html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 JacobiDN

 http://functions.wolfram.com/09.29.16.0147.01

 Input Form

 Sum[JacobiSN[z + 4 k (EllipticK[m]/p), m]^4 (JacobiSN[z + 4 (k + r) (EllipticK[m]/p), m] + JacobiSN[z + 4 (k - r) (EllipticK[m]/p), m]), {k, 0, p - 1}] == (-(2/m)) JacobiCS[4 r (EllipticK[m]/p), m] JacobiDS[4 r (EllipticK[m]/p), m] Sum[JacobiSN[z + 4 k (EllipticK[m]/p), m]^3, {k, 0, p - 1}] + (2/m^2) JacobiNS[4 r (EllipticK[m]/p), m]^2 (JacobiNS[4 r (EllipticK[m]/p), m]^2 - JacobiCS[4 r (EllipticK[m]/p), m] JacobiDS[4 r (EllipticK[m]/p), m]) Sum[JacobiSN[z + 4 k (EllipticK[m]/p), m], {k, 0, p - 1}] /; Element[p, Integers] && p >= 1 && Element[r, Integers] && Inequality[1, LessEqual, r, Less, p]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["p", "-", "1"]]], RowBox[List[SuperscriptBox[RowBox[List["JacobiSN", "[", RowBox[List[RowBox[List["z", "+", RowBox[List["4", "k", " ", RowBox[List[RowBox[List["EllipticK", "[", "m", "]"]], "/", "p"]]]]]], ",", "m"]], "]"]], "4"], RowBox[List["(", RowBox[List[RowBox[List["JacobiSN", "[", RowBox[List[RowBox[List["z", "+", RowBox[List["4", RowBox[List["(", RowBox[List["k", "+", "r"]], ")"]], " ", RowBox[List[RowBox[List["EllipticK", "[", "m", "]"]], "/", "p"]]]]]], ",", "m"]], "]"]], "+", RowBox[List["JacobiSN", "[", RowBox[List[RowBox[List["z", "+", RowBox[List["4", RowBox[List["(", RowBox[List["k", "-", "r"]], ")"]], RowBox[List[RowBox[List["EllipticK", "[", "m", "]"]], "/", "p"]]]]]], ",", "m"]], "]"]]]], ")"]]]]]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["2", "m"]]], RowBox[List["JacobiCS", "[", RowBox[List[RowBox[List["4", "r", " ", RowBox[List[RowBox[List["EllipticK", "[", "m", "]"]], "/", "p"]]]], ",", "m"]], "]"]], RowBox[List["JacobiDS", "[", RowBox[List[RowBox[List["4", "r", " ", RowBox[List[RowBox[List["EllipticK", "[", "m", "]"]], "/", "p"]]]], ",", "m"]], "]"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["p", "-", "1"]]], SuperscriptBox[RowBox[List["JacobiSN", "[", RowBox[List[RowBox[List["z", "+", RowBox[List["4", "k", " ", RowBox[List[RowBox[List["EllipticK", "[", "m", "]"]], "/", "p"]]]]]], ",", "m"]], "]"]], "3"]]]]], "+", RowBox[List[FractionBox["2", SuperscriptBox["m", "2"]], SuperscriptBox[RowBox[List["JacobiNS", "[", RowBox[List[RowBox[List["4", "r", " ", RowBox[List[RowBox[List["EllipticK", "[", "m", "]"]], "/", "p"]]]], ",", "m"]], "]"]], "2"], RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["JacobiNS", "[", RowBox[List[RowBox[List["4", "r", " ", RowBox[List[RowBox[List["EllipticK", "[", "m", "]"]], "/", "p"]]]], ",", "m"]], "]"]], "2"], "-", RowBox[List[RowBox[List["JacobiCS", "[", RowBox[List[RowBox[List["4", "r", " ", RowBox[List[RowBox[List["EllipticK", "[", "m", "]"]], "/", "p"]]]], ",", "m"]], "]"]], RowBox[List["JacobiDS", "[", RowBox[List[RowBox[List["4", "r", " ", RowBox[List[RowBox[List["EllipticK", "[", "m", "]"]], "/", "p"]]]], ",", "m"]], "]"]]]]]], ")"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["p", "-", "1"]]], RowBox[List["JacobiSN", "[", RowBox[List[RowBox[List["z", "+", RowBox[List["4", "k", " ", RowBox[List[RowBox[List["EllipticK", "[", "m", "]"]], "/", "p"]]]]]], ",", "m"]], "]"]]]]]]]]]], "/;", RowBox[List[RowBox[List["p", "\[Element]", "Integers"]], "\[And]", RowBox[List["p", "\[GreaterEqual]", "1"]], "\[And]", RowBox[List["r", "\[Element]", "Integers"]], "\[And]", RowBox[List["1", "\[LessEqual]", "r", "<", "p"]]]]]]]]

