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JacobiDN






Mathematica Notation

Traditional Notation









Elliptic Functions > JacobiDN[z,m] > Identities involving the group of functions > Cyclic Identities of rank 3 > p=3





http://functions.wolfram.com/09.29.18.0018.01









  


  










Input Form





JacobiDN[z, m] JacobiCN[z + (4 EllipticK[m])/3, m] JacobiCN[z + (8 EllipticK[m])/3, m] + JacobiDN[z + (4 EllipticK[m])/3, m] JacobiCN[z + (8 EllipticK[m])/3, m] JacobiCN[z, m] + JacobiDN[z + (8 EllipticK[m])/3, m] JacobiCN[z, m] JacobiCN[z + (4 EllipticK[m])/3, m] == (-(JacobiDN[(2 EllipticK[m])/3, m]^2/(1 + JacobiDN[(2 EllipticK[m])/3, m])^ 2)) (JacobiDN[z, m] + JacobiDN[z + (4 EllipticK[m])/3, m] + JacobiDN[z + (8 EllipticK[m])/3, m])










Standard Form





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MathML Form







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</mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mi> K </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> <mn> 3 </mn> </mfrac> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mfrac> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <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> + </mo> <mrow> <mi> dn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> + </mo> <mfrac> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mrow> <mi> K </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> <mn> 3 </mn> </mfrac> </mrow> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> dn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> + </mo> <mfrac> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <mrow> <mi> K </mi> <mo> &#8289; 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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List[RowBox[List["JacobiDN", "[", RowBox[List["z_", ",", "m_"]], "]"]], " ", RowBox[List["JacobiCN", "[", RowBox[List[RowBox[List["z_", "+", FractionBox[RowBox[List["4", " ", RowBox[List["EllipticK", "[", "m_", "]"]]]], "3"]]], ",", "m_"]], "]"]], " ", RowBox[List["JacobiCN", "[", RowBox[List[RowBox[List["z_", "+", FractionBox[RowBox[List["8", " ", RowBox[List["EllipticK", "[", "m_", "]"]]]], "3"]]], ",", "m_"]], "]"]]]], "+", RowBox[List[RowBox[List["JacobiDN", "[", RowBox[List[RowBox[List["z_", "+", FractionBox[RowBox[List["4", " ", RowBox[List["EllipticK", "[", "m_", "]"]]]], "3"]]], ",", "m_"]], "]"]], " ", RowBox[List["JacobiCN", "[", RowBox[List[RowBox[List["z_", "+", FractionBox[RowBox[List["8", " ", RowBox[List["EllipticK", "[", "m_", "]"]]]], "3"]]], ",", "m_"]], "]"]], " ", RowBox[List["JacobiCN", "[", RowBox[List["z_", ",", "m_"]], "]"]]]], "+", RowBox[List[RowBox[List["JacobiDN", "[", RowBox[List[RowBox[List["z_", "+", FractionBox[RowBox[List["8", " ", RowBox[List["EllipticK", "[", "m_", "]"]]]], "3"]]], ",", "m_"]], "]"]], " ", RowBox[List["JacobiCN", "[", RowBox[List["z_", ",", "m_"]], "]"]], " ", RowBox[List["JacobiCN", "[", RowBox[List[RowBox[List["z_", "+", FractionBox[RowBox[List["4", " ", RowBox[List["EllipticK", "[", "m_", "]"]]]], "3"]]], ",", "m_"]], "]"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["JacobiDN", "[", RowBox[List[FractionBox[RowBox[List["2", " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "3"], ",", "m"]], "]"]], "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["JacobiDN", "[", RowBox[List["z", ",", "m"]], "]"]], "+", RowBox[List["JacobiDN", "[", RowBox[List[RowBox[List["z", "+", FractionBox[RowBox[List["4", " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "3"]]], ",", "m"]], "]"]], "+", RowBox[List["JacobiDN", "[", RowBox[List[RowBox[List["z", "+", FractionBox[RowBox[List["8", " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "3"]]], ",", "m"]], "]"]]]], ")"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["JacobiDN", "[", RowBox[List[FractionBox[RowBox[List["2", " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "3"], ",", "m"]], "]"]]]], ")"]], "2"]]]]]]]]










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





2002-03-07