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JacobiDN






Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/09.29.16.0167.01









  


  










Input Form





Sum[Product[JacobiSN[z + 2 (j + k) (EllipticK[m]/p), m], {k, 0, r - 1}], {j, 0, p - 1}] == (p/(2 EllipticK[m])) Integrate[Product[JacobiSN[t + 2 k (EllipticK[m]/p), m], {k, 0, r - 1}], {t, 0, 2 EllipticK[m]}] /; Element[p, Integers] && p >= 3 && Element[r/2, Integers] && Inequality[2, LessEqual, r, Less, p - 1]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["p", "-", "1"]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "0"]], RowBox[List["r", "-", "1"]]], RowBox[List["JacobiSN", "[", RowBox[List[RowBox[List["z", "+", RowBox[List["2", RowBox[List["(", RowBox[List["j", "+", "k"]], ")"]], RowBox[List[RowBox[List["EllipticK", "[", "m", "]"]], "/", "p"]]]]]], ",", "m"]], "]"]]]]]], "\[Equal]", RowBox[List[FractionBox["p", RowBox[List["2", RowBox[List["EllipticK", "[", "m", "]"]]]]], RowBox[List[SubsuperscriptBox["\[Integral]", "0", RowBox[List["2", RowBox[List["EllipticK", "[", "m", "]"]]]]], RowBox[List[RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "0"]], RowBox[List["r", "-", "1"]]], RowBox[List["JacobiSN", "[", RowBox[List[RowBox[List["t", "+", RowBox[List["2", "k", " ", RowBox[List[RowBox[List["EllipticK", "[", "m", "]"]], "/", "p"]]]]]], ",", "m"]], "]"]]]], ")"]], " ", RowBox[List["\[DifferentialD]", "t"]]]]]]]]]], "/;", RowBox[List[RowBox[List["p", "\[Element]", "Integers"]], "\[And]", RowBox[List["p", "\[GreaterEqual]", "3"]], "\[And]", RowBox[List[RowBox[List["r", "/", "2"]], "\[Element]", "Integers"]], "\[And]", RowBox[List["2", "\[LessEqual]", "r", "<", RowBox[List["p", "-", "1"]]]]]]]]]]










MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["p_", "-", "1"]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "0"]], RowBox[List["r_", "-", "1"]]], RowBox[List["JacobiSN", "[", RowBox[List[RowBox[List["z_", "+", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List["j", "+", "k"]], ")"]], " ", RowBox[List["EllipticK", "[", "m_", "]"]]]], "p_"]]], ",", "m_"]], "]"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List["p", " ", RowBox[List[SubsuperscriptBox["\[Integral]", "0", RowBox[List["2", " ", RowBox[List["EllipticK", "[", "m", "]"]]]]], RowBox[List[RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "0"]], RowBox[List["r", "-", "1"]]], RowBox[List["JacobiSN", "[", RowBox[List[RowBox[List["t", "+", FractionBox[RowBox[List["2", " ", "k", " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "p"]]], ",", "m"]], "]"]]]], RowBox[List["\[DifferentialD]", "t"]]]]]]]], RowBox[List["2", " ", RowBox[List["EllipticK", "[", "m", "]"]]]]], "/;", RowBox[List[RowBox[List["p", "\[Element]", "Integers"]], "&&", RowBox[List["p", "\[GreaterEqual]", "3"]], "&&", RowBox[List[FractionBox["r", "2"], "\[Element]", "Integers"]], "&&", RowBox[List["2", "\[LessEqual]", "r", "<", RowBox[List["p", "-", "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