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InverseJacobiSC






Mathematica Notation

Traditional Notation









Elliptic Functions > InverseJacobiSC[z,m] > Representations through more general functions > Through other functions > Involving some hypergeometric-type functions





http://functions.wolfram.com/09.46.26.0003.01









  


  










Input Form





InverseJacobiSC[z, m] == (z/Sqrt[z^2 + 1]) AppellF1[1/2, 1/2, 1/2, 3/2, z^2/(z^2 + 1), (m z^2)/(z^2 + 1)] /; Element[z, Reals] && (m - 1) z^2 < 1










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["InverseJacobiSC", "[", RowBox[List["z", ",", "m"]], "]"]], "\[Equal]", RowBox[List[FractionBox["z", SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "+", "1"]]]], " ", RowBox[List["AppellF1", "[", RowBox[List[FractionBox["1", "2"], ",", FractionBox["1", "2"], ",", FractionBox["1", "2"], ",", FractionBox["3", "2"], ",", FractionBox[SuperscriptBox["z", "2"], RowBox[List[SuperscriptBox["z", "2"], "+", "1"]]], ",", FractionBox[RowBox[List["m", " ", SuperscriptBox["z", "2"]]], RowBox[List[SuperscriptBox["z", "2"], "+", "1"]]]]], "]"]]]]]], "/;", RowBox[List[RowBox[List["z", "\[Element]", "Reals"]], "\[And]", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], SuperscriptBox["z", "2"]]], "<", "1"]]]]]]]]










MathML Form







<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> sc </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> z </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mi> z </mi> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mfrac> <mo> &#8290; </mo> <mrow> <semantics> <msub> <mi> F </mi> <mn> 1 </mn> </msub> <annotation-xml encoding='MathML-Content'> <ci> AppellF1 </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ; </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ; </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> <mo> ; </mo> <mfrac> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> </mrow> <mo> , </mo> <mfrac> <mrow> <mi> m </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> z </mi> <mo> &#8712; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[Reals]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> &lt; </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> InverseJacobiSC </ci> <ci> z </ci> <ci> m </ci> </apply> <apply> <times /> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> AppellF1 </ci> <cn type='rational'> 1 <sep /> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> <cn type='rational'> 3 <sep /> 2 </cn> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <ci> m </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> z </ci> <reals /> </apply> <apply> <lt /> <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> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["InverseJacobiSC", "[", RowBox[List["z_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List["z", " ", RowBox[List["AppellF1", "[", RowBox[List[FractionBox["1", "2"], ",", FractionBox["1", "2"], ",", FractionBox["1", "2"], ",", FractionBox["3", "2"], ",", FractionBox[SuperscriptBox["z", "2"], RowBox[List[SuperscriptBox["z", "2"], "+", "1"]]], ",", FractionBox[RowBox[List["m", " ", SuperscriptBox["z", "2"]]], RowBox[List[SuperscriptBox["z", "2"], "+", "1"]]]]], "]"]]]], SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "+", "1"]]]], "/;", RowBox[List[RowBox[List["z", "\[Element]", "Reals"]], "&&", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], " ", SuperscriptBox["z", "2"]]], "<", "1"]]]]]]]]]]










Date Added to functions.wolfram.com (modification date)





2001-10-29