Jacobi elliptic function dn: Identities involving the group of functions (formula 09.29.18.0023)

JacobiDN

$Mathematica Notation$

Elliptic Functions

JacobiDN[z,m]

Identities involving the group of functions

Cyclic Identities of rank 3

p=3

http://functions.wolfram.com/09.29.18.0023.01

Input Form

JacobiCN[z, m] (JacobiCN[z + (4 EllipticK[m])/3, m] JacobiSN[z + (4 EllipticK[m])/3, m] + JacobiCN[z + (8 EllipticK[m])/3, m] JacobiSN[z + (8 EllipticK[m])/3, m]) + JacobiCN[z + (4 EllipticK[m])/3, m] (JacobiCN[z + (8 EllipticK[m])/3, m] JacobiSN[z + (8 EllipticK[m])/3, m] + JacobiCN[z, m] JacobiSN[z, m]) + JacobiCN[z + (8 EllipticK[m])/3, m] (JacobiCN[z, m] JacobiSN[z, m] + JacobiCN[z + (4 EllipticK[m])/3, m] JacobiSN[z + (4 EllipticK[m])/3, m]) == -2 JacobiDN[(2 EllipticK[m])/3, m] ((JacobiDN[(2 EllipticK[m])/3, m] + 2)/ ((1 + JacobiDN[(2 EllipticK[m])/3, m]) (1 - JacobiDN[(2 EllipticK[m])/3, m]^2))) (JacobiSN[z, m] + JacobiSN[z + (4 EllipticK[m])/3, m] + JacobiSN[z + (8 EllipticK[m])/3, m])

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List["JacobiCN", "[", RowBox[List["z", ",", "m"]], "]"]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["JacobiCN", "[", RowBox[List[RowBox[List["z", "+", FractionBox[RowBox[List["4", RowBox[List["EllipticK", "[", "m", "]"]]]], "3"]]], ",", "m"]], "]"]], RowBox[List["JacobiSN", "[", RowBox[List[RowBox[List["z", "+", FractionBox[RowBox[List["4", RowBox[List["EllipticK", "[", "m", "]"]]]], "3"]]], ",", "m"]], "]"]]]], "+", RowBox[List[RowBox[List["JacobiCN", "[", RowBox[List[RowBox[List["z", "+", FractionBox[RowBox[List["8", RowBox[List["EllipticK", "[", "m", "]"]]]], "3"]]], ",", "m"]], "]"]], RowBox[List["JacobiSN", "[", RowBox[List[RowBox[List["z", "+", FractionBox[RowBox[List["8", RowBox[List["EllipticK", "[", "m", "]"]]]], "3"]]], ",", "m"]], "]"]]]]]], ")"]]]], "+", RowBox[List[RowBox[List["JacobiCN", "[", RowBox[List[RowBox[List["z", "+", FractionBox[RowBox[List["4", RowBox[List["EllipticK", "[", "m", "]"]]]], "3"]]], ",", "m"]], "]"]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["JacobiCN", "[", RowBox[List[RowBox[List["z", "+", FractionBox[RowBox[List["8", RowBox[List["EllipticK", "[", "m", "]"]]]], "3"]]], ",", "m"]], "]"]], RowBox[List["JacobiSN", "[", RowBox[List[RowBox[List["z", "+", FractionBox[RowBox[List["8", RowBox[List["EllipticK", "[", "m", "]"]]]], "3"]]], ",", "m"]], "]"]]]], "+", RowBox[List[RowBox[List["JacobiCN", "[", RowBox[List["z", ",", "m"]], "]"]], RowBox[List["JacobiSN", "[", RowBox[List["z", ",", "m"]], "]"]]]]]], ")"]]]], "+", RowBox[List[RowBox[List["JacobiCN", "[", RowBox[List[RowBox[List["z", "+", FractionBox[RowBox[List["8", RowBox[List["EllipticK", "[", "m", "]"]]]], "3"]]], ",", "m"]], "]"]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["JacobiCN", "[", RowBox[List["z", ",", "m"]], "]"]], RowBox[List["JacobiSN", "[", RowBox[List["z", ",", "m"]], "]"]]]], "+", RowBox[List[RowBox[List["JacobiCN", "[", RowBox[List[RowBox[List["z", "+", FractionBox[RowBox[List["4", RowBox[List["EllipticK", "[", "m", "]"]]]], "3"]]], ",", "m"]], "]"]], RowBox[List["JacobiSN", "[", RowBox[List[RowBox[List["z", "+", FractionBox[RowBox[List["4", RowBox[List["EllipticK", "[", "m", "]"]]]], "3"]]], ",", "m"]], "]"]]]]]], ")"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", "2"]], RowBox[List["JacobiDN", "[", RowBox[List[FractionBox[RowBox[List["2", RowBox[List["EllipticK", "[", "m", "]"]]]], "3"], ",", "m"]], "]"]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["JacobiDN", "[", RowBox[List[FractionBox[RowBox[List["2", RowBox[List["EllipticK", "[", "m", "]"]]]], "3"], ",", "m"]], "]"]], "+", "2"]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["JacobiDN", "[", RowBox[List[FractionBox[RowBox[List["2", RowBox[List["EllipticK", "[", "m", "]"]]]], "3"], ",", "m"]], "]"]]]], ")"]], RowBox[List["(", RowBox[List["1", "-", SuperscriptBox[RowBox[List["JacobiDN", "[", RowBox[List[FractionBox[RowBox[List["2", RowBox[List["EllipticK", "[", "m", "]"]]]], "3"], ",", "m"]], "]"]], "2"]]], ")"]]]], ")"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["JacobiSN", "[", RowBox[List["z", ",", "m"]], "]"]], "+", RowBox[List["JacobiSN", "[", RowBox[List[RowBox[List["z", "+", FractionBox[RowBox[List["4", RowBox[List["EllipticK", "[", "m", "]"]]]], "3"]]], ",", "m"]], "]"]], "+", RowBox[List["JacobiSN", "[", RowBox[List[RowBox[List["z", "+", FractionBox[RowBox[List["8", RowBox[List["EllipticK", "[", "m", "]"]]]], "3"]]], ",", "m"]], "]"]]]], ")"]]]]]]]]

