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JacobiDN




Mathematica Notation

Traditional Notation









Elliptic Functions > JacobiDN[z,m] > Identities involving the group of functions > Local identities of rank 3 > Rank 3 identities with 3 distinct arguments





http://functions.wolfram.com/09.29.18.0057.01









  


  








Input Form





JacobiDN[z, m] JacobiDN[z + a, m] JacobiCN[z + b, m] == (-JacobiCS[a, m]) JacobiDS[b, m] JacobiCN[z, m] - JacobiCS[a, m] JacobiDS[a - b, m] JacobiCN[z + a, m] + JacobiCS[b, m] JacobiCS[a - b, m] JacobiCN[z + b, m]








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> <mrow> <mi> dn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> dn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> + </mo> <mi> z </mi> </mrow> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> cn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> b </mi> <mo> + </mo> <mi> z </mi> </mrow> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mrow> <mi> cn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> b </mi> <mo> + </mo> <mi> z </mi> </mrow> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> cs </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> </mrow> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> cs </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> b </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mrow> <mi> cn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> + </mo> <mi> z </mi> </mrow> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> cs </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> ds </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> </mrow> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mrow> <mi> cn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> cs </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> ds </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> b </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <times /> <apply> <ci> JacobiDN </ci> <ci> z </ci> <ci> m </ci> </apply> <apply> <ci> JacobiDN </ci> <apply> <plus /> <ci> a </ci> <ci> z </ci> </apply> <ci> m </ci> </apply> <apply> <ci> JacobiCN </ci> <apply> <plus /> <ci> b </ci> <ci> z </ci> </apply> <ci> m </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <ci> JacobiCN </ci> <apply> <plus /> <ci> b </ci> <ci> z </ci> </apply> <ci> m </ci> </apply> <apply> <ci> JacobiCS </ci> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <ci> m </ci> </apply> <apply> <ci> JacobiCS </ci> <ci> b </ci> <ci> m </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> JacobiCN </ci> <apply> <plus /> <ci> a </ci> <ci> z </ci> </apply> <ci> m </ci> </apply> <apply> <ci> JacobiCS </ci> <ci> a </ci> <ci> m </ci> </apply> <apply> <ci> JacobiDS </ci> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <ci> m </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> JacobiCN </ci> <ci> z </ci> <ci> m </ci> </apply> <apply> <ci> JacobiCS </ci> <ci> a </ci> <ci> m </ci> </apply> <apply> <ci> JacobiDS </ci> <ci> b </ci> <ci> m </ci> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["JacobiDN", "[", RowBox[List["z_", ",", "m_"]], "]"]], " ", RowBox[List["JacobiDN", "[", RowBox[List[RowBox[List["z_", "+", "a_"]], ",", "m_"]], "]"]], " ", RowBox[List["JacobiCN", "[", RowBox[List[RowBox[List["z_", "+", "b_"]], ",", "m_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["JacobiCS", "[", RowBox[List["a", ",", "m"]], "]"]]]], " ", RowBox[List["JacobiDS", "[", RowBox[List["b", ",", "m"]], "]"]], " ", RowBox[List["JacobiCN", "[", RowBox[List["z", ",", "m"]], "]"]]]], "-", RowBox[List[RowBox[List["JacobiCS", "[", RowBox[List["a", ",", "m"]], "]"]], " ", RowBox[List["JacobiDS", "[", RowBox[List[RowBox[List["a", "-", "b"]], ",", "m"]], "]"]], " ", RowBox[List["JacobiCN", "[", RowBox[List[RowBox[List["z", "+", "a"]], ",", "m"]], "]"]]]], "+", RowBox[List[RowBox[List["JacobiCS", "[", RowBox[List["b", ",", "m"]], "]"]], " ", RowBox[List["JacobiCS", "[", RowBox[List[RowBox[List["a", "-", "b"]], ",", "m"]], "]"]], " ", RowBox[List["JacobiCN", "[", RowBox[List[RowBox[List["z", "+", "b"]], ",", "m"]], "]"]]]]]]]]]]










References





A. Khare, A. Lakshminarayan, U. Sukhatme, "Local Identities Involving Jacobi Elliptic Functions", math-ph/0306028, (2003) http://arXiv.org/abs/math-ph/0306028


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





2003-08-21