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JacobiSC




Mathematica Notation

Traditional Notation









Elliptic Functions > JacobiSC[z,m] > Complex characteristics > Conjugate value





http://functions.wolfram.com/09.34.19.0005.01









  


  








Input Form





Conjugate[JacobiSC[x + I y, m]] == (JacobiSN[x, m] JacobiDN[y, 1 - m] - I JacobiCN[x, m] JacobiDN[x, m] JacobiCN[y, 1 - m] JacobiSN[y, 1 - m])/ (JacobiCN[x, m] JacobiCN[y, 1 - m] + I JacobiSN[x, m] JacobiDN[x, m] JacobiSN[y, 1 - m] JacobiDN[y, 1 - m]) /; Element[{x, y, m}, Reals]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Conjugate", "[", RowBox[List["JacobiSC", "[", RowBox[List[RowBox[List["x", "+", RowBox[List["\[ImaginaryI]", " ", "y"]]]], ",", "m"]], "]"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["JacobiSN", "[", RowBox[List["x", ",", "m"]], "]"]], " ", RowBox[List["JacobiDN", "[", RowBox[List["y", ",", RowBox[List["1", "-", "m"]]]], "]"]]]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["JacobiCN", "[", RowBox[List["x", ",", "m"]], "]"]], " ", RowBox[List["JacobiDN", "[", RowBox[List["x", ",", "m"]], "]"]], " ", RowBox[List["JacobiCN", "[", RowBox[List["y", ",", RowBox[List["1", "-", "m"]]]], "]"]], " ", RowBox[List["JacobiSN", "[", RowBox[List["y", ",", RowBox[List["1", "-", "m"]]]], "]"]]]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["JacobiCN", "[", RowBox[List["x", ",", "m"]], "]"]], " ", RowBox[List["JacobiCN", "[", RowBox[List["y", ",", RowBox[List["1", "-", "m"]]]], "]"]]]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["JacobiSN", "[", RowBox[List["x", ",", "m"]], "]"]], " ", RowBox[List["JacobiDN", "[", RowBox[List["x", ",", "m"]], "]"]], " ", RowBox[List["JacobiSN", "[", RowBox[List["y", ",", RowBox[List["1", "-", "m"]]]], "]"]], " ", RowBox[List["JacobiDN", "[", RowBox[List["y", ",", RowBox[List["1", "-", "m"]]]], "]"]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["{", RowBox[List["x", ",", "y", ",", "m"]], "}"]], "\[Element]", "Reals"]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mover> <mrow> <mi> sc </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> x </mi> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> y </mi> </mrow> </mrow> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> _ </mo> </mover> <mo> &#10869; </mo> <mfrac> <mrow> <mrow> <mrow> <mi> dn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> y </mi> <mo> &#10072; </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> m </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> sn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> x </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mi> cn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> x </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> cn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> y </mi> <mo> &#10072; </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> m </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> dn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> x </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> sn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> y </mi> <mo> &#10072; </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> m </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mrow> <mrow> <mi> cn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> x </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> cn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> y </mi> <mo> &#10072; </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> m </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mi> sn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> x </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> dn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> x </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> sn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> y </mi> <mo> &#10072; </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> m </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> dn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> y </mi> <mo> &#10072; </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> m </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mfrac> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> { </mo> <mrow> <mi> x </mi> <mo> , </mo> <mi> y </mi> <mo> , </mo> <mi> m </mi> </mrow> <mo> } </mo> </mrow> <mo> &#8712; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[List[], Reals]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> OverBar </ci> <apply> <ci> JacobiSC </ci> <apply> <plus /> <ci> x </ci> <apply> <times /> <imaginaryi /> <ci> y </ci> </apply> </apply> <ci> m </ci> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <ci> JacobiDN </ci> <ci> y </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> <apply> <ci> JacobiSN </ci> <ci> x </ci> <ci> m </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <ci> JacobiCN </ci> <ci> x </ci> <ci> m </ci> </apply> <apply> <ci> JacobiCN </ci> <ci> y </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> <apply> <ci> JacobiDN </ci> <ci> x </ci> <ci> m </ci> </apply> <apply> <ci> JacobiSN </ci> <ci> y </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <ci> JacobiCN </ci> <ci> x </ci> <ci> m </ci> </apply> <apply> <ci> JacobiCN </ci> <ci> y </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> </apply> <apply> <times /> <imaginaryi /> <apply> <ci> JacobiSN </ci> <ci> x </ci> <ci> m </ci> </apply> <apply> <ci> JacobiDN </ci> <ci> x </ci> <ci> m </ci> </apply> <apply> <ci> JacobiSN </ci> <ci> y </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> <apply> <ci> JacobiDN </ci> <ci> y </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <in /> <list> <ci> x </ci> <ci> y </ci> <ci> m </ci> </list> <reals /> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Conjugate", "[", RowBox[List["JacobiSC", "[", RowBox[List[RowBox[List["x_", "+", RowBox[List["\[ImaginaryI]", " ", "y_"]]]], ",", "m_"]], "]"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List[RowBox[List["JacobiSN", "[", RowBox[List["x", ",", "m"]], "]"]], " ", RowBox[List["JacobiDN", "[", RowBox[List["y", ",", RowBox[List["1", "-", "m"]]]], "]"]]]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["JacobiCN", "[", RowBox[List["x", ",", "m"]], "]"]], " ", RowBox[List["JacobiDN", "[", RowBox[List["x", ",", "m"]], "]"]], " ", RowBox[List["JacobiCN", "[", RowBox[List["y", ",", RowBox[List["1", "-", "m"]]]], "]"]], " ", RowBox[List["JacobiSN", "[", RowBox[List["y", ",", RowBox[List["1", "-", "m"]]]], "]"]]]]]], RowBox[List[RowBox[List[RowBox[List["JacobiCN", "[", RowBox[List["x", ",", "m"]], "]"]], " ", RowBox[List["JacobiCN", "[", RowBox[List["y", ",", RowBox[List["1", "-", "m"]]]], "]"]]]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["JacobiSN", "[", RowBox[List["x", ",", "m"]], "]"]], " ", RowBox[List["JacobiDN", "[", RowBox[List["x", ",", "m"]], "]"]], " ", RowBox[List["JacobiSN", "[", RowBox[List["y", ",", RowBox[List["1", "-", "m"]]]], "]"]], " ", RowBox[List["JacobiDN", "[", RowBox[List["y", ",", RowBox[List["1", "-", "m"]]]], "]"]]]]]]], "/;", RowBox[List[RowBox[List["{", RowBox[List["x", ",", "y", ",", "m"]], "}"]], "\[Element]", "Reals"]]]]]]]]


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





2007-05-02