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JacobiNS






Mathematica Notation

Traditional Notation









Elliptic Functions > JacobiNS[z,m] > Transformations > Multiple arguments > Multiple angle formulas





http://functions.wolfram.com/09.33.16.0029.01









  


  










Input Form





JacobiNS[((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)) (ModularLambda[(n/(Pi I)) Log[EllipticNomeQ[m]]]^(1/4)/(m^(1/4))^n) Product[JacobiNS[((2 EllipticK[m])/Pi) (z + (r Pi)/n), m], {r, 0, n - 1}] /; Element[n, Integers] && n > 0










Standard Form





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MathML Form







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</mo> <mi> z </mi> </mrow> <mo> &#10072; </mo> <semantics> <mrow> <mi> &#955; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mi> n </mi> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </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[&quot;\[Lambda]&quot;, &quot;(&quot;, TagBox[RowBox[List[FractionBox[&quot;n&quot;, RowBox[List[&quot;\[Pi]&quot;, &quot; &quot;, &quot;\[ImaginaryI]&quot;]]], &quot; &quot;, RowBox[List[&quot;log&quot;, &quot;(&quot;, RowBox[List[InterpretationBox[&quot;q&quot;, EllipticNomeQ, Rule[Editable, False], Rule[Selectable, False]], &quot;(&quot;, &quot;m&quot;, &quot;)&quot;]], &quot;)&quot;]]]], Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; 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</mo> <mi> &#960; </mi> </mrow> <mi> n </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <msup> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> <mo> + </mo> </msup> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> JacobiNS </ci> <apply> <times /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> EllipticK </ci> <apply> <ci> ModularLambda </ci> <apply> <times /> <apply> <times /> <ci> n </ci> <apply> <power /> <apply> <times /> <pi /> <imaginaryi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ln /> <apply> <ci> EllipticNomeQ </ci> <ci> m </ci> </apply> </apply> </apply> </apply> </apply> <ci> z </ci> </apply> <apply> <ci> ModularLambda </ci> <apply> <times /> <apply> <times /> <ci> n </ci> <apply> <power /> <apply> <times /> <pi /> <imaginaryi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ln /> <apply> <ci> EllipticNomeQ </ci> <ci> m </ci> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <power /> <apply> <ci> EllipticNomeQ </ci> <ci> m </ci> </apply> <ci> n </ci> </apply> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <ci> EllipticNomeQ </ci> <ci> m </ci> </apply> <apply> <times /> <ci> n </ci> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <ci> ModularLambda </ci> <apply> <times /> <apply> <times /> <ci> n </ci> <apply> <power /> <apply> <times /> <pi /> <imaginaryi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ln /> <apply> <ci> EllipticNomeQ </ci> <ci> m </ci> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <power /> <ci> m </ci> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <product /> <bvar> <ci> r </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <ci> JacobiNS </ci> <apply> <times /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <ci> z </ci> <apply> <times /> <ci> r </ci> <pi /> <apply> <power /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <ci> m </ci> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <apply> <ci> SuperPlus </ci> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["JacobiNS", "[", 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["ModularLambda", "[", FractionBox[RowBox[List["n", " ", RowBox[List["Log", "[", RowBox[List["EllipticNomeQ", "[", "m", "]"]], "]"]]]], RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]]], "]"]], RowBox[List["1", "/", "4"]]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["r", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List["JacobiNS", "[", 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["(", SuperscriptBox["m", RowBox[List["1", "/", "4"]]], ")"]], "n"]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]]










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





2001-10-29