MathML Form

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</mo> <mfrac> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> <mn> 3 </mn> </mfrac> </mrow> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ⁢ </mo> <mrow> <mi> dn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> <mn> 3 </mn> </mfrac> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> dn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> <mn> 3 </mn> </mfrac> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> / </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mi> dn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> <mn> 3 </mn> </mfrac> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mrow> <mi> dn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> <mn> 3 </mn> </mfrac> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> sn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> sn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> + </mo> <mfrac> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> <mn> 3 </mn> </mfrac> </mrow> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> sn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> + </mo> <mfrac> <mrow> <mn> 8 </mn> <mo> ⁢ </mo> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> <mn> 3 </mn> </mfrac> </mrow> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <plus /> <apply> <times /> <apply> <ci> JacobiCN </ci> <ci> z </ci> <ci> m </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <ci> JacobiCN </ci> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <ci> m </ci> </apply> <apply> <ci> JacobiSN </ci> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <ci> m </ci> </apply> </apply> <apply> <times /> <apply> <ci> JacobiCN </ci> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <ci> m </ci> </apply> <apply> <ci> JacobiSN </ci> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <ci> m </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <ci> JacobiCN </ci> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <ci> m </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <ci> JacobiCN </ci> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <ci> m </ci> </apply> <apply> <ci> JacobiSN </ci> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <ci> m </ci> </apply> </apply> <apply> <times /> <apply> <ci> JacobiCN </ci> <ci> z </ci> <ci> m </ci> </apply> <apply> <ci> JacobiSN </ci> <ci> z </ci> <ci> m </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <ci> JacobiCN </ci> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <ci> m </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <ci> JacobiCN </ci> <ci> z </ci> <ci> m </ci> </apply> <apply> <ci> JacobiSN </ci> <ci> z </ci> <ci> m </ci> </apply> </apply> <apply> <times /> <apply> <ci> JacobiCN </ci> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <ci> m </ci> </apply> <apply> <ci> JacobiSN </ci> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <ci> m </ci> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -2 </cn> <apply> <ci> JacobiDN </ci> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <ci> m </ci> </apply> <apply> <times /> <apply> <plus /> <apply> <ci> JacobiDN </ci> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <ci> m </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <ci> JacobiDN </ci> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <ci> m </ci> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <ci> JacobiDN </ci> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <ci> m </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <ci> JacobiSN </ci> <ci> z </ci> <ci> m </ci> </apply> <apply> <ci> JacobiSN </ci> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <ci> m </ci> </apply> <apply> <ci> JacobiSN </ci> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <ci> m </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

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

2002-03-07