Inverse elliptic nome: Differentiation (formula 09.52.20.0005)

InverseEllipticNomeQ

$Mathematica Notation$

Elliptic Functions

InverseEllipticNomeQ[z]

Differentiation

Low-order differentiation

http://functions.wolfram.com/09.52.20.0005.01

Input Form

D[InverseEllipticNomeQ[z], {z, 5}] == (-(32/(Pi^10 z^5))) EllipticK[InverseEllipticNomeQ[z]]^2 (-1 + InverseEllipticNomeQ[z]) InverseEllipticNomeQ[z] (3 Pi^8 - 25 Pi^6 EllipticE[InverseEllipticNomeQ[z]] EllipticK[InverseEllipticNomeQ[z]] + 5 Pi^4 EllipticK[InverseEllipticNomeQ[z]]^2 (21 EllipticE[InverseEllipticNomeQ[z]]^2 + 5 Pi^2 InverseEllipticNomeQ[z]) - 30 Pi^2 EllipticE[InverseEllipticNomeQ[z]] EllipticK[InverseEllipticNomeQ[z]]^3 (8 EllipticE[InverseEllipticNomeQ[z]]^2 + 7 Pi^2 InverseEllipticNomeQ[z]) - 160 EllipticE[InverseEllipticNomeQ[z]] EllipticK[InverseEllipticNomeQ[z]]^7 (1 + InverseEllipticNomeQ[z]) (-2 + InverseEllipticNomeQ[z] + 2 InverseEllipticNomeQ[z]^2) - 240 EllipticE[InverseEllipticNomeQ[z]] EllipticK[InverseEllipticNomeQ[z]]^ 5 (-Pi^2 + Pi^2 InverseEllipticNomeQ[z] + 4 EllipticE[InverseEllipticNomeQ[z]]^2 InverseEllipticNomeQ[z] + 2 Pi^2 InverseEllipticNomeQ[z]^2) + 40 EllipticK[InverseEllipticNomeQ[z]]^6 (1 + InverseEllipticNomeQ[z]) (-2 Pi^2 - 12 EllipticE[InverseEllipticNomeQ[z]]^2 + Pi^2 InverseEllipticNomeQ[z] + 24 EllipticE[InverseEllipticNomeQ[z]]^2 InverseEllipticNomeQ[z] + 2 Pi^2 InverseEllipticNomeQ[z]^2) + 5 EllipticK[InverseEllipticNomeQ[z]]^4 (-7 Pi^4 + 48 EllipticE[InverseEllipticNomeQ[z]]^4 + 7 Pi^4 InverseEllipticNomeQ[z] + 144 Pi^2 EllipticE[InverseEllipticNomeQ[z]]^2 InverseEllipticNomeQ[z] + 14 Pi^4 InverseEllipticNomeQ[z]^2) + 16 EllipticK[InverseEllipticNomeQ[z]]^8 (-3 - 4 InverseEllipticNomeQ[z] + InverseEllipticNomeQ[z]^2 + 6 InverseEllipticNomeQ[z]^3 + 2 InverseEllipticNomeQ[z]^4))

