Jacobi elliptic function dn: Representations through equivalent functions (formula 09.29.27.0031)

JacobiDN

$Mathematica Notation$

Elliptic Functions

JacobiDN[z,m]

Representations through equivalent functions

With related functions

Involving theta functions

http://functions.wolfram.com/09.29.27.0031.01

Input Form

JacobiZeta[JacobiAmplitude[u, m], m] + JacobiZeta[JacobiAmplitude[v, m], m] - JacobiZeta[JacobiAmplitude[u + v, m], m] == ((m JacobiSN[v, m])/JacobiDN[v, m]) (JacobiCN[v, m] - JacobiCN[u, m] JacobiCN[u + v, m])

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["JacobiZeta", "[", RowBox[List[RowBox[List["JacobiAmplitude", "[", RowBox[List["u", ",", "m"]], "]"]], ",", "m"]], "]"]], "+", RowBox[List["JacobiZeta", "[", RowBox[List[RowBox[List["JacobiAmplitude", "[", RowBox[List["v", ",", "m"]], "]"]], ",", "m"]], "]"]], "-", RowBox[List["JacobiZeta", "[", RowBox[List[RowBox[List["JacobiAmplitude", "[", RowBox[List[RowBox[List["u", "+", "v"]], ",", "m"]], "]"]], ",", "m"]], "]"]]]], "\[Equal]", RowBox[List[FractionBox[RowBox[List["m", " ", RowBox[List["JacobiSN", "[", RowBox[List["v", ",", "m"]], "]"]]]], RowBox[List["JacobiDN", "[", RowBox[List["v", ",", "m"]], "]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["JacobiCN", "[", RowBox[List["v", ",", "m"]], "]"]], "-", RowBox[List[RowBox[List["JacobiCN", "[", RowBox[List["u", ",", "m"]], "]"]], " ", RowBox[List["JacobiCN", "[", RowBox[List[RowBox[List["u", "+", "v"]], ",", "m"]], "]"]]]]]], ")"]]]]]]]]

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> Ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> am </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> u </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> Ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> am </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> v </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> Ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> am </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> u </mi> <mo> + </mo> <mi> v </mi> </mrow> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ⩵ </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> ⁢ </mo> <mrow> <mi> sn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> v </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> cn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> v </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mrow> <mi> cn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> u </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> cn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> u </mi> <mo> + </mo> <mi> v </mi> </mrow> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> dn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> v </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mfrac> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <plus /> <apply> <ci> JacobiZeta </ci> <apply> <ci> JacobiAmplitude </ci> <ci> u </ci> <ci> m </ci> </apply> <ci> m </ci> </apply> <apply> <ci> JacobiZeta </ci> <apply> <ci> JacobiAmplitude </ci> <ci> v </ci> <ci> m </ci> </apply> <ci> m </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> JacobiZeta </ci> <apply> <ci> JacobiAmplitude </ci> <apply> <plus /> <ci> u </ci> <ci> v </ci> </apply> <ci> m </ci> </apply> <ci> m </ci> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <ci> m </ci> <apply> <ci> JacobiSN </ci> <ci> v </ci> <ci> m </ci> </apply> </apply> <apply> <plus /> <apply> <ci> JacobiCN </ci> <ci> v </ci> <ci> m </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> JacobiCN </ci> <ci> u </ci> <ci> m </ci> </apply> <apply> <ci> JacobiCN </ci> <apply> <plus /> <ci> u </ci> <ci> v </ci> </apply> <ci> m </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <ci> JacobiDN </ci> <ci> v </ci> <ci> m </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["JacobiZeta", "[", RowBox[List[RowBox[List["JacobiAmplitude", "[", RowBox[List["u_", ",", "m_"]], "]"]], ",", "m_"]], "]"]], "+", RowBox[List["JacobiZeta", "[", RowBox[List[RowBox[List["JacobiAmplitude", "[", RowBox[List["v_", ",", "m_"]], "]"]], ",", "m_"]], "]"]], "-", RowBox[List["JacobiZeta", "[", RowBox[List[RowBox[List["JacobiAmplitude", "[", RowBox[List[RowBox[List["u_", "+", "v_"]], ",", "m_"]], "]"]], ",", "m_"]], "]"]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["m", " ", RowBox[List["JacobiSN", "[", RowBox[List["v", ",", "m"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["JacobiCN", "[", RowBox[List["v", ",", "m"]], "]"]], "-", RowBox[List[RowBox[List["JacobiCN", "[", RowBox[List["u", ",", "m"]], "]"]], " ", RowBox[List["JacobiCN", "[", RowBox[List[RowBox[List["u", "+", "v"]], ",", "m"]], "]"]]]]]], ")"]]]], RowBox[List["JacobiDN", "[", RowBox[List["v", ",", "m"]], "]"]]]]]]]

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

2001-10-29