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http://functions.wolfram.com/09.47.20.0011.01
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D[InverseJacobiSD[z, m], {z, 2}] ==
((Sqrt[1 + m z^2] JacobiCN[InverseJacobiSD[z, m], m])/
Sqrt[1 + (-1 + m) z^2]) D[1/(Sqrt[m z^2 + 1] Sqrt[1 - (1 - m) z^2]), z]
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Cell[BoxData[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "2"]], "}"]]], RowBox[List["InverseJacobiSD", "[", RowBox[List["z", ",", "m"]], "]"]]]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SqrtBox[RowBox[List["1", "+", RowBox[List["m", " ", SuperscriptBox["z", "2"]]]]]], " ", RowBox[List["JacobiCN", "[", RowBox[List[RowBox[List["InverseJacobiSD", "[", RowBox[List["z", ",", "m"]], "]"]], ",", "m"]], "]"]]]], SqrtBox[RowBox[List["1", "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]], " ", SuperscriptBox["z", "2"]]]]]]], RowBox[List["D", "[", RowBox[List[FractionBox["1", RowBox[List[SqrtBox[RowBox[List[RowBox[List["m", " ", SuperscriptBox["z", "2"]]], "+", "1"]]], " ", SqrtBox[RowBox[List["1", "-", RowBox[List[RowBox[List["(", RowBox[List["1", "-", "m"]], ")"]], " ", SuperscriptBox["z", "2"]]]]]]]]], ",", "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> <mi> sd </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> ∂ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo>  </mo> <mrow> <mfrac> <mrow> <msqrt> <mrow> <mrow> <mi> m </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> ⁢ </mo> <mrow> <mi> cn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <msup> <mi> sd </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <msqrt> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mfrac> <mo> ⁢ </mo> <mfrac> <mrow> <mo> ∂ </mo> <mfrac> <mn> 1 </mn> <mrow> <msqrt> <mrow> <mrow> <mi> m </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </mrow> </msqrt> </mrow> </mfrac> </mrow> <mrow> <mo> ∂ </mo> <mi> z </mi> </mrow> </mfrac> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> z </ci> <degree> <cn type='integer'> 2 </cn> </degree> </bvar> <apply> <ci> InverseJacobiSD </ci> <ci> z </ci> <ci> m </ci> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <ci> m </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> JacobiCN </ci> <apply> <ci> InverseJacobiSD </ci> <ci> z </ci> <ci> m </ci> </apply> <ci> m </ci> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <partialdiff /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <ci> m </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <apply> <power /> <ci> z </ci> <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> </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["InverseJacobiSD", "[", RowBox[List["z_", ",", "m_"]], "]"]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["1", "+", RowBox[List["m", " ", SuperscriptBox["z", "2"]]]]]], " ", RowBox[List["JacobiCN", "[", RowBox[List[RowBox[List["InverseJacobiSD", "[", RowBox[List["z", ",", "m"]], "]"]], ",", "m"]], "]"]]]], ")"]], " ", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["z"]]], FractionBox["1", RowBox[List[SqrtBox[RowBox[List[RowBox[List["m", " ", SuperscriptBox["z", "2"]]], "+", "1"]]], " ", SqrtBox[RowBox[List["1", "-", RowBox[List[RowBox[List["(", RowBox[List["1", "-", "m"]], ")"]], " ", SuperscriptBox["z", "2"]]]]]]]]]]]]], SqrtBox[RowBox[List["1", "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]], " ", SuperscriptBox["z", "2"]]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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