassZeta[1, WeierstrassInvariants[{1, Subscript[\[Omega], 3]}]]^ 2)










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", SubscriptBox["\[Omega]", "3"]], RowBox[List["WeierstrassZeta", "[", RowBox[List["z", ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox[SubscriptBox["\[Omega]", "3"], RowBox[List["6", " ", "\[Pi]", " ", SqrtBox[RowBox[List["-", SubsuperscriptBox["\[Omega]", "3", "2"]]]]]]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["WeierstrassInvariants", "[", RowBox[List["{", RowBox[List["1", ",", SubscriptBox["\[Omega]", "3"]]], "}"]], "]"]], "[", RowBox[List["[", "1", "]"]], "]"]], "-", RowBox[List["6", " ", RowBox[List["WeierstrassPPrime", "[", RowBox[List["1", ",", RowBox[List["WeierstrassInvariants", "[", RowBox[List["{", RowBox[List["1", ",", SubscriptBox["\[Omega]", "3"]]], "}"]], "]"]]]], "]"]]]], "-", RowBox[List["12", " ", SuperscriptBox[RowBox[List["WeierstrassZeta", "[", RowBox[List["1", ",", RowBox[List["WeierstrassInvariants", "[", RowBox[List["{", RowBox[List["1", ",", SubscriptBox["\[Omega]", "3"]]], "}"]], "]"]]]], "]"]], "2"]]]]], ")"]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mfrac> <mrow> <mo> &#8706; </mo> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <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> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Zeta]&quot;, &quot;(&quot;, RowBox[List[RowBox[List[TagBox[&quot;z&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> <mo> &#8706; </mo> <msub> <mi> &#969; </mi> <mn> 3 </mn> </msub> </mrow> </mfrac> <mo> &#10869; </mo> <mrow> <mrow> <mo> - </mo> <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> </mrow> <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> <msub> <mi> &#969; </mi> <mn> 1 </mn> </msub> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> &#8472; </mi> <mo> &#8242; </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> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <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> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Zeta]&quot;, &quot;(&quot;, RowBox[List[RowBox[List[TagBox[&quot;z&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> <mo> &#8290; </mo> <mrow> <mi> &#8472; </mi> <mo> &#8289; </mo> <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> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 6 </mn> </mfrac> <mo> &#8290; </mo> <msub> <mi> g </mi> <mn> 2 </mn> </msub> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msub> <mi> &#951; </mi> <mn> 1 </mn> </msub> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <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> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Zeta]&quot;, &quot;(&quot;, RowBox[List[RowBox[List[TagBox[&quot;z&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> <mo> - </mo> <mrow> <mi> z </mi> <mo> &#8290; 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</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 /> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <apply> <partialdiff /> <list> <cn type='integer'> 1 </cn> </list> <ci> &#8472; </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> <times /> <cn type='integer'> 2 </cn> <apply> <ci> WeierstrassZeta </ci> <ci> z </ci> <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> <ci> WeierstrassP </ci> <ci> z </ci> <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> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 6 </cn> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 2 </cn> </apply> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Subscript </ci> <ci> &#951; </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <ci> WeierstrassZeta </ci> <ci> z </ci> <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> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> z </ci> <apply> <ci> WeierstrassP </ci> <ci> z </ci> <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> </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"]]]], RowBox[List["WeierstrassZeta", "[", RowBox[List["z_", ",", RowBox[List["{", RowBox[List[SubscriptBox["g_", "2"], ",", SubscriptBox["g_", "3"]]], "}"]]]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List[SubscriptBox["\[Omega]\[Omega]", "3"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["WeierstrassInvariants", "[", RowBox[List["{", RowBox[List["1", ",", SubscriptBox["\[Omega]\[Omega]", "3"]]], "}"]], "]"]], "\[LeftDoubleBracket]", "1", "\[RightDoubleBracket]"]], "-", RowBox[List["6", " ", RowBox[List["WeierstrassPPrime", "[", RowBox[List["1", ",", RowBox[List["WeierstrassInvariants", "[", RowBox[List["{", RowBox[List["1", ",", SubscriptBox["\[Omega]\[Omega]", "3"]]], "}"]], "]"]]]], "]"]]]], "-", RowBox[List["12", " ", SuperscriptBox[RowBox[List["WeierstrassZeta", "[", RowBox[List["1", ",", RowBox[List["WeierstrassInvariants", "[", RowBox[List["{", RowBox[List["1", ",", SubscriptBox["\[Omega]\[Omega]", "3"]]], "}"]], "]"]]]], "]"]], "2"]]]]], ")"]]]], RowBox[List["6", " ", "\[Pi]", " ", SqrtBox[RowBox[List["-", SubsuperscriptBox["\[Omega]\[Omega]", "3", "2"]]]]]]]]]]]]]










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





2002-12-18





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