 MathML Form

 k = 0 p - 1 sn ( z + 4 k K ( m ) p m ) 4 ( sn ( z + 4 ( k - r ) K ( m ) p m ) + sn ( z + 4 ( k + r ) K ( m ) p m ) ) - 2 m cs ( 4 r K ( m ) p m ) ds ( 4 r K ( m ) p m ) k = 0 p - 1 sn ( z + 4 k K ( m ) p m ) 3 + 2 m 2 ns ( 4 r K ( m ) p m ) 2 ( ns ( 4 r K ( m ) p m ) 2 - cs ( 4 r K ( m ) p m ) ds ( 4 r K ( m ) p m ) ) k = 0 p - 1 sn ( z + 4 k K ( m ) p m ) /; p + r + r < p Condition k 0 p -1 JacobiSN z 4 k EllipticK m p -1 m 4 JacobiSN z 4 k -1 r EllipticK m p -1 m JacobiSN z 4 k r EllipticK m p -1 m -1 2 m -1 JacobiCS 4 r EllipticK m p -1 m JacobiDS 4 r EllipticK m p -1 m k 0 p -1 JacobiSN z 4 k EllipticK m p -1 m 3 2 m 2 -1 JacobiNS 4 r EllipticK m p -1 m 2 JacobiNS 4 r EllipticK m p -1 m 2 -1 JacobiCS 4 r EllipticK m p -1 m JacobiDS 4 r EllipticK m p -1 m k 0 p -1 JacobiSN z 4 k EllipticK m p -1 m p SuperPlus r SuperPlus r p [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["p_", "-", "1"]]], RowBox[List[SuperscriptBox[RowBox[List["JacobiSN", "[", RowBox[List[RowBox[List["z_", "+", FractionBox[RowBox[List["4", " ", "k", " ", RowBox[List["EllipticK", "[", "m_", "]"]]]], "p_"]]], ",", "m_"]], "]"]], "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["JacobiSN", "[", RowBox[List[RowBox[List["z_", "+", FractionBox[RowBox[List["4", " ", RowBox[List["(", RowBox[List["k", "+", "r_"]], ")"]], " ", RowBox[List["EllipticK", "[", "m_", "]"]]]], "p_"]]], ",", "m_"]], "]"]], "+", RowBox[List["JacobiSN", "[", RowBox[List[RowBox[List["z_", "+", FractionBox[RowBox[List["4", " ", RowBox[List["(", RowBox[List["k", "-", "r_"]], ")"]], " ", RowBox[List["EllipticK", "[", "m_", "]"]]]], "p_"]]], ",", "m_"]], "]"]]]], ")"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[RowBox[List["2", " ", RowBox[List["JacobiCS", "[", RowBox[List[FractionBox[RowBox[List["4", " ", "r", " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "p"], ",", "m"]], "]"]], " ", RowBox[List["JacobiDS", "[", RowBox[List[FractionBox[RowBox[List["4", " ", "r", " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "p"], ",", "m"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["p", "-", "1"]]], SuperscriptBox[RowBox[List["JacobiSN", "[", RowBox[List[RowBox[List["z", "+", FractionBox[RowBox[List["4", " ", "k", " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "p"]]], ",", "m"]], "]"]], "3"]]]]], "m"]]], "+", FractionBox[RowBox[List["2", " ", SuperscriptBox[RowBox[List["JacobiNS", "[", RowBox[List[FractionBox[RowBox[List["4", " ", "r", " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "p"], ",", "m"]], "]"]], "2"], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["JacobiNS", "[", RowBox[List[FractionBox[RowBox[List["4", " ", "r", " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "p"], ",", "m"]], "]"]], "2"], "-", RowBox[List[RowBox[List["JacobiCS", "[", RowBox[List[FractionBox[RowBox[List["4", " ", "r", " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "p"], ",", "m"]], "]"]], " ", RowBox[List["JacobiDS", "[", RowBox[List[FractionBox[RowBox[List["4", " ", "r", " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "p"], ",", "m"]], "]"]]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["p", "-", "1"]]], RowBox[List["JacobiSN", "[", RowBox[List[RowBox[List["z", "+", FractionBox[RowBox[List["4", " ", "k", " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "p"]]], ",", "m"]], "]"]]]]]], SuperscriptBox["m", "2"]]]], "/;", RowBox[List[RowBox[List["p", "\[Element]", "Integers"]], "&&", RowBox[List["p", "\[GreaterEqual]", "1"]], "&&", RowBox[List["r", "\[Element]", "Integers"]], "&&", RowBox[List["1", "\[LessEqual]", "r", "<", "p"]]]]]]]]]]

 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