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http://functions.wolfram.com/09.53.20.0005.01
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D[EllipticNomeQ[m], {m, 4}] ==
(Pi^2 (Pi^6 + 8 EllipticK[m] (3 m Pi^4 EllipticK[m] -
24 EllipticE[m]^3 EllipticK[m]^2 + 2 (2 + m (-2 + 11 m)) Pi^2
EllipticK[m]^3 + 4 (-1 + 4 m) (2 + m (-1 + 3 m)) EllipticK[m]^5 +
18 EllipticE[m]^2 EllipticK[m] (Pi^2 + 4 m EllipticK[m]^2) +
EllipticE[m] (-3 Pi^4 - 36 m Pi^2 EllipticK[m]^2 -
8 (2 + m (-2 + 11 m)) EllipticK[m]^4))) EllipticNomeQ[m])/
(256 (-1 + m)^4 m^4 EllipticK[m]^8)
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Cell[BoxData[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["m", ",", "4"]], "}"]]], RowBox[List["EllipticNomeQ", "[", "m", "]"]]]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[Pi]", "2"], " ", RowBox[List["(", RowBox[List[SuperscriptBox["\[Pi]", "6"], "+", RowBox[List["8", " ", RowBox[List["EllipticK", "[", "m", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", "m", " ", SuperscriptBox["\[Pi]", "4"], " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "-", RowBox[List["24", " ", SuperscriptBox[RowBox[List["EllipticE", "[", "m", "]"]], "3"], " ", SuperscriptBox[RowBox[List["EllipticK", "[", "m", "]"]], "2"]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List["2", "+", RowBox[List["m", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", RowBox[List["11", " ", "m"]]]], ")"]]]]]], ")"]], " ", SuperscriptBox["\[Pi]", "2"], " ", SuperscriptBox[RowBox[List["EllipticK", "[", "m", "]"]], "3"]]], "+", RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["4", " ", "m"]]]], ")"]], " ", RowBox[List["(", RowBox[List["2", "+", RowBox[List["m", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["3", " ", "m"]]]], ")"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["EllipticK", "[", "m", "]"]], "5"]]], "+", RowBox[List["18", " ", SuperscriptBox[RowBox[List["EllipticE", "[", "m", "]"]], "2"], " ", RowBox[List["EllipticK", "[", "m", "]"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["\[Pi]", "2"], "+", RowBox[List["4", " ", "m", " ", SuperscriptBox[RowBox[List["EllipticK", "[", "m", "]"]], "2"]]]]], ")"]]]], "+", RowBox[List[RowBox[List["EllipticE", "[", "m", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "3"]], " ", SuperscriptBox["\[Pi]", "4"]]], "-", RowBox[List["36", " ", "m", " ", SuperscriptBox["\[Pi]", "2"], " ", SuperscriptBox[RowBox[List["EllipticK", "[", "m", "]"]], "2"]]], "-", RowBox[List["8", " ", RowBox[List["(", RowBox[List["2", "+", RowBox[List["m", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", RowBox[List["11", " ", "m"]]]], ")"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["EllipticK", "[", "m", "]"]], "4"]]]]], ")"]]]]]], ")"]]]]]], ")"]], " ", RowBox[List["EllipticNomeQ", "[", "m", "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List["256", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]], "4"], " ", SuperscriptBox["m", "4"], " ", SuperscriptBox[RowBox[List["EllipticK", "[", "m", "]"]], "8"]]], ")"]]]]]]]]
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<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> ∂ </mo> <mn> 4 </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> ∂ </mo> <msup> <mi> m </mi> <mn> 4 </mn> </msup> </mrow> </mfrac> <mo>  </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 256 </mn> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> <mo> ⁢ </mo> <msup> <mi> m </mi> <mn> 4 </mn> </msup> <mo> ⁢ </mo> <msup> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <mn> 8 </mn> </msup> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> π </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 8 </mn> <mo> ⁢ </mo> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> m </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> m </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mi> m </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <mn> 5 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> m </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 11 </mn> <mo> ⁢ </mo> <mi> m </mi> </mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> π </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msup> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 24 </mn> <mo> ⁢ </mo> <msup> <mrow> <mi> E </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> <mo> ⁢ </mo> <msup> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mi> m </mi> <mo> ⁢ </mo> <msup> <mi> π </mi> <mn> 4 </mn> </msup> <mo> ⁢ </mo> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 18 </mn> <mo> ⁢ </mo> <msup> <mrow> <mi> E </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> m </mi> <mo> ⁢ </mo> <msup> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <msup> <mi> π </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mi> E </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 8 </mn> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> m </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 11 </mn> <mo> ⁢ </mo> <mi> m </mi> </mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 36 </mn> <mo> ⁢ </mo> <mi> m </mi> <mo> ⁢ </mo> <msup> <mi> π </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msup> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <msup> <mi> π </mi> <mn> 4 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <msup> <mi> π </mi> <mn> 6 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </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'> 4 </cn> </degree> </bvar> <apply> <ci> EllipticNomeQ </ci> <ci> m </ci> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 256 </cn> <apply> <power /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <ci> m </ci> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> <cn type='integer'> 8 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> m </ci> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <apply> <times /> <ci> m </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> m </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <ci> m </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 11 </cn> <ci> m </ci> </apply> <cn type='integer'> -2 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <pi /> <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'> -1 </cn> <apply> <times /> <cn type='integer'> 24 </cn> <apply> <power /> <apply> <ci> EllipticE </ci> <ci> m </ci> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 3 </cn> <ci> m </ci> <apply> <power /> <pi /> <cn type='integer'> 4 </cn> </apply> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 18 </cn> <apply> <power /> <apply> <ci> EllipticE </ci> <ci> m </ci> </apply> <cn type='integer'> 2 </cn> </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> <ci> EllipticK </ci> <ci> m </ci> </apply> </apply> <apply> <times /> <apply> <ci> EllipticE </ci> <ci> m </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -8 </cn> <apply> <plus /> <apply> <times /> <ci> m </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 11 </cn> <ci> m </ci> </apply> <cn type='integer'> -2 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 36 </cn> <ci> m </ci> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <pi /> <cn type='integer'> 4 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <pi /> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <ci> EllipticNomeQ </ci> <ci> m </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["m_", ",", "4"]], "}"]]]]], RowBox[List["EllipticNomeQ", "[", "m_", "]"]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox["\[Pi]", "2"], " ", RowBox[List["(", RowBox[List[SuperscriptBox["\[Pi]", "6"], "+", RowBox[List["8", " ", RowBox[List["EllipticK", "[", "m", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", "m", " ", SuperscriptBox["\[Pi]", "4"], " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "-", RowBox[List["24", " ", SuperscriptBox[RowBox[List["EllipticE", "[", "m", "]"]], "3"], " ", SuperscriptBox[RowBox[List["EllipticK", "[", "m", "]"]], "2"]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List["2", "+", RowBox[List["m", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", RowBox[List["11", " ", "m"]]]], ")"]]]]]], ")"]], " ", SuperscriptBox["\[Pi]", "2"], " ", SuperscriptBox[RowBox[List["EllipticK", "[", "m", "]"]], "3"]]], "+", RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["4", " ", "m"]]]], ")"]], " ", RowBox[List["(", RowBox[List["2", "+", RowBox[List["m", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["3", " ", "m"]]]], ")"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["EllipticK", "[", "m", "]"]], "5"]]], "+", RowBox[List["18", " ", SuperscriptBox[RowBox[List["EllipticE", "[", "m", "]"]], "2"], " ", RowBox[List["EllipticK", "[", "m", "]"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["\[Pi]", "2"], "+", RowBox[List["4", " ", "m", " ", SuperscriptBox[RowBox[List["EllipticK", "[", "m", "]"]], "2"]]]]], ")"]]]], "+", RowBox[List[RowBox[List["EllipticE", "[", "m", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "3"]], " ", SuperscriptBox["\[Pi]", "4"]]], "-", RowBox[List["36", " ", "m", " ", SuperscriptBox["\[Pi]", "2"], " ", SuperscriptBox[RowBox[List["EllipticK", "[", "m", "]"]], "2"]]], "-", RowBox[List["8", " ", RowBox[List["(", RowBox[List["2", "+", RowBox[List["m", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", RowBox[List["11", " ", "m"]]]], ")"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["EllipticK", "[", "m", "]"]], "4"]]]]], ")"]]]]]], ")"]]]]]], ")"]], " ", RowBox[List["EllipticNomeQ", "[", "m", "]"]]]], RowBox[List["256", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]], "4"], " ", SuperscriptBox["m", "4"], " ", SuperscriptBox[RowBox[List["EllipticK", "[", "m", "]"]], "8"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
