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JacobiDN






Mathematica Notation

Traditional Notation









Elliptic Functions > JacobiDN[z,m] > Identities involving the group of functions > Local identities of rank 4 > Rank 4 identities with 2 distinct arguments





http://functions.wolfram.com/09.29.18.0135.01









  


  










Input Form





m JacobiSN[z, m] JacobiCN[z, m] JacobiDN[z + a, m] JacobiSN[z + a, m] == JacobiDS[a, m] (JacobiCS[a, m]^2 + JacobiNS[a, m]^2) JacobiSN[z, m] - JacobiCS[a, m] (JacobiDS[a, m]^2 + JacobiNS[a, m]^2) JacobiSN[z + a, m] + JacobiDS[a, m] JacobiNS[a, m] JacobiCN[z + a, m] JacobiDN[z + a, m] + JacobiCS[a, m] JacobiNS[a, m] JacobiCN[z, m] JacobiDN[z, m]










Standard Form





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MathML Form







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</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> 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> ns </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </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> 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> ds </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; 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</mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mrow> <mi> cs </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mrow> <mi> ns </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> sn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <times /> <ci> m </ci> <apply> <ci> JacobiSN </ci> <ci> z </ci> <ci> m </ci> </apply> <apply> <ci> JacobiCN </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> JacobiSN </ci> <apply> <plus /> <ci> a </ci> <ci> z </ci> </apply> <ci> m </ci> </apply> </apply> <apply> <plus /> <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> JacobiDN </ci> <ci> z </ci> <ci> m </ci> </apply> <apply> <ci> JacobiNS </ci> <ci> a </ci> <ci> m </ci> </apply> </apply> <apply> <times /> <apply> <ci> JacobiCN </ci> <apply> <plus /> <ci> a </ci> <ci> z </ci> </apply> <ci> m </ci> </apply> <apply> <ci> JacobiDN </ci> <apply> <plus /> <ci> a </ci> <ci> z </ci> </apply> <ci> m </ci> </apply> <apply> <ci> JacobiDS </ci> <ci> a </ci> <ci> m </ci> </apply> <apply> <ci> JacobiNS </ci> <ci> a </ci> <ci> m </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> JacobiCS </ci> <ci> a </ci> <ci> m </ci> </apply> <apply> <plus /> <apply> <power /> <apply> <ci> JacobiDS </ci> <ci> a </ci> <ci> m </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ci> JacobiNS </ci> <ci> a </ci> <ci> m </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <ci> JacobiSN </ci> <apply> <plus /> <ci> a </ci> <ci> z </ci> </apply> <ci> m </ci> </apply> </apply> </apply> <apply> <times /> <apply> <ci> JacobiDS </ci> <ci> a </ci> <ci> m </ci> </apply> <apply> <plus /> <apply> <power /> <apply> <ci> JacobiCS </ci> <ci> a </ci> <ci> m </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ci> JacobiNS </ci> <ci> a </ci> <ci> m </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <ci> JacobiSN </ci> <ci> z </ci> <ci> m </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["m_", " ", RowBox[List["JacobiSN", "[", RowBox[List["z_", ",", "m_"]], "]"]], " ", RowBox[List["JacobiCN", "[", RowBox[List["z_", ",", "m_"]], "]"]], " ", RowBox[List["JacobiDN", "[", RowBox[List[RowBox[List["z_", "+", "a_"]], ",", "m_"]], "]"]], " ", RowBox[List["JacobiSN", "[", RowBox[List[RowBox[List["z_", "+", "a_"]], ",", "m_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["JacobiDS", "[", RowBox[List["a", ",", "m"]], "]"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["JacobiCS", "[", RowBox[List["a", ",", "m"]], "]"]], "2"], "+", SuperscriptBox[RowBox[List["JacobiNS", "[", RowBox[List["a", ",", "m"]], "]"]], "2"]]], ")"]], " ", RowBox[List["JacobiSN", "[", RowBox[List["z", ",", "m"]], "]"]]]], "-", RowBox[List[RowBox[List["JacobiCS", "[", RowBox[List["a", ",", "m"]], "]"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["JacobiDS", "[", RowBox[List["a", ",", "m"]], "]"]], "2"], "+", SuperscriptBox[RowBox[List["JacobiNS", "[", RowBox[List["a", ",", "m"]], "]"]], "2"]]], ")"]], " ", RowBox[List["JacobiSN", "[", RowBox[List[RowBox[List["z", "+", "a"]], ",", "m"]], "]"]]]], "+", RowBox[List[RowBox[List["JacobiDS", "[", RowBox[List["a", ",", "m"]], "]"]], " ", RowBox[List["JacobiNS", "[", RowBox[List["a", ",", "m"]], "]"]], " ", RowBox[List["JacobiCN", "[", RowBox[List[RowBox[List["z", "+", "a"]], ",", "m"]], "]"]], " ", RowBox[List["JacobiDN", "[", RowBox[List[RowBox[List["z", "+", "a"]], ",", "m"]], "]"]]]], "+", RowBox[List[RowBox[List["JacobiCS", "[", RowBox[List["a", ",", "m"]], "]"]], " ", RowBox[List["JacobiNS", "[", RowBox[List["a", ",", "m"]], "]"]], " ", RowBox[List["JacobiCN", "[", RowBox[List["z", ",", "m"]], "]"]], " ", RowBox[List["JacobiDN", "[", RowBox[List["z", ",", "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





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