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http://functions.wolfram.com/08.05.20.0004.01
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D[EllipticF[z, m], {m, 2}] == (1/(16 (m - 1)^2 m^2))
(8 (2 m - 1) EllipticE[z, m] + 4 (m - 1) (3 m - 2) EllipticF[z, m] +
(m Sin[2 z] (2 - 7 m + m^2 - (m - 1) m Cos[2 z] + 2 m (3 m - 1) Sin[z]^2))/
(1 - m Sin[z]^2)^(3/2))
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Cell[BoxData[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["m", ",", "2"]], "}"]]], RowBox[List["EllipticF", "[", RowBox[List["z", ",", "m"]], "]"]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["16", " ", SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], "2"], " ", SuperscriptBox["m", "2"]]]], RowBox[List["(", RowBox[List[RowBox[List["8", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "m"]], "-", "1"]], ")"]], " ", RowBox[List["EllipticE", "[", RowBox[List["z", ",", "m"]], "]"]]]], "+", RowBox[List["4", " ", RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", "m"]], "-", "2"]], ")"]], " ", RowBox[List["EllipticF", "[", RowBox[List["z", ",", "m"]], "]"]]]], "+", FractionBox[RowBox[List["m", " ", RowBox[List["Sin", "[", RowBox[List["2", " ", "z"]], "]"]], " ", RowBox[List["(", RowBox[List["2", "-", RowBox[List["7", " ", "m"]], "+", SuperscriptBox["m", "2"], "-", RowBox[List[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], " ", "m", " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "z"]], "]"]]]], "+", RowBox[List["2", " ", "m", " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", "m"]], "-", "1"]], ")"]], " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]], ")"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]], ")"]], RowBox[List["3", "/", "2"]]]]]], ")"]]]]]]]]
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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> <mi> F </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> ∂ </mo> <msup> <mi> m </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 16 </mn> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msup> <mi> m </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 8 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> m </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> E </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mi> m </mi> </mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> F </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mfrac> <mrow> <mi> m </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> m </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mrow> <mn> 2 </mn> <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> <mo> ⁢ </mo> <mrow> <msup> <mi> sin </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> m </mi> </mrow> <mo> - </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> m </mi> </mrow> <mo> - </mo> <mrow> <mn> 7 </mn> <mo> ⁢ </mo> <mi> m </mi> </mrow> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> m </mi> <mo> ⁢ </mo> <mrow> <msup> <mi> sin </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> m </ci> <degree> <cn type='integer'> 2 </cn> </degree> </bvar> <apply> <ci> EllipticF </ci> <ci> z </ci> <ci> m </ci> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 16 </cn> <apply> <power /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> m </ci> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <ci> EllipticE </ci> <ci> z </ci> <ci> m </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> m </ci> </apply> <cn type='integer'> -2 </cn> </apply> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <apply> <ci> EllipticF </ci> <ci> z </ci> <ci> m </ci> </apply> </apply> <apply> <times /> <ci> m </ci> <apply> <plus /> <apply> <power /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> m </ci> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <sin /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> <ci> m </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <apply> <cos /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> <ci> m </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 7 </cn> <ci> m </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <sin /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> m </ci> <apply> <power /> <apply> <sin /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 3 <sep /> 2 </cn> </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["m_", ",", "2"]], "}"]]]]], RowBox[List["EllipticF", "[", RowBox[List["z_", ",", "m_"]], "]"]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["8", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "m"]], "-", "1"]], ")"]], " ", RowBox[List["EllipticE", "[", RowBox[List["z", ",", "m"]], "]"]]]], "+", RowBox[List["4", " ", RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", "m"]], "-", "2"]], ")"]], " ", RowBox[List["EllipticF", "[", RowBox[List["z", ",", "m"]], "]"]]]], "+", FractionBox[RowBox[List["m", " ", RowBox[List["Sin", "[", RowBox[List["2", " ", "z"]], "]"]], " ", RowBox[List["(", RowBox[List["2", "-", RowBox[List["7", " ", "m"]], "+", SuperscriptBox["m", "2"], "-", RowBox[List[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], " ", "m", " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "z"]], "]"]]]], "+", RowBox[List["2", " ", "m", " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", "m"]], "-", "1"]], ")"]], " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]], ")"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]], ")"]], RowBox[List["3", "/", "2"]]]]]], RowBox[List["16", " ", SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], "2"], " ", SuperscriptBox["m", "2"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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