Jacobi elliptic function dn: Identities involving the group of functions (formula 09.29.18.0083)

JacobiDN

$Mathematica Notation$

Elliptic Functions

JacobiDN[z,m]

Identities involving the group of functions

Local identities of rank 4

Rank 4 identities with 4 distinct arguments

http://functions.wolfram.com/09.29.18.0083.01

Input Form

m^2 JacobiCN[z, m] JacobiCN[z + a, m] JacobiCN[z + b, m] JacobiSN[z + c, m] == (-JacobiDS[a, m]) JacobiDS[b, m] (JacobiNS[c, m] JacobiDN[z, m] - JacobiDS[c, m] JacobiDN[z + c, m]) - JacobiDS[a, m] JacobiDS[a - b, m] (JacobiNS[c - a, m] JacobiDN[z + a, m] - JacobiDS[c - a, m] JacobiDN[z + c, m]) + JacobiDS[b, m] JacobiDS[a - b, m] (JacobiNS[c - b, m] JacobiDN[z + b, m] - JacobiDS[c - b, m] JacobiDN[z + c, m])

Standard Form

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

Rule Form

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