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EllipticNomeQ






Mathematica Notation

Traditional Notation









Elliptic Functions > EllipticNomeQ[m] > Differentiation > Low-order differentiation





http://functions.wolfram.com/09.53.20.0004.01









  


  










Input Form





D[EllipticNomeQ[m], {m, 3}] == (-((Pi^2 EllipticNomeQ[m])/(64 (-1 + m)^3 m^3 EllipticK[m]^6))) (Pi^4 + 4 EllipticK[m] (3 m Pi^2 EllipticK[m] + 6 EllipticE[m]^2 EllipticK[m] + (2 - 2 m + 8 m^2) EllipticK[m]^3 - 3 EllipticE[m] (Pi^2 + 4 m EllipticK[m]^2)))










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["m", ",", "3"]], "}"]]], RowBox[List["EllipticNomeQ", "[", "m", "]"]]]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["\[Pi]", "2"], " ", RowBox[List["EllipticNomeQ", "[", "m", "]"]]]], RowBox[List["64", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]], "3"], " ", SuperscriptBox["m", "3"], " ", SuperscriptBox[RowBox[List["EllipticK", "[", "m", "]"]], "6"]]]]]], RowBox[List["(", RowBox[List[SuperscriptBox["\[Pi]", "4"], "+", RowBox[List["4", " ", RowBox[List["EllipticK", "[", "m", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", "m", " ", SuperscriptBox["\[Pi]", "2"], " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "+", RowBox[List["6", " ", SuperscriptBox[RowBox[List["EllipticE", "[", "m", "]"]], "2"], " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["2", "-", RowBox[List["2", " ", "m"]], "+", RowBox[List["8", " ", SuperscriptBox["m", "2"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["EllipticK", "[", "m", "]"]], "3"]]], "-", RowBox[List["3", " ", RowBox[List["EllipticE", "[", "m", "]"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["\[Pi]", "2"], "+", RowBox[List["4", " ", "m", " ", SuperscriptBox[RowBox[List["EllipticK", "[", "m", "]"]], "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> <msup> <mo> &#8706; </mo> <mn> 3 </mn> </msup> <mrow> <semantics> <mi> q </mi> <annotation-xml encoding='MathML-Content'> <ci> EllipticNomeQ </ci> </annotation-xml> </semantics> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> &#8706; </mo> <msup> <mi> m </mi> <mn> 3 </mn> </msup> </mrow> </mfrac> <mo> &#63449; </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mrow> <mn> 64 </mn> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> m </mi> <mn> 3 </mn> </msup> <mo> &#8290; </mo> <msup> <mrow> <mi> K </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <mn> 6 </mn> </msup> </mrow> </mfrac> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mrow> <mi> K </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <msup> <mi> m </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> m </mi> </mrow> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mi> K </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> m </mi> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <mi> K </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 6 </mn> <mo> &#8290; </mo> <msup> <mrow> <mi> E </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <mi> K </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mrow> <mi> E </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> m </mi> <mo> &#8290; </mo> <msup> <mrow> <mi> K </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <msup> <mi> &#960; </mi> <mn> 4 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <semantics> <mi> q </mi> <annotation-xml encoding='MathML-Content'> <ci> EllipticNomeQ </ci> </annotation-xml> </semantics> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> m </ci> <degree> <cn type='integer'> 3 </cn> </degree> </bvar> <apply> <ci> EllipticNomeQ </ci> <ci> m </ci> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 64 </cn> <apply> <power /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <ci> m </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> <cn type='integer'> 6 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> m </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 3 </cn> <ci> m </ci> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 6 </cn> <apply> <power /> <apply> <ci> EllipticE </ci> <ci> m </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <ci> EllipticE </ci> <ci> m </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> m </ci> <apply> <power /> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <pi /> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <ci> EllipticNomeQ </ci> <ci> m </ci> </apply> </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["m_", ",", "3"]], "}"]]]]], RowBox[List["EllipticNomeQ", "[", "m_", "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[Pi]", "2"], " ", RowBox[List["EllipticNomeQ", "[", "m", "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["\[Pi]", "4"], "+", RowBox[List["4", " ", RowBox[List["EllipticK", "[", "m", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", "m", " ", SuperscriptBox["\[Pi]", "2"], " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "+", RowBox[List["6", " ", SuperscriptBox[RowBox[List["EllipticE", "[", "m", "]"]], "2"], " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["2", "-", RowBox[List["2", " ", "m"]], "+", RowBox[List["8", " ", SuperscriptBox["m", "2"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["EllipticK", "[", "m", "]"]], "3"]]], "-", RowBox[List["3", " ", RowBox[List["EllipticE", "[", "m", "]"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["\[Pi]", "2"], "+", RowBox[List["4", " ", "m", " ", SuperscriptBox[RowBox[List["EllipticK", "[", "m", "]"]], "2"]]]]], ")"]]]]]], ")"]]]]]], ")"]]]], RowBox[List["64", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]], "3"], " ", SuperscriptBox["m", "3"], " ", SuperscriptBox[RowBox[List["EllipticK", "[", "m", "]"]], "6"]]]]]]]]]]










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





2007-05-02





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