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http://functions.wolfram.com/09.32.21.0002.01
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Integrate[JacobiND[z, m]^2, z] ==
(1/(1 - m)) ((EllipticE[JacobiAmplitude[z, m], m] JacobiDN[z, m])/
Sqrt[1 - m JacobiSN[z, m]^2]) -
(m/(1 - m)) ((JacobiCN[z, m] JacobiSN[z, m])/JacobiDN[z, m])
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Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox[RowBox[List["JacobiND", "[", RowBox[List["z", ",", "m"]], "]"]], "2"], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List[FractionBox["1", RowBox[List["1", "-", "m"]]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["EllipticE", "[", RowBox[List[RowBox[List["JacobiAmplitude", "[", RowBox[List["z", ",", "m"]], "]"]], ",", "m"]], "]"]], " ", RowBox[List["JacobiDN", "[", RowBox[List["z", ",", "m"]], "]"]]]], ")"]], "/", RowBox[List["(", SqrtBox[RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["JacobiSN", "[", RowBox[List["z", ",", "m"]], "]"]], "2"]]]]]], ")"]]]]]], "-", RowBox[List[FractionBox["m", RowBox[List["1", "-", "m"]]], " ", FractionBox[RowBox[List[RowBox[List["JacobiCN", "[", RowBox[List["z", ",", "m"]], "]"]], " ", RowBox[List["JacobiSN", "[", RowBox[List["z", ",", "m"]], "]"]]]], RowBox[List["JacobiDN", "[", RowBox[List["z", ",", "m"]], "]"]]]]]]]]]]]
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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> <mrow> <mo> ∫ </mo> <mrow> <msup> <mrow> <mi> nd </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <mrow> <mi> E </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> am </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> dn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> m </mi> <mo> ⁢ </mo> <msup> <mrow> <mi> sn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> </mrow> </msqrt> </mrow> </mfrac> <mo> - </mo> <mfrac> <mrow> <mi> m </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> cn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> sn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> dn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <power /> <apply> <ci> JacobiND </ci> <ci> z </ci> <ci> m </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <ci> EllipticE </ci> <apply> <ci> JacobiAmplitude </ci> <ci> z </ci> <ci> m </ci> </apply> <ci> m </ci> </apply> <apply> <ci> JacobiDN </ci> <ci> z </ci> <ci> m </ci> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> m </ci> <apply> <power /> <apply> <ci> JacobiSN </ci> <ci> z </ci> <ci> m </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> m </ci> <apply> <times /> <apply> <ci> JacobiCN </ci> <ci> z </ci> <ci> m </ci> </apply> <apply> <ci> JacobiSN </ci> <ci> z </ci> <ci> m </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <apply> <ci> JacobiDN </ci> <ci> z </ci> <ci> m </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[SuperscriptBox[RowBox[List["JacobiND", "[", RowBox[List["z_", ",", "m_"]], "]"]], "2"], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["EllipticE", "[", RowBox[List[RowBox[List["JacobiAmplitude", "[", RowBox[List["z", ",", "m"]], "]"]], ",", "m"]], "]"]], " ", RowBox[List["JacobiDN", "[", RowBox[List["z", ",", "m"]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["1", "-", "m"]], ")"]], " ", SqrtBox[RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["JacobiSN", "[", RowBox[List["z", ",", "m"]], "]"]], "2"]]]]]]]]], "-", FractionBox[RowBox[List["m", " ", RowBox[List["(", RowBox[List[RowBox[List["JacobiCN", "[", RowBox[List["z", ",", "m"]], "]"]], " ", RowBox[List["JacobiSN", "[", RowBox[List["z", ",", "m"]], "]"]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["1", "-", "m"]], ")"]], " ", RowBox[List["JacobiDN", "[", RowBox[List["z", ",", "m"]], "]"]]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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