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http://functions.wolfram.com/09.36.16.0060.01
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JacobiSN[((2 n)/Pi) EllipticK[ModularLambda[(n/(Pi I))
Log[EllipticNomeQ[m]]]] z, ModularLambda[
(n/(Pi I)) Log[EllipticNomeQ[m]]]] ==
((EllipticNomeQ[m]^n)^(1/4)/EllipticNomeQ[m]^(n/4))
((m^(1/4))^n/ModularLambda[(n/(Pi I)) Log[EllipticNomeQ[m]]]^(1/4))
Product[JacobiSN[((2 EllipticK[m])/Pi) (z + (r Pi)/n), m],
{r, 0, n - 1}] /; Element[n, Integers] && n > 0
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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> sn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> <mi> π </mi> </mfrac> <mo> ⁢ </mo> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mi> n </mi> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> q </mi> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox[RowBox[List[FractionBox["n", RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]]], " ", RowBox[List["log", "(", RowBox[List[InterpretationBox["q", EllipticNomeQ, Rule[Editable, False], Rule[Selectable, False]], "(", "m", ")"]], ")"]]]], Rule[Editable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ❘ </mo> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mi> n </mi> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> q </mi> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox[RowBox[List[FractionBox["n", RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]]], " ", RowBox[List["log", "(", RowBox[List[InterpretationBox["q", EllipticNomeQ, Rule[Editable, False], Rule[Selectable, False]], "(", "m", ")"]], ")"]]]], Rule[Editable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mroot> <msup> <mrow> <semantics> <mi> q </mi> <annotation-xml encoding='MathML-Content'> <ci> EllipticNomeQ </ci> </annotation-xml> </semantics> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <mi> n </mi> </msup> <mn> 4 </mn> </mroot> <msup> <mrow> <semantics> <mi> q </mi> <annotation-xml encoding='MathML-Content'> <ci> EllipticNomeQ </ci> </annotation-xml> </semantics> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <mrow> <mi> n </mi> <mo> / </mo> <mn> 4 </mn> </mrow> </msup> </mfrac> <mo> ⁢ </mo> <mfrac> <msup> <mrow> <mo> ( </mo> <mroot> <mi> m </mi> <mn> 4 </mn> </mroot> <mo> ) </mo> </mrow> <mi> n </mi> </msup> <mroot> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mi> n </mi> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> q </mi> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox[RowBox[List[FractionBox["n", RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]]], " ", RowBox[List["log", "(", RowBox[List[InterpretationBox["q", EllipticNomeQ, Rule[Editable, False], Rule[Selectable, False]], "(", "m", ")"]], ")"]]]], Rule[Editable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mn> 4 </mn> </mroot> </mfrac> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> r </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mi> sn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> <mi> π </mi> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mfrac> <mrow> <mi> r </mi> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mi> n </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; 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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["JacobiSN", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", " ", "n_"]], ")"]], " ", RowBox[List["EllipticK", "[", RowBox[List["ModularLambda", "[", FractionBox[RowBox[List["n_", " ", RowBox[List["Log", "[", RowBox[List["EllipticNomeQ", "[", "m_", "]"]], "]"]]]], RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]]], "]"]], "]"]], " ", "z_"]], "\[Pi]"], ",", RowBox[List["ModularLambda", "[", FractionBox[RowBox[List["n_", " ", RowBox[List["Log", "[", RowBox[List["EllipticNomeQ", "[", "m_", "]"]], "]"]]]], RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]]], "]"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["EllipticNomeQ", "[", "m", "]"]], "n"], ")"]], RowBox[List["1", "/", "4"]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["m", RowBox[List["1", "/", "4"]]], ")"]], "n"], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["r", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List["JacobiSN", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", " ", RowBox[List["EllipticK", "[", "m", "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["z", "+", FractionBox[RowBox[List["r", " ", "\[Pi]"]], "n"]]], ")"]]]], "\[Pi]"], ",", "m"]], "]"]]]]]], RowBox[List[SuperscriptBox[RowBox[List["EllipticNomeQ", "[", "m", "]"]], RowBox[List["n", "/", "4"]]], " ", SuperscriptBox[RowBox[List["ModularLambda", "[", FractionBox[RowBox[List["n", " ", RowBox[List["Log", "[", RowBox[List["EllipticNomeQ", "[", "m", "]"]], "]"]]]], RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]]], "]"]], RowBox[List["1", "/", "4"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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