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JacobiDS




Mathematica Notation

Traditional Notation









Elliptic Functions > JacobiDS[z,m] > Representations through equivalent functions > With related functions > Involving theta functions





http://functions.wolfram.com/09.30.27.0024.01









  


  








Input Form





JacobiDS[z, m] == (Pi/(2 EllipticK[m])) (EllipticThetaPrime[1, 0, EllipticNomeQ[m]]/EllipticTheta[3, 0, EllipticNomeQ[m]]) (EllipticTheta[3, z/EllipticTheta[3, 0, EllipticNomeQ[m]]^2, EllipticNomeQ[m]]/EllipticTheta[1, z/EllipticTheta[3, 0, EllipticNomeQ[m]]^2, EllipticNomeQ[m]])








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["JacobiDS", "[", RowBox[List["z", ",", "m"]], "]"]], "\[Equal]", RowBox[List[FractionBox["\[Pi]", RowBox[List["2", RowBox[List["EllipticK", "[", "m", "]"]]]]], " ", FractionBox[RowBox[List["EllipticThetaPrime", "[", RowBox[List["1", ",", "0", ",", RowBox[List["EllipticNomeQ", "[", "m", "]"]]]], "]"]], RowBox[List["EllipticTheta", "[", RowBox[List["3", ",", "0", ",", RowBox[List["EllipticNomeQ", "[", "m", "]"]]]], "]"]]], " ", RowBox[List[RowBox[List["EllipticTheta", "[", RowBox[List["3", ",", FractionBox["z", SuperscriptBox[RowBox[List["EllipticTheta", "[", RowBox[List["3", ",", "0", ",", RowBox[List["EllipticNomeQ", "[", "m", "]"]]]], "]"]], "2"]], ",", RowBox[List["EllipticNomeQ", "[", "m", "]"]]]], "]"]], "/", RowBox[List["EllipticTheta", "[", RowBox[List["1", ",", FractionBox["z", SuperscriptBox[RowBox[List["EllipticTheta", "[", RowBox[List["3", ",", "0", ",", RowBox[List["EllipticNomeQ", "[", "m", "]"]]]], "]"]], "2"]], ",", RowBox[List["EllipticNomeQ", "[", "m", "]"]]]], "]"]]]]]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mi> ds </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mi> &#960; </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mi> K </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mfrac> <mrow> <msubsup> <mi> &#977; </mi> <mn> 1 </mn> <mo> &#8242; </mo> </msubsup> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mrow> <semantics> <mi> q </mi> <annotation-xml encoding='MathML-Content'> <ci> EllipticNomeQ </ci> </annotation-xml> </semantics> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <msub> <mi> &#977; </mi> <mn> 3 </mn> </msub> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mrow> <semantics> <mi> q </mi> <annotation-xml encoding='MathML-Content'> <ci> EllipticNomeQ </ci> </annotation-xml> </semantics> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> &#8290; </mo> <mfrac> <mrow> <msub> <mi> &#977; </mi> <mn> 3 </mn> </msub> <mo> ( </mo> <mrow> <mfrac> <mi> z </mi> <msup> <mrow> <msub> <mi> &#977; </mi> <mn> 3 </mn> </msub> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mrow> <semantics> <mi> q </mi> <annotation-xml encoding='MathML-Content'> <ci> EllipticNomeQ </ci> </annotation-xml> </semantics> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mfrac> <mo> , </mo> <mrow> <semantics> <mi> q </mi> <annotation-xml encoding='MathML-Content'> <ci> EllipticNomeQ </ci> </annotation-xml> </semantics> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <msub> <mi> &#977; </mi> <mn> 1 </mn> </msub> <mo> ( </mo> <mrow> <mfrac> <mi> z </mi> <msup> <mrow> <msub> <mi> &#977; </mi> <mn> 3 </mn> </msub> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mrow> <semantics> <mi> q </mi> <annotation-xml encoding='MathML-Content'> <ci> EllipticNomeQ </ci> </annotation-xml> </semantics> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mfrac> <mo> , </mo> <mrow> <semantics> <mi> q </mi> <annotation-xml encoding='MathML-Content'> <ci> EllipticNomeQ </ci> </annotation-xml> </semantics> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mfrac> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> JacobiDS </ci> <ci> z </ci> <ci> m </ci> </apply> <apply> <times /> <apply> <times /> <pi /> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <apply> <partialdiff /> <list> <cn type='integer'> 1 </cn> </list> <apply> <ci> Subscript </ci> <ci> &#977; </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> 0 </cn> <apply> <ci> EllipticNomeQ </ci> <ci> m </ci> </apply> </apply> <apply> <power /> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 3 </cn> <cn type='integer'> 0 </cn> <apply> <ci> EllipticNomeQ </ci> <ci> m </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 3 </cn> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <power /> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 3 </cn> <cn type='integer'> 0 </cn> <apply> <ci> EllipticNomeQ </ci> <ci> m </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> EllipticNomeQ </ci> <ci> m </ci> </apply> </apply> <apply> <power /> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 1 </cn> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <power /> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 3 </cn> <cn type='integer'> 0 </cn> <apply> <ci> EllipticNomeQ </ci> <ci> m </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> EllipticNomeQ </ci> <ci> m </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["JacobiDS", "[", RowBox[List["z_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List["\[Pi]", " ", RowBox[List["EllipticThetaPrime", "[", RowBox[List["1", ",", "0", ",", RowBox[List["EllipticNomeQ", "[", "m", "]"]]]], "]"]], " ", RowBox[List["EllipticTheta", "[", RowBox[List["3", ",", FractionBox["z", SuperscriptBox[RowBox[List["EllipticTheta", "[", RowBox[List["3", ",", "0", ",", RowBox[List["EllipticNomeQ", "[", "m", "]"]]]], "]"]], "2"]], ",", RowBox[List["EllipticNomeQ", "[", "m", "]"]]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["2", " ", RowBox[List["EllipticK", "[", "m", "]"]]]], ")"]], " ", RowBox[List["EllipticTheta", "[", RowBox[List["3", ",", "0", ",", RowBox[List["EllipticNomeQ", "[", "m", "]"]]]], "]"]], " ", RowBox[List["EllipticTheta", "[", RowBox[List["1", ",", FractionBox["z", SuperscriptBox[RowBox[List["EllipticTheta", "[", RowBox[List["3", ",", "0", ",", RowBox[List["EllipticNomeQ", "[", "m", "]"]]]], "]"]], "2"]], ",", RowBox[List["EllipticNomeQ", "[", "m", "]"]]]], "]"]]]]]]]]]


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





2001-10-29