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "5"]], "}"]]], RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]]]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["32", RowBox[List[SuperscriptBox["\[Pi]", "10"], " ", SuperscriptBox["z", "5"]]]]]], 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[RowBox[List["3", " ", SuperscriptBox["\[Pi]", "8"]]], "-", RowBox[List["25", " ", SuperscriptBox["\[Pi]", "6"], " ", RowBox[List["EllipticE", "[", RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]], "]"]], " ", RowBox[List["EllipticK", "[", RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]], "]"]]]], "+", RowBox[List["5", " ", SuperscriptBox["\[Pi]", "4"], " ", SuperscriptBox[RowBox[List["EllipticK", "[", RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]], "]"]], "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["21", " ", SuperscriptBox[RowBox[List["EllipticE", "[", RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]], "]"]], "2"]]], "+", RowBox[List["5", " ", SuperscriptBox["\[Pi]", "2"], " ", RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]]]]]], ")"]]]], "-", RowBox[List["30", " ", SuperscriptBox["\[Pi]", "2"], " ", RowBox[List["EllipticE", "[", RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]], "]"]], " ", SuperscriptBox[RowBox[List["EllipticK", "[", RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]], "]"]], "3"], " ", RowBox[List["(", RowBox[List[RowBox[List["8", " ", SuperscriptBox[RowBox[List["EllipticE", "[", RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]], "]"]], "2"]]], "+", RowBox[List["7", " ", SuperscriptBox["\[Pi]", "2"], " ", RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]]]]]], ")"]]]], "-", RowBox[List["160", " ", RowBox[List["EllipticE", "[", RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]], "]"]], " ", SuperscriptBox[RowBox[List["EllipticK", "[", RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]], "]"]], "7"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]], "+", RowBox[List["2", " ", SuperscriptBox[RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]], "2"]]]]], ")"]]]], "-", RowBox[List["240", " ", RowBox[List["EllipticE", "[", RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]], "]"]], " ", SuperscriptBox[RowBox[List["EllipticK", "[", RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]], "]"]], "5"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox["\[Pi]", "2"]]], "+", RowBox[List[SuperscriptBox["\[Pi]", "2"], " ", RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]]]], "+", RowBox[List["4", " ", SuperscriptBox[RowBox[List["EllipticE", "[", RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]], "]"]], "2"], " ", RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]]]], "+", RowBox[List["2", " ", SuperscriptBox["\[Pi]", "2"], " ", SuperscriptBox[RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]], "2"]]]]], ")"]]]], "+", RowBox[List["40", " ", SuperscriptBox[RowBox[List["EllipticK", "[", RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]], "]"]], "6"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", SuperscriptBox["\[Pi]", "2"]]], "-", RowBox[List["12", " ", SuperscriptBox[RowBox[List["EllipticE", "[", RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]], "]"]], "2"]]], "+", RowBox[List[SuperscriptBox["\[Pi]", "2"], " ", RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]]]], "+", RowBox[List["24", " ", SuperscriptBox[RowBox[List["EllipticE", "[", RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]], "]"]], "2"], " ", RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]]]], "+", RowBox[List["2", " ", SuperscriptBox["\[Pi]", "2"], " ", SuperscriptBox[RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]], "2"]]]]], ")"]]]], "+", RowBox[List["5", " ", SuperscriptBox[RowBox[List["EllipticK", "[", RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]], "]"]], "4"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "7"]], " ", SuperscriptBox["\[Pi]", "4"]]], "+", RowBox[List["48", " ", SuperscriptBox[RowBox[List["EllipticE", "[", RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]], "]"]], "4"]]], "+", RowBox[List["7", " ", SuperscriptBox["\[Pi]", "4"], " ", RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]]]], "+", RowBox[List["144", " ", SuperscriptBox["\[Pi]", "2"], " ", SuperscriptBox[RowBox[List["EllipticE", "[", RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]], "]"]], "2"], " ", RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]]]], "+", RowBox[List["14", " ", SuperscriptBox["\[Pi]", "4"], " ", SuperscriptBox[RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]], "2"]]]]], ")"]]]], "+", RowBox[List["16", " ", SuperscriptBox[RowBox[List["EllipticK", "[", RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]], "]"]], "8"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "3"]], "-", RowBox[List["4", " ", RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]]]], "+", SuperscriptBox[RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]], "2"], "+", RowBox[List["6", " ", SuperscriptBox[RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]], "3"]]], "+", RowBox[List["2", " ", SuperscriptBox[RowBox[List["InverseEllipticNomeQ", "[", "z", "]"]], "4"]]]]], ")"]]]]]], ")"]]]]]]]]

MathML Form

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</mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <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> <mn> 2 </mn> </msup> </mrow> <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> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mi> K </mi> <mo> ⁡ </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> 7 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 40 </mn> <mo> ⁢ </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> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 24 </mn> <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> <msup> <mrow> <mi> E </mi> <mo> ⁡ </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> 12 </mn> <mo> ⁢ </mo> <msup> <mrow> <mi> E </mi> <mo> ⁡ </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> 2 </mn> <mo> ⁢ </mo> <msup> <mi> π </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mi> π </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <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> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <msup> <mi> π </mi> <mn> 2 </mn> </msup> <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> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mi> K </mi> <mo> ⁡ </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> 6 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 240 </mn> <mo> ⁢ </mo> <mrow> <mi> E </mi> <mo> ⁡ </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> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> 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</mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <msup> <mi> π </mi> <mn> 2 </mn> </msup> <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> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mi> K </mi> <mo> ⁡ </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> 5 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 5 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 48 </mn> <mo> ⁢ </mo> <msup> <mrow> <mi> E </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msup> <semantics> <mi> q </mi> <annotation-xml encoding='MathML-Content'> <ci> EllipticNomeQ 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<apply> <ci> InverseEllipticNomeQ </ci> <ci> z </ci> </apply> <apply> <power /> <apply> <ci> EllipticE </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> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ci> InverseEllipticNomeQ </ci> <ci> z </ci> </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'> 5 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 5 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 48 </cn> <apply> <power /> <apply> <ci> EllipticE </ci> <apply> <ci> InverseEllipticNomeQ </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 144 </cn> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <ci> InverseEllipticNomeQ </ci> <ci> z </ci> </apply> <apply> <power /> <apply> <ci> EllipticE </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'> 7 </cn> <apply> <power /> <pi /> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 14 </cn> <apply> <power /> <pi /> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <apply> <ci> InverseEllipticNomeQ </ci> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 7 </cn> <apply> <power /> <pi 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Rule Form

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

2007-05-02