Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site ContributeEmail CommentsSign the Guestbook

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

Download All Introductions For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











WeierstrassHalfPeriods






Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/09.18.20.0004.01









  


  










Input Form





D[Subscript[\[Omega], 3], Subscript[g, 3]] == (9 Subscript[g, 3] Subscript[\[Omega], 3] - 6 Subscript[g, 2] Subscript[\[Eta], 3])/(2 (Subscript[g, 2]^3 - 27 Subscript[g, 3]^2)) /; {Subscript[\[Omega], 1], Subscript[\[Omega], 3]} == WeierstrassHalfPeriods[{Subscript[g, 2], Subscript[g, 3]}] && Subscript[\[Eta], n] == WeierstrassZeta[Subscript[\[Omega], n], {Subscript[g, 2], Subscript[g, 3]}] && Element[n, {1, 2, 3}]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", SubscriptBox["g", "3"]], SubscriptBox["\[Omega]", "3"]]], "\[Equal]", FractionBox[RowBox[List[RowBox[List["9", SubscriptBox["g", "3"], SubscriptBox["\[Omega]", "3"]]], "-", RowBox[List["6", SubscriptBox["g", "2"], SubscriptBox["\[Eta]", "3"]]]]], RowBox[List["2", RowBox[List["(", RowBox[List[SubsuperscriptBox["g", "2", "3"], "-", RowBox[List["27", SubsuperscriptBox["g", "3", "2"]]]]], ")"]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["\[Omega]", "1"], ",", SubscriptBox["\[Omega]", "3"]]], "}"]], "\[Equal]", RowBox[List["WeierstrassHalfPeriods", "[", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]], "]"]]]], "\[And]", RowBox[List[SubscriptBox["\[Eta]", "n"], "\[Equal]", RowBox[List["WeierstrassZeta", "[", RowBox[List[SubscriptBox["\[Omega]", "n"], ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]]]], "\[And]", RowBox[List["n", "\[Element]", RowBox[List["{", RowBox[List["1", ",", "2", ",", "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> &#969; </mi> <mn> 3 </mn> </msub> </mrow> <mrow> <mo> &#8706; </mo> <msub> <mi> g </mi> <mn> 3 </mn> </msub> </mrow> </mfrac> <mo> &#10869; </mo> <mfrac> <mrow> <mrow> <mn> 9 </mn> <mo> &#8290; </mo> <msub> <mi> g </mi> <mn> 3 </mn> </msub> <mo> &#8290; </mo> <msub> <mi> &#969; </mi> <mn> 3 </mn> </msub> </mrow> <mo> - </mo> <mrow> <mn> 6 </mn> <mo> &#8290; </mo> <msub> <mi> g </mi> <mn> 2 </mn> </msub> <mo> &#8290; </mo> <msub> <mi> &#951; </mi> <mn> 3 </mn> </msub> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msubsup> <mi> g </mi> <mn> 2 </mn> <mn> 3 </mn> </msubsup> <mo> - </mo> <mrow> <mn> 27 </mn> <mo> &#8290; </mo> <msubsup> <mi> g </mi> <mn> 3 </mn> <mn> 2 </mn> </msubsup> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <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> &#10869; </mo> <mrow> <mo> { </mo> <mrow> <mrow> <msub> <mi> &#969; </mi> <mn> 1 </mn> </msub> <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> , </mo> <mrow> <msub> <mi> &#969; </mi> <mn> 3 </mn> </msub> <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> </mrow> <mo> } </mo> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> &#951; </mi> <mi> n </mi> </msub> <mo> &#10869; </mo> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <msub> <mi> &#969; </mi> <mi> n </mi> </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;n&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> <mo> &#8743; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <mrow> <mo> { </mo> <mrow> <mn> 1 </mn> <mo> , </mo> <mn> 2 </mn> <mo> , </mo> <mn> 3 </mn> </mrow> <mo> } </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> D </ci> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <cn type='integer'> 3 </cn> </apply> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 9 </cn> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 3 </cn> </apply> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 6 </cn> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> &#951; </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 27 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <and /> <apply> <eq /> <list> <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> </list> <list> <apply> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <cn type='integer'> 1 </cn> </apply> <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> </apply> <apply> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <cn type='integer'> 3 </cn> </apply> <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> </apply> </list> </apply> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> &#951; </ci> <ci> n </ci> </apply> <apply> <ci> WeierstrassZeta </ci> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <ci> n </ci> </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> <in /> <ci> n </ci> <list> <cn type='integer'> 1 </cn> <cn type='integer'> 2 </cn> <cn type='integer'> 3 </cn> </list> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[SubscriptBox["g_", "3"]]]], SubscriptBox["\[Omega]_", "3"]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["9", " ", SubscriptBox["gg", "3"], " ", SubscriptBox["\[Omega]\[Omega]", "3"]]], "-", RowBox[List["6", " ", SubscriptBox["gg", "2"], " ", SubscriptBox["\[Eta]", "3"]]]]], RowBox[List["2", " ", RowBox[List["(", RowBox[List[SubsuperscriptBox["gg", "2", "3"], "-", RowBox[List["27", " ", SubsuperscriptBox["gg", "3", "2"]]]]], ")"]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["\[Omega]", "1"], ",", SubscriptBox["\[Omega]\[Omega]", "3"]]], "}"]], "\[Equal]", RowBox[List["WeierstrassHalfPeriods", "[", RowBox[List["{", RowBox[List[SubscriptBox["gg", "2"], ",", SubscriptBox["gg", "3"]]], "}"]], "]"]]]], "&&", RowBox[List[SubscriptBox["\[Eta]", "n"], "\[Equal]", RowBox[List["WeierstrassZeta", "[", RowBox[List[SubscriptBox["\[Omega]", "n"], ",", RowBox[List["{", RowBox[List[SubscriptBox["gg", "2"], ",", SubscriptBox["gg", "3"]]], "}"]]]], "]"]]]], "&&", RowBox[List["n", "\[Element]", RowBox[List["{", RowBox[List["1", ",", "2", ",", "3"]], "}"]]]]]]]]]]]]










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





2002-12-18





© 1998-2014 Wolfram Research, Inc.