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InverseEllipticNomeQ






Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/09.52.20.0002.02









  


  










Input Form





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










Standard Form





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










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> 2 </mn> </msup> <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> &#8706; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> &#10869; </mo> <mfrac> <mrow> <mn> 4 </mn> <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> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <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> <mn> 1 </mn> </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> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <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> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 4 </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> <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> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <msup> <mi> &#960; </mi> <mn> 4 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> z </ci> <degree> <cn type='integer'> 2 </cn> </degree> </bvar> <apply> <ci> InverseEllipticNomeQ </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 4 </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> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <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'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <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'> 2 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <pi /> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </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_", ",", "2"]], "}"]]]]], RowBox[List["InverseEllipticNomeQ", "[", "z_", "]"]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List["4", " ", 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]", "2"], "-", RowBox[List["4", " ", RowBox[List["EllipticE", "[", RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]], "]"]], " ", RowBox[List["EllipticK", "[", RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]], "]"]]]], "+", RowBox[List["4", " ", SuperscriptBox[RowBox[List["EllipticK", "[", RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]], "]"]], "2"], " ", RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]]]]]], ")"]]]], RowBox[List[SuperscriptBox["\[Pi]", "4"], " ", SuperscriptBox["z", "2"]]]]]]]]










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





2001-10-29





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