Jacobi elliptic function dn: Transformations (formula 09.29.16.0040)

JacobiDN

$Mathematica Notation$

Multiple angle formulas

http://functions.wolfram.com/09.29.16.0040.01

Input Form

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

Standard Form

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MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mi> dn </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> <mfrac> <mrow> <mi> n </mi> <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> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> </mrow> </mfrac> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox[FractionBox[RowBox[List["n", " ", RowBox[List["log", "(", RowBox[List[InterpretationBox["q", EllipticNomeQ, Rule[Editable, False], Rule[Selectable, False]], "(", "m", ")"]], ")"]]]], RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]]], Rule[Editable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> x </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> <mrow> <mn> 1 </mn> <mo> - </mo> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mi> n </mi> <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> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> </mrow> </mfrac> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox[FractionBox[RowBox[List["n", " ", RowBox[List["log", "(", RowBox[List[InterpretationBox["q", EllipticNomeQ, Rule[Editable, False], Rule[Selectable, False]], "(", "m", ")"]], ")"]]]], RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]]], Rule[Editable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> </mrow> <mn> 4 </mn> </mroot> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> n </mi> <mo> / </mo> <mn> 4 </mn> </mrow> </msup> </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> dn </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> <mfrac> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> r </mi> </mrow> <mi> n </mi> </mfrac> <mo> + </mo> <mi> x </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <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> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> JacobiDN </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 /> <ci> n </ci> <apply> <ln /> <apply> <ci> EllipticNomeQ </ci> <ci> m </ci> </apply> </apply> <apply> <power /> <apply> <times /> <pi /> <imaginaryi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <ci> x </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> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> ModularLambda </ci> <apply> <times /> <ci> n </ci> <apply> <ln /> <apply> <ci> EllipticNomeQ </ci> <ci> m </ci> </apply> </apply> <apply> <power /> <apply> <times /> <pi /> <imaginaryi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </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> <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> JacobiDN </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 /> <apply> <times /> <pi /> <ci> r </ci> <apply> <power /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <ci> x </ci> </apply> </apply> <ci> m </ci> </apply> </apply> </apply> </apply> <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> </annotation-xml> </semantics> </math>

Rule Form

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

2001-10-29