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http://functions.wolfram.com/09.37.06.0003.02
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InverseJacobiCD[z, m] \[Proportional]
ArcCos[z] + (1/4) (z Sqrt[1 - z^2] + ArcCos[z]) m +
(3/64) (z Sqrt[1 - z^2] (3 + 2 z^2) + 3 ArcCos[z]) m^2 + \[Ellipsis] /;
(m -> 0)
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["InverseJacobiCD", "[", RowBox[List["z", ",", "m"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List["ArcCos", "[", "z", "]"]], "+", RowBox[List[FractionBox["1", "4"], RowBox[List["(", RowBox[List[RowBox[List["z", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]]]], "+", RowBox[List["ArcCos", "[", "z", "]"]]]], ")"]], "m"]], "+", RowBox[List[FractionBox["3", "64"], " ", RowBox[List["(", RowBox[List[RowBox[List["z", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", SuperscriptBox["z", "2"]]]]], ")"]]]], "+", RowBox[List["3", " ", RowBox[List["ArcCos", "[", "z", "]"]]]]]], ")"]], SuperscriptBox["m", "2"]]], "+", "\[Ellipsis]"]]]], "/;", RowBox[List["(", RowBox[List["m", "\[Rule]", "0"]], ")"]]]]]]
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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> <mrow> <msup> <mi> cd </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> <mrow> <mrow> <msup> <mi> cos </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mrow> <msup> <mi> cos </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> m </mi> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 3 </mn> <mn> 64 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> ⁢ </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mrow> <msup> <mi> cos </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> m </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mo> … </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> InverseJacobiCD </ci> <ci> z </ci> <ci> m </ci> </apply> <apply> <plus /> <apply> <arccos /> <ci> z </ci> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 4 </cn> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> </apply> <apply> <arccos /> <ci> z </ci> </apply> </apply> <ci> m </ci> </apply> <apply> <times /> <cn type='rational'> 3 <sep /> 64 </cn> <apply> <plus /> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <arccos /> <ci> z </ci> </apply> </apply> </apply> <apply> <power /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> </apply> <ci> … </ci> </apply> </apply> <apply> <ci> Rule </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["InverseJacobiCD", "[", RowBox[List["z_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["ArcCos", "[", "z", "]"]], "+", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["z", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]]]], "+", RowBox[List["ArcCos", "[", "z", "]"]]]], ")"]], " ", "m"]], "+", RowBox[List[FractionBox["3", "64"], " ", RowBox[List["(", RowBox[List[RowBox[List["z", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", SuperscriptBox["z", "2"]]]]], ")"]]]], "+", RowBox[List["3", " ", RowBox[List["ArcCos", "[", "z", "]"]]]]]], ")"]], " ", SuperscriptBox["m", "2"]]], "+", "\[Ellipsis]"]], "/;", RowBox[List["(", RowBox[List["m", "\[Rule]", "0"]], ")"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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