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/09.29.18.0036.01

 Input Form

 Sum[JacobiCN[z + (4 k EllipticK[m])/p, m] JacobiDN[z + (4 (k + n) EllipticK[m])/p, m] JacobiDN[z + (4 (k - n) EllipticK[m])/p, m], {k, 0, p - 1}]/ Sum[JacobiCN[z + (4 k EllipticK[m])/p, m], {k, 0, p - 1}] == Sum[JacobiCN[(4 k EllipticK[m])/p, m] JacobiDN[(4 (k + n) EllipticK[m])/p, m] JacobiDN[(4 (k - n) EllipticK[m])/p, m], {k, 0, p - 1}]/ Sum[JacobiCN[(4 k EllipticK[m])/p, m], {k, 0, p - 1}] /; Element[(p - 1)/2, Integers] && p >= 3 && Element[n, Integers] && Inequality[1, LessEqual, n, Less, p]

 Standard Form

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

 MathML Form

 k = 0 p - 1 cn ( z + 4 k K ( m ) p m ) dn ( z + 4 ( k + n ) K ( m ) p m ) dn ( z + 4 ( k - n ) K ( m ) p m ) k = 0 p - 1 cn ( z + 4 k K ( m ) p m ) k = 0 p - 1 cn ( 4 k K ( m ) p m ) dn ( 4 ( k + n ) K ( m ) p m ) dn ( 4 ( k - n ) K ( m ) p m ) k = 0 p - 1 cn ( 4 k K ( m ) p m ) /; p - 1 2 + n TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] 1 n < p Condition k 0 p -1 JacobiCN z 4 k EllipticK m p -1 m JacobiDN z 4 k n EllipticK m p -1 m JacobiDN z 4 k -1 n EllipticK m p -1 m k 0 p -1 JacobiCN z 4 k EllipticK m p -1 m -1 k 0 p -1 JacobiCN 4 k EllipticK m p -1 m JacobiDN 4 k n EllipticK m p -1 m JacobiDN 4 k -1 n EllipticK m p -1 m k 0 p -1 JacobiCN 4 k EllipticK m p -1 m -1 p -1 2 -1 SuperPlus n Inequality 1 n p [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["p_", "-", "1"]]], RowBox[List[RowBox[List["JacobiCN", "[", RowBox[List[RowBox[List["z_", "+", FractionBox[RowBox[List["4", " ", "k", " ", RowBox[List["EllipticK", "[", "m_", "]"]]]], "p_"]]], ",", "m_"]], "]"]], " ", RowBox[List["JacobiDN", "[", RowBox[List[RowBox[List["z_", "+", FractionBox[RowBox[List["4", " ", RowBox[List["(", RowBox[List["k", "+", "n_"]], ")"]], " ", RowBox[List["EllipticK", "[", "m_", "]"]]]], "p_"]]], ",", "m_"]], "]"]], " ", RowBox[List["JacobiDN", "[", RowBox[List[RowBox[List["z_", "+", FractionBox[RowBox[List["4", " ", RowBox[List["(", RowBox[List["k", "-", "n_"]], ")"]], " ", RowBox[List["EllipticK", "[", "m_", "]"]]]], "p_"]]], ",", "m_"]], "]"]]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["p_", "-", "1"]]], RowBox[List["JacobiCN", "[", RowBox[List[RowBox[List["z_", "+", FractionBox[RowBox[List["4", " ", "k", " ", RowBox[List["EllipticK", "[", "m_", "]"]]]], "p_"]]], ",", "m_"]], "]"]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["p", "-", "1"]]], RowBox[List[RowBox[List["JacobiCN", "[", RowBox[List[FractionBox[RowBox[List["4", " ", "k", " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "p"], ",", "m"]], "]"]], " ", RowBox[List["JacobiDN", "[", RowBox[List[FractionBox[RowBox[List["4", " ", RowBox[List["(", RowBox[List["k", "+", "n"]], ")"]], " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "p"], ",", "m"]], "]"]], " ", RowBox[List["JacobiDN", "[", RowBox[List[FractionBox[RowBox[List["4", " ", RowBox[List["(", RowBox[List["k", "-", "n"]], ")"]], " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "p"], ",", "m"]], "]"]]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["p", "-", "1"]]], RowBox[List["JacobiCN", "[", RowBox[List[FractionBox[RowBox[List["4", " ", "k", " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "p"], ",", "m"]], "]"]]]]], "/;", RowBox[List[RowBox[List[FractionBox[RowBox[List["p", "-", "1"]], "2"], "\[Element]", "Integers"]], "&&", RowBox[List["p", "\[GreaterEqual]", "3"]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["1", "\[LessEqual]", "n", "<", "p"]]]]]]]]]]

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

 2002-03-07

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