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JacobiDN




Mathematica Notation

Traditional Notation









Elliptic Functions > JacobiDN[z,m] > Product representations





http://functions.wolfram.com/09.29.08.0001.01









  


  








Input Form





JacobiDN[z, m] == (1 - m)^(1/4) Product[(1 + 2 EllipticNomeQ[m]^(2 n - 1) Cos[(Pi z)/EllipticK[m]] + EllipticNomeQ[m]^(4 n - 2))/(1 - 2 EllipticNomeQ[m]^(2 n - 1) Cos[(Pi z)/EllipticK[m]] + EllipticNomeQ[m]^(4 n - 2)), {n, 1, Infinity}]








Standard Form





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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> dn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mroot> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> m </mi> </mrow> <mn> 4 </mn> </mroot> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> n </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mfrac> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <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> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mrow> <mi> K </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <semantics> <mi> q </mi> <annotation-xml encoding='MathML-Content'> <ci> EllipticNomeQ </ci> </annotation-xml> </semantics> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> - </mo> <mn> 2 </mn> </mrow> </msup> </mrow> </mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <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> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mrow> <mi> K </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <semantics> <mi> q </mi> <annotation-xml encoding='MathML-Content'> <ci> EllipticNomeQ </ci> </annotation-xml> </semantics> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> - </mo> <mn> 2 </mn> </mrow> </msup> </mrow> </mrow> </mfrac> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> JacobiDN </ci> <ci> z </ci> <ci> m </ci> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <apply> <product /> <bvar> <ci> n </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <ci> EllipticNomeQ </ci> <ci> m </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <cos /> <apply> <times /> <pi /> <ci> z </ci> <apply> <power /> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <ci> EllipticNomeQ </ci> <apply> <power /> <ci> m </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> n </ci> </apply> <cn type='integer'> -2 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <ci> EllipticNomeQ </ci> <ci> m </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <cos /> <apply> <times /> <pi /> <ci> z </ci> <apply> <power /> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <ci> EllipticNomeQ </ci> <apply> <power /> <ci> m </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> n </ci> </apply> <cn type='integer'> -2 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["JacobiDN", "[", RowBox[List["z_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "m"]], ")"]], RowBox[List["1", "/", "4"]]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["n", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List["1", "+", RowBox[List["2", " ", SuperscriptBox[RowBox[List["EllipticNomeQ", "[", "m", "]"]], RowBox[List[RowBox[List["2", " ", "n"]], "-", "1"]]], " ", RowBox[List["Cos", "[", FractionBox[RowBox[List["\[Pi]", " ", "z"]], RowBox[List["EllipticK", "[", "m", "]"]]], "]"]]]], "+", SuperscriptBox[RowBox[List["EllipticNomeQ", "[", "m", "]"]], RowBox[List[RowBox[List["4", " ", "n"]], "-", "2"]]]]], RowBox[List["1", "-", RowBox[List["2", " ", SuperscriptBox[RowBox[List["EllipticNomeQ", "[", "m", "]"]], RowBox[List[RowBox[List["2", " ", "n"]], "-", "1"]]], " ", RowBox[List["Cos", "[", FractionBox[RowBox[List["\[Pi]", " ", "z"]], RowBox[List["EllipticK", "[", "m", "]"]]], "]"]]]], "+", SuperscriptBox[RowBox[List["EllipticNomeQ", "[", "m", "]"]], RowBox[List[RowBox[List["4", " ", "n"]], "-", "2"]]]]]]]]]]]]]]


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





2001-10-29