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JacobiDN




Mathematica Notation

Traditional Notation









Elliptic Functions > JacobiDN[z,m] > Transformations > Sums over products of three Jacobi functions





http://functions.wolfram.com/09.29.16.0055.01









  


  








Input Form





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








Standard Form





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MathML Form







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Rule Form





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










References





A. Khare, U. Sukhatme, "Cyclic Identities Involving Jacobi Elliptic Functions", math-ph/0201004, (2002) http://arXiv.org/abs/math-ph/0201004

A. Khare, U. Sukhatme, "Cyclic Identities Involving Jacobi Elliptic Functions", Journal of Mathematical Physics, v. 43, issue 7, pp. 3798-3806 (2002)


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





2002-03-07