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 JacobiCD

 http://functions.wolfram.com/09.25.16.0015.01

 Input Form

 JacobiCD[z/M + EllipticK[m]/(n M), l] == (-(1/M)) ((JacobiDN[z, m] JacobiSN[z, m])/JacobiCN[z, m]) Product[(1 - JacobiNS[(2 r EllipticK[m])/n, m]^2 JacobiSN[z, m]^2)/ (1 - m JacobiSN[(2 r EllipticK[m])/n, m]^2 JacobiSN[z, m]^2), {r, 1, n/2}] /; Element[n/2, Integers] && n > 0 && l == m^n Product[JacobiSN[((2 r - 1) EllipticK[m])/n, m]^8, {r, 1, n/2}] && M == Product[JacobiSN[((2 r - 1) EllipticK[m])/n, m]^2/ JacobiSN[(2 r EllipticK[m])/n, m]^2, {r, 1, n/2}]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["JacobiCD", "[", RowBox[List[RowBox[List[FractionBox["z", "M"], "+", FractionBox[RowBox[List["EllipticK", "[", "m", "]"]], RowBox[List["n", " ", "M"]]]]], ",", "l"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["1", "M"]]], " ", FractionBox[RowBox[List[RowBox[List["JacobiDN", "[", RowBox[List["z", ",", "m"]], "]"]], " ", RowBox[List["JacobiSN", "[", RowBox[List["z", ",", "m"]], "]"]]]], RowBox[List["JacobiCN", "[", RowBox[List["z", ",", "m"]], "]"]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["r", "=", "1"]], FractionBox["n", "2"]], FractionBox[RowBox[List["1", "-", RowBox[List[SuperscriptBox[RowBox[List["JacobiNS", "[", RowBox[List[FractionBox[RowBox[List["2", "r", " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "n"], ",", "m"]], "]"]], "2"], " ", SuperscriptBox[RowBox[List["JacobiSN", "[", RowBox[List["z", ",", "m"]], "]"]], "2"]]]]], RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["JacobiSN", "[", RowBox[List[FractionBox[RowBox[List["2", "r", " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "n"], ",", "m"]], "]"]], "2"], " ", SuperscriptBox[RowBox[List["JacobiSN", "[", RowBox[List["z", ",", "m"]], "]"]], "2"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List[FractionBox["n", "2"], "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "0"]], "\[And]", RowBox[List["l", "\[Equal]", RowBox[List[SuperscriptBox["m", "n"], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["r", "=", "1"]], FractionBox["n", "2"]], SuperscriptBox[RowBox[List["JacobiSN", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", "r"]], "-", "1"]], ")"]], " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "n"], ",", "m"]], "]"]], "8"]]]]]]], "\[And]", RowBox[List["M", "\[Equal]", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["r", "=", "1"]], FractionBox["n", "2"]], FractionBox[SuperscriptBox[RowBox[List["JacobiSN", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", "r"]], "-", "1"]], ")"]], " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "n"], ",", "m"]], "]"]], "2"], SuperscriptBox[RowBox[List["JacobiSN", "[", RowBox[List[FractionBox[RowBox[List["2", "r", " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "n"], ",", "m"]], "]"]], "2"]]]]]]]]]]]]

 MathML Form

 cd ( z M + K ( m ) n M l ) - 1 M dn ( z m ) sn ( z m ) cn ( z m ) r = 1 n 2 1 - ns ( 2 r K ( m ) n m ) 2 sn ( z m ) 2 1 - m sn ( 2 r K ( m ) n m ) 2 sn ( z m ) 2 /; n 2 TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] + l m n r = 1 n 2 sn ( ( 2 r - 1 ) K ( m ) n m ) 8 M r = 1 n 2 sn ( ( 2 r - 1 ) K ( m ) n m ) 2 sn ( 2 r K ( m ) n m ) 2 Condition JacobiCD z M -1 EllipticK m n M -1 l -1 1 M -1 JacobiDN z m JacobiSN z m JacobiCN z m -1 r 1 n 2 -1 1 -1 JacobiNS 2 r EllipticK m n -1 m 2 JacobiSN z m 2 1 -1 m JacobiSN 2 r EllipticK m n -1 m 2 JacobiSN z m 2 -1 n 2 -1 SuperPlus l m n r 1 n 2 -1 JacobiSN 2 r -1 EllipticK m n -1 m 8 M r 1 n 2 -1 JacobiSN 2 r -1 EllipticK m n -1 m 2 JacobiSN 2 r EllipticK m n -1 m 2 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["JacobiCD", "[", RowBox[List[RowBox[List[FractionBox["z_", "M_"], "+", FractionBox[RowBox[List["EllipticK", "[", "m_", "]"]], RowBox[List["n_", " ", "M_"]]]]], ",", "l_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["JacobiDN", "[", RowBox[List["z", ",", "m"]], "]"]], " ", RowBox[List["JacobiSN", "[", RowBox[List["z", ",", "m"]], "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["r", "=", "1"]], FractionBox["n", "2"]], FractionBox[RowBox[List["1", "-", RowBox[List[SuperscriptBox[RowBox[List["JacobiNS", "[", RowBox[List[FractionBox[RowBox[List["2", " ", "r", " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "n"], ",", "m"]], "]"]], "2"], " ", SuperscriptBox[RowBox[List["JacobiSN", "[", RowBox[List["z", ",", "m"]], "]"]], "2"]]]]], RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["JacobiSN", "[", RowBox[List[FractionBox[RowBox[List["2", " ", "r", " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "n"], ",", "m"]], "]"]], "2"], " ", SuperscriptBox[RowBox[List["JacobiSN", "[", RowBox[List["z", ",", "m"]], "]"]], "2"]]]]]]]]]], RowBox[List["M", " ", RowBox[List["JacobiCN", "[", RowBox[List["z", ",", "m"]], "]"]]]]]]], "/;", RowBox[List[RowBox[List[FractionBox["n", "2"], "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]], "&&", RowBox[List["l", "\[Equal]", RowBox[List[SuperscriptBox["m", "n"], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["r", "=", "1"]], FractionBox["n", "2"]], SuperscriptBox[RowBox[List["JacobiSN", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "r"]], "-", "1"]], ")"]], " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "n"], ",", "m"]], "]"]], "8"]]]]]]], "&&", RowBox[List["M", "\[Equal]", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["r", "=", "1"]], FractionBox["n", "2"]], FractionBox[SuperscriptBox[RowBox[List["JacobiSN", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "r"]], "-", "1"]], ")"]], " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "n"], ",", "m"]], "]"]], "2"], SuperscriptBox[RowBox[List["JacobiSN", "[", RowBox[List[FractionBox[RowBox[List["2", " ", "r", " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "n"], ",", "m"]], "]"]], "2"]]]]]]]]]]]]]]

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

 2001-10-29