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http://functions.wolfram.com/09.52.20.0002.02
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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]))
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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", "]"]]]]]], ")"]]]], ")"]]]]]]]]
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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> 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> ∂ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> ⩵ </mo> <mfrac> <mrow> <mn> 4 </mn> <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> 2 </mn> </msup> <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> <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> <mrow> <mn> 4 </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> 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> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 4 </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> <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> </mrow> <mo> + </mo> <msup> <mi> π </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <msup> <mi> π </mi> <mn> 4 </mn> </msup> <mo> ⁢ </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>
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| 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"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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