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JacobiCN






Mathematica Notation

Traditional Notation









Elliptic Functions > JacobiCN[z,m] > Transformations > Sums over products of two Jacobi functions





http://functions.wolfram.com/09.26.16.0054.01









  


  










Input Form





Sum[JacobiSN[z + 2 k (EllipticK[m]/p), m] JacobiSN[z + 2 (k + r) (EllipticK[m]/p), m], {k, 0, p - 1}] == p JacobiCN[2 r (EllipticK[m]/p), m] (JacobiZeta[ArcSin[JacobiSN[2 r (EllipticK[m]/p), m]], m]/ (m Abs[JacobiSN[2 r (EllipticK[m]/p), m]] Abs[JacobiCN[2 r (EllipticK[m]/p), m]])) /; Element[p, Integers] && p >= 2 && Element[r, Integers] && Inequality[1, LessEqual, r, Less, p - 1] && Element[m, Reals] && m < 1










Standard Form





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MathML Form







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</ci> </apply> <apply> <in /> <ci> r </ci> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> <apply> <lt /> <ci> r </ci> <apply> <plus /> <ci> p </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <in /> <ci> m </ci> <reals /> </apply> <apply> <lt /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["p_", "-", "1"]]], RowBox[List[RowBox[List["JacobiSN", "[", RowBox[List[RowBox[List["z_", "+", FractionBox[RowBox[List["2", " ", "k", " ", RowBox[List["EllipticK", "[", "m_", "]"]]]], "p_"]]], ",", "m_"]], "]"]], " ", RowBox[List["JacobiSN", "[", RowBox[List[RowBox[List["z_", "+", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List["k", "+", "r_"]], ")"]], " ", RowBox[List["EllipticK", "[", "m_", "]"]]]], "p_"]]], ",", "m_"]], "]"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List["p", " ", RowBox[List["JacobiCN", "[", RowBox[List[FractionBox[RowBox[List["2", " ", "r", " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "p"], ",", "m"]], "]"]], " ", RowBox[List["JacobiZeta", "[", RowBox[List[RowBox[List["ArcSin", "[", RowBox[List["JacobiSN", "[", RowBox[List[FractionBox[RowBox[List["2", " ", "r", " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "p"], ",", "m"]], "]"]], "]"]], ",", "m"]], "]"]]]], RowBox[List["m", " ", RowBox[List["Abs", "[", RowBox[List["JacobiSN", "[", RowBox[List[FractionBox[RowBox[List["2", " ", "r", " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "p"], ",", "m"]], "]"]], "]"]], " ", RowBox[List["Abs", "[", RowBox[List["JacobiCN", "[", RowBox[List[FractionBox[RowBox[List["2", " ", "r", " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "p"], ",", "m"]], "]"]], "]"]]]]], "/;", RowBox[List[RowBox[List["p", "\[Element]", "Integers"]], "&&", RowBox[List["p", "\[GreaterEqual]", "2"]], "&&", RowBox[List["r", "\[Element]", "Integers"]], "&&", RowBox[List["1", "\[LessEqual]", "r", "<", RowBox[List["p", "-", "1"]]]], "&&", RowBox[List["m", "\[Element]", "Reals"]], "&&", RowBox[List["m", "<", "1"]]]]]]]]]]










References





A. Khare, A. Lakshminarayan, U. Sukhatme, "Cyclic Identities Involving Jacobi Elliptic Functions. II", math-ph/0207019, (2002) http://arXiv.org/abs/math-ph/0207019

A. Khare, A. Lakshminarayan, U. Sukhatme, "Cyclic Identities Involving Jacobi Elliptic Functions", Journal of Mathematical Physics, v. 44, issue 4, pp. 1822-1841 (2003)










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





2002-12-18





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