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http://functions.wolfram.com/09.30.16.0014.01
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JacobiDS[z/M, l] == M JacobiDS[z, m]
Product[(1 - m JacobiSN[(2 r - 1) (EllipticK[m]/n), m]^2 JacobiSN[z, m]^2)/
(1 - JacobiNS[2 r (EllipticK[m]/n), m]^2 JacobiSN[z, m]^2),
{r, 1, (n - 1)/2}] /; Element[(n + 1)/2, Integers] && n > 0 &&
l == m^n Product[JacobiSN[((2 r - 1) EllipticK[m])/n, m]^8,
{r, 1, (n - 1)/2}] &&
M == Product[JacobiSN[((2 r - 1) EllipticK[m])/n, m]^2/
JacobiSN[(2 r EllipticK[m])/n, m]^2, {r, 1, (n - 1)/2}]
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["JacobiDS", "[", RowBox[List[FractionBox["z", "M"], ",", "l"]], "]"]], "\[Equal]", RowBox[List["M", " ", RowBox[List["JacobiDS", "[", RowBox[List["z", ",", "m"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["r", "=", "1"]], FractionBox[RowBox[List["n", "-", "1"]], "2"]], RowBox[List[RowBox[List["(", RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["JacobiSN", "[", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", "r"]], "-", "1"]], ")"]], " ", RowBox[List[RowBox[List["EllipticK", "[", "m", "]"]], "/", "n"]]]], ",", "m"]], "]"]], "2"], " ", SuperscriptBox[RowBox[List["JacobiSN", "[", RowBox[List["z", ",", "m"]], "]"]], "2"]]]]], ")"]], "/", RowBox[List["(", RowBox[List["1", "-", RowBox[List[SuperscriptBox[RowBox[List["JacobiNS", "[", RowBox[List[RowBox[List["2", "r", " ", RowBox[List[RowBox[List["EllipticK", "[", "m", "]"]], "/", "n"]]]], ",", "m"]], "]"]], "2"], " ", SuperscriptBox[RowBox[List["JacobiSN", "[", RowBox[List["z", ",", "m"]], "]"]], "2"]]]]], ")"]]]]]]]]]], "/;", RowBox[List[RowBox[List[FractionBox[RowBox[List["n", "+", "1"]], "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[RowBox[List["n", "-", "1"]], "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[RowBox[List["n", "-", "1"]], "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"]]]]]]]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mi> ds </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mi> z </mi> <mi> M </mi> </mfrac> <mo> ❘ </mo> <mi> l </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mi> M </mi> <mo> ⁢ </mo> <mrow> <mi> ds </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> r </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mfrac> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> </munderover> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> m </mi> <mo> ⁢ </mo> <msup> <mrow> <mi> sn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> r </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> <mi> n </mi> </mfrac> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msup> <mrow> <mi> sn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> </mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <msup> <mrow> <mi> ns </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> r </mi> <mo> ⁢ </mo> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> <mi> n </mi> </mfrac> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msup> <mrow> <mi> sn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> </mrow> </mfrac> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mfrac> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> ∈ </mo> <msup> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> <mo> + </mo> </msup> </mrow> <mo> ∧ </mo> <mrow> <mi> l </mi> <mo> ⩵ </mo> <mrow> <msup> <mi> m </mi> <mi> n </mi> </msup> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> r </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mfrac> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> </munderover> <msup> <mrow> <mi> sn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> r </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> <mi> n </mi> </mfrac> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mn> 8 </mn> </msup> </mrow> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mi> M </mi> <mo> ⩵ </mo> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> r </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mfrac> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> </munderover> <mfrac> <msup> <mrow> <mi> sn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> r </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> <mi> n </mi> </mfrac> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <msup> <mrow> <mi> sn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> r </mi> <mo> ⁢ </mo> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> <mi> n </mi> </mfrac> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mfrac> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> JacobiDS </ci> <apply> <times /> <ci> z </ci> <apply> <power /> <ci> M </ci> <cn type='integer'> -1 </cn> </apply> </apply> <ci> l </ci> </apply> <apply> <times /> <ci> M </ci> <apply> <ci> JacobiDS </ci> <ci> z </ci> <ci> m </ci> </apply> <apply> <product /> <bvar> <ci> r </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <apply> <times /> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </uplimit> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> m </ci> <apply> <power /> <apply> <ci> JacobiSN </ci> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> r </ci> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> <apply> <power /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <ci> m </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ci> JacobiSN </ci> <ci> z </ci> <ci> m </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <ci> JacobiNS </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> r </ci> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> <apply> <power /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <ci> m </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ci> JacobiSN </ci> <ci> z </ci> <ci> m </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <apply> <times /> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> SuperPlus </ci> <integers /> </apply> </apply> <apply> <eq /> <ci> l </ci> <apply> <times /> <apply> <power /> <ci> m </ci> <ci> n </ci> </apply> <apply> <product /> <bvar> <ci> r </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <apply> <times /> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </uplimit> <apply> <power /> <apply> <ci> JacobiSN </ci> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> r </ci> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> <apply> <power /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <ci> m </ci> </apply> <cn type='integer'> 8 </cn> </apply> </apply> </apply> </apply> <apply> <eq /> <ci> M </ci> <apply> <product /> <bvar> <ci> r </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <apply> <times /> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </uplimit> <apply> <times /> <apply> <power /> <apply> <ci> JacobiSN </ci> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> r </ci> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> <apply> <power /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <ci> m </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <ci> JacobiSN </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> r </ci> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> <apply> <power /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <ci> m </ci> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["JacobiDS", "[", RowBox[List[FractionBox["z_", "M_"], ",", "l_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["M", " ", RowBox[List["JacobiDS", "[", RowBox[List["z", ",", "m"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["r", "=", "1"]], FractionBox[RowBox[List["n", "-", "1"]], "2"]], FractionBox[RowBox[List["1", "-", RowBox[List["m", " ", 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["z", ",", "m"]], "]"]], "2"]]]]], 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[RowBox[List[FractionBox[RowBox[List["n", "+", "1"]], "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[RowBox[List["n", "-", "1"]], "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[RowBox[List["n", "-", "1"]], "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"]]]]]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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