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InverseEllipticNomeQ






Mathematica Notation

Traditional Notation









Elliptic Functions > InverseEllipticNomeQ[z] > Differentiation > Low-order differentiation





http://functions.wolfram.com/09.52.20.0003.01









  


  










Input Form





D[InverseEllipticNomeQ[z], {z, 3}] == (-(8/(Pi^6 z^3))) EllipticK[InverseEllipticNomeQ[z]]^2 (-1 + InverseEllipticNomeQ[z]) InverseEllipticNomeQ[z] (Pi^4 - 6 Pi^2 EllipticE[InverseEllipticNomeQ[z]] EllipticK[InverseEllipticNomeQ[z]] - 24 EllipticE[InverseEllipticNomeQ[z]] EllipticK[InverseEllipticNomeQ[z]]^3 InverseEllipticNomeQ[z] + 4 EllipticK[InverseEllipticNomeQ[z]]^4 (1 + InverseEllipticNomeQ[z]) (-1 + 2 InverseEllipticNomeQ[z]) + 6 EllipticK[InverseEllipticNomeQ[z]]^2 (2 EllipticE[InverseEllipticNomeQ[z]]^2 + Pi^2 InverseEllipticNomeQ[z]))










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "3"]], "}"]]], RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]]]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["8", RowBox[List[SuperscriptBox["\[Pi]", "6"], " ", SuperscriptBox["z", "3"]]]]]], SuperscriptBox[RowBox[List["EllipticK", "[", RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]], "]"]], "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]]]], ")"]], " ", RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]], RowBox[List["(", RowBox[List[SuperscriptBox["\[Pi]", "4"], "-", RowBox[List["6", " ", SuperscriptBox["\[Pi]", "2"], " ", RowBox[List["EllipticE", "[", RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]], "]"]], " ", RowBox[List["EllipticK", "[", RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]], "]"]]]], "-", RowBox[List["24", " ", RowBox[List["EllipticE", "[", RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]], "]"]], " ", SuperscriptBox[RowBox[List["EllipticK", "[", RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]], "]"]], "3"], " ", RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]]]], "+", RowBox[List["4", " ", SuperscriptBox[RowBox[List["EllipticK", "[", RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]], "]"]], "4"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]]]]]], ")"]]]], "+", RowBox[List["6", " ", SuperscriptBox[RowBox[List["EllipticK", "[", RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]], "]"]], "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", SuperscriptBox[RowBox[List["EllipticE", "[", RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]], "]"]], "2"]]], "+", RowBox[List[SuperscriptBox["\[Pi]", "2"], " ", RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]]]]]], ")"]]]]]], ")"]]]]]]]]










MathML Form







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</mo> <msup> <mrow> <mi> K </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msup> <semantics> <mi> q </mi> <annotation-xml encoding='MathML-Content'> <ci> EllipticNomeQ </ci> </annotation-xml> </semantics> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 24 </mn> <mo> &#8290; </mo> <mrow> <mi> E </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msup> <semantics> <mi> q </mi> <annotation-xml encoding='MathML-Content'> <ci> EllipticNomeQ </ci> </annotation-xml> </semantics> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <semantics> <mi> q </mi> <annotation-xml encoding='MathML-Content'> <ci> EllipticNomeQ </ci> </annotation-xml> </semantics> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mi> K </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msup> <semantics> <mi> q </mi> <annotation-xml encoding='MathML-Content'> <ci> EllipticNomeQ </ci> </annotation-xml> </semantics> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 6 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mrow> <mi> E </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msup> <semantics> <mi> q </mi> <annotation-xml encoding='MathML-Content'> <ci> EllipticNomeQ </ci> </annotation-xml> </semantics> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <msup> <semantics> <mi> q </mi> <annotation-xml encoding='MathML-Content'> <ci> EllipticNomeQ </ci> </annotation-xml> </semantics> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mi> K </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msup> <semantics> <mi> q </mi> <annotation-xml encoding='MathML-Content'> <ci> EllipticNomeQ </ci> </annotation-xml> </semantics> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 6 </mn> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <mi> E </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msup> <semantics> <mi> q </mi> <annotation-xml encoding='MathML-Content'> <ci> EllipticNomeQ </ci> </annotation-xml> </semantics> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> K </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msup> <semantics> <mi> q </mi> <annotation-xml encoding='MathML-Content'> <ci> EllipticNomeQ </ci> </annotation-xml> </semantics> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <msup> <mi> &#960; </mi> <mn> 4 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> z </ci> <degree> <cn type='integer'> 3 </cn> </degree> </bvar> <apply> <ci> InverseEllipticNomeQ </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <power /> <apply> <ci> EllipticK </ci> <apply> <ci> InverseEllipticNomeQ </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <ci> InverseEllipticNomeQ </ci> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <ci> InverseEllipticNomeQ </ci> <ci> z </ci> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <pi /> <cn type='integer'> 6 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <plus /> <apply> <ci> InverseEllipticNomeQ </ci> <ci> z </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> InverseEllipticNomeQ </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <ci> EllipticK </ci> <apply> <ci> InverseEllipticNomeQ </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 24 </cn> <apply> <ci> EllipticE </ci> <apply> <ci> InverseEllipticNomeQ </ci> <ci> z </ci> </apply> </apply> <apply> <ci> InverseEllipticNomeQ </ci> <ci> z </ci> </apply> <apply> <power /> <apply> <ci> EllipticK </ci> <apply> <ci> InverseEllipticNomeQ </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 6 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <ci> EllipticE </ci> <apply> <ci> InverseEllipticNomeQ </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <ci> InverseEllipticNomeQ </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <power /> <apply> <ci> EllipticK </ci> <apply> <ci> InverseEllipticNomeQ </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 6 </cn> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <ci> EllipticE </ci> <apply> <ci> InverseEllipticNomeQ </ci> <ci> z </ci> </apply> </apply> <apply> <ci> EllipticK </ci> <apply> <ci> InverseEllipticNomeQ </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <power /> <pi /> <cn type='integer'> 4 </cn> </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["z_", ",", "3"]], "}"]]]]], RowBox[List["InverseEllipticNomeQ", "[", "z_", "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List["8", " ", SuperscriptBox[RowBox[List["EllipticK", "[", RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]], "]"]], "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]]]], ")"]], " ", RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["\[Pi]", "4"], "-", RowBox[List["6", " ", SuperscriptBox["\[Pi]", "2"], " ", RowBox[List["EllipticE", "[", RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]], "]"]], " ", RowBox[List["EllipticK", "[", RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]], "]"]]]], "-", RowBox[List["24", " ", RowBox[List["EllipticE", "[", RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]], "]"]], " ", SuperscriptBox[RowBox[List["EllipticK", "[", RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]], "]"]], "3"], " ", RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]]]], "+", RowBox[List["4", " ", SuperscriptBox[RowBox[List["EllipticK", "[", RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]], "]"]], "4"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]]]]]], ")"]]]], "+", RowBox[List["6", " ", SuperscriptBox[RowBox[List["EllipticK", "[", RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]], "]"]], "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", SuperscriptBox[RowBox[List["EllipticE", "[", RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]], "]"]], "2"]]], "+", RowBox[List[SuperscriptBox["\[Pi]", "2"], " ", RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]]]]]], ")"]]]]]], ")"]]]], RowBox[List[SuperscriptBox["\[Pi]", "6"], " ", SuperscriptBox["z", "3"]]]]]]]]]]










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





2007-05-02





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