Jacobi elliptic function dn: Identities involving the group of functions (formula 09.29.18.0139)

JacobiDN

$Mathematica Notation$

Elliptic Functions

JacobiDN[z,m]

Identities involving the group of functions

Higher order local identities

Examples of local identities of rank 5

http://functions.wolfram.com/09.29.18.0139.01

Input Form

m JacobiDN[z, m] JacobiSN[z, m] JacobiCN[z, m] JacobiDN[z + a, m]^2 == JacobiCS[a, m] JacobiDS[a, m] JacobiNS[a, m] (JacobiDN[z + a, m]^2 + 2 JacobiDN[z, m]^2 - JacobiDN[a, m]^2 - 2) - m JacobiCS[a, m]^2 JacobiDN[z, m] JacobiSN[z, m] JacobiCN[z, m] + (JacobiNS[a, m]^2 (JacobiDS[a, m]^2 + JacobiCS[a, m]^2) + JacobiCS[a, m]^2 JacobiDS[a, m]^2) (JacobiZeta[JacobiAmplitude[z + a, m], m] - JacobiZeta[JacobiAmplitude[z, m], m] - JacobiZeta[JacobiAmplitude[a, m], m])

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["m", " ", RowBox[List["JacobiDN", "[", RowBox[List["z", ",", "m"]], "]"]], RowBox[List["JacobiSN", "[", RowBox[List["z", ",", "m"]], "]"]], RowBox[List["JacobiCN", "[", RowBox[List["z", ",", "m"]], "]"]], SuperscriptBox[RowBox[List["JacobiDN", "[", RowBox[List[RowBox[List["z", "+", "a"]], ",", "m"]], "]"]], "2"]]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["JacobiCS", "[", RowBox[List["a", ",", "m"]], "]"]], RowBox[List["JacobiDS", "[", RowBox[List["a", ",", "m"]], "]"]], RowBox[List["JacobiNS", "[", RowBox[List["a", ",", "m"]], "]"]], RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["JacobiDN", "[", RowBox[List[RowBox[List["z", "+", "a"]], ",", "m"]], "]"]], "2"], "+", RowBox[List["2", SuperscriptBox[RowBox[List["JacobiDN", "[", RowBox[List["z", ",", "m"]], "]"]], "2"]]], "-", SuperscriptBox[RowBox[List["JacobiDN", "[", RowBox[List["a", ",", "m"]], "]"]], "2"], "-", "2"]], ")"]]]], "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["JacobiCS", "[", RowBox[List["a", ",", "m"]], "]"]], "2"], RowBox[List["JacobiDN", "[", RowBox[List["z", ",", "m"]], "]"]], RowBox[List["JacobiSN", "[", RowBox[List["z", ",", "m"]], "]"]], RowBox[List["JacobiCN", "[", RowBox[List["z", ",", "m"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["JacobiNS", "[", RowBox[List["a", ",", "m"]], "]"]], "2"], RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["JacobiDS", "[", RowBox[List["a", ",", "m"]], "]"]], "2"], "+", SuperscriptBox[RowBox[List["JacobiCS", "[", RowBox[List["a", ",", "m"]], "]"]], "2"]]], ")"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["JacobiCS", "[", RowBox[List["a", ",", "m"]], "]"]], "2"], SuperscriptBox[RowBox[List["JacobiDS", "[", RowBox[List["a", ",", "m"]], "]"]], "2"]]]]], ")"]], RowBox[List["(", RowBox[List[RowBox[List["JacobiZeta", "[", RowBox[List[RowBox[List["JacobiAmplitude", "[", RowBox[List[RowBox[List["z", "+", "a"]], ",", "m"]], "]"]], ",", "m"]], "]"]], "-", RowBox[List["JacobiZeta", "[", RowBox[List[RowBox[List["JacobiAmplitude", "[", RowBox[List["z", ",", "m"]], "]"]], ",", "m"]], "]"]], "-", RowBox[List["JacobiZeta", "[", RowBox[List[RowBox[List["JacobiAmplitude", "[", RowBox[List["a", ",", "m"]], "]"]], ",", "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> <mi> m </mi> <mo> ⁢ </mo> <mrow> <mi> dn </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> <mi> z </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> cn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mi> dn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> + </mo> <mi> z </mi> </mrow> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mi> m </mi> </mrow> <mo> ⁢ </mo> <mrow> <mi> dn </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> <mi> z </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> cn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mi> cs </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <msup> <mrow> <mi> dn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <msup> <mrow> <mi> dn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> + </mo> <mi> z </mi> </mrow> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mrow> <mi> dn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> ds </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> ns </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> cs </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mrow> <mi> cs </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msup> <mrow> <mi> ds </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <msup> <mrow> <mi> cs </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mrow> <mi> ds </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mi> ns </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mrow> <mi> Ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> am </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> Ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> am </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> + </mo> <mi> z </mi> </mrow> <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> z </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <times /> <ci> m </ci> <apply> <ci> JacobiDN </ci> <ci> z </ci> <ci> m </ci> </apply> <apply> <ci> JacobiSN </ci> <ci> z </ci> <ci> m </ci> </apply> <apply> <ci> JacobiCN </ci> <ci> z </ci> <ci> m </ci> </apply> <apply> <power /> <apply> <ci> JacobiDN </ci> <apply> <plus /> <ci> a </ci> <ci> z </ci> </apply> <ci> m </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <apply> <ci> JacobiDN </ci> <ci> z </ci> <ci> m </ci> </apply> <apply> <ci> JacobiSN </ci> <ci> z </ci> <ci> m </ci> </apply> <apply> <ci> JacobiCN </ci> <ci> z </ci> <ci> m </ci> </apply> <apply> <power /> <apply> <ci> JacobiCS </ci> <ci> a </ci> <ci> m </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <ci> JacobiDN </ci> <ci> a </ci> <ci> m </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <apply> <ci> JacobiDN </ci> <apply> <plus /> <ci> a </ci> <ci> z </ci> </apply> <ci> m </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <ci> JacobiDN </ci> <ci> z </ci> <ci> m </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -2 </cn> </apply> <apply> <ci> JacobiDS </ci> <ci> a </ci> <ci> m </ci> </apply> <apply> <ci> JacobiNS </ci> <ci> a </ci> <ci> m </ci> </apply> <apply> <ci> JacobiCS </ci> <ci> a </ci> <ci> m </ci> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <ci> JacobiCS </ci> <ci> a </ci> <ci> m </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ci> JacobiDS </ci> <ci> a </ci> <ci> m </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <power /> <apply> <ci> JacobiCS </ci> <ci> a </ci> <ci> m </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ci> JacobiDS </ci> <ci> a </ci> <ci> m </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <apply> <ci> JacobiNS </ci> <ci> a </ci> <ci> m </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> JacobiZeta </ci> <apply> <ci> JacobiAmplitude </ci> <ci> a </ci> <ci> m </ci> </apply> <ci> m </ci> </apply> </apply> <apply> <ci> JacobiZeta </ci> <apply> <ci> JacobiAmplitude </ci> <apply> <plus /> <ci> a </ci> <ci> z </ci> </apply> <ci> m </ci> </apply> <ci> m </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> JacobiZeta </ci> <apply> <ci> JacobiAmplitude </ci> <ci> z </ci> <ci> m </ci> </apply> <ci> m </ci> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["m_", " ", RowBox[List["JacobiDN", "[", RowBox[List["z_", ",", "m_"]], "]"]], " ", RowBox[List["JacobiSN", "[", RowBox[List["z_", ",", "m_"]], "]"]], " ", RowBox[List["JacobiCN", "[", RowBox[List["z_", ",", "m_"]], "]"]], " ", SuperscriptBox[RowBox[List["JacobiDN", "[", RowBox[List[RowBox[List["z_", "+", "a_"]], ",", "m_"]], "]"]], "2"]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["JacobiCS", "[", RowBox[List["a", ",", "m"]], "]"]], " ", RowBox[List["JacobiDS", "[", RowBox[List["a", ",", "m"]], "]"]], " ", RowBox[List["JacobiNS", "[", RowBox[List["a", ",", "m"]], "]"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["JacobiDN", "[", RowBox[List[RowBox[List["z", "+", "a"]], ",", "m"]], "]"]], "2"], "+", RowBox[List["2", " ", SuperscriptBox[RowBox[List["JacobiDN", "[", RowBox[List["z", ",", "m"]], "]"]], "2"]]], "-", SuperscriptBox[RowBox[List["JacobiDN", "[", RowBox[List["a", ",", "m"]], "]"]], "2"], "-", "2"]], ")"]]]], "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["JacobiCS", "[", RowBox[List["a", ",", "m"]], "]"]], "2"], " ", RowBox[List["JacobiDN", "[", RowBox[List["z", ",", "m"]], "]"]], " ", RowBox[List["JacobiSN", "[", RowBox[List["z", ",", "m"]], "]"]], " ", RowBox[List["JacobiCN", "[", RowBox[List["z", ",", "m"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["JacobiNS", "[", RowBox[List["a", ",", "m"]], "]"]], "2"], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["JacobiDS", "[", RowBox[List["a", ",", "m"]], "]"]], "2"], "+", SuperscriptBox[RowBox[List["JacobiCS", "[", RowBox[List["a", ",", "m"]], "]"]], "2"]]], ")"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["JacobiCS", "[", RowBox[List["a", ",", "m"]], "]"]], "2"], " ", SuperscriptBox[RowBox[List["JacobiDS", "[", RowBox[List["a", ",", "m"]], "]"]], "2"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["JacobiZeta", "[", RowBox[List[RowBox[List["JacobiAmplitude", "[", RowBox[List[RowBox[List["z", "+", "a"]], ",", "m"]], "]"]], ",", "m"]], "]"]], "-", RowBox[List["JacobiZeta", "[", RowBox[List[RowBox[List["JacobiAmplitude", "[", RowBox[List["z", ",", "m"]], "]"]], ",", "m"]], "]"]], "-", RowBox[List["JacobiZeta", "[", RowBox[List[RowBox[List["JacobiAmplitude", "[", RowBox[List["a", ",", "m"]], "]"]], ",", "m"]], "]"]]]], ")"]]]]]]]]]]

References

A. Khare, A. Lakshminarayan, U. Sukhatme, "Local Identities Involving Jacobi Elliptic Functions", math-ph/0306028, (2003) http://arXiv.org/abs/math-ph/0306028

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

2003-08-21