Jacobi elliptic function dn: Identities involving the group of functions (formula 09.29.18.0077)

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/09.29.18.0077.01

Input Form

m^2 JacobiCN[z, m] JacobiCN[z + a, m] JacobiCN[z + b, m] JacobiCN[z + c, m] == (-JacobiDS[a, m]) JacobiDS[b, m] (m JacobiCN[c, m] + JacobiDS[c, m] (JacobiZeta[JacobiAmplitude[z + c, m], m] - JacobiZeta[JacobiAmplitude[z, m], m] - JacobiZeta[JacobiAmplitude[c, m], m])) - JacobiDS[a, m] JacobiDS[a - b, m] (m JacobiCN[c - a, m] + JacobiDS[c - a, m] (JacobiZeta[JacobiAmplitude[z + c, m], m] - JacobiZeta[JacobiAmplitude[z + a, m], m] - JacobiZeta[JacobiAmplitude[c - a, m], m])) + JacobiDS[b, m] JacobiDS[a - b, m] (m JacobiCN[c - b, m] + JacobiDS[c - b, m] (JacobiZeta[JacobiAmplitude[z + c, m], m] - JacobiZeta[JacobiAmplitude[z + b, m], m] - JacobiZeta[JacobiAmplitude[c - b, m], 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["JacobiCN", "[", 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["m", " ", RowBox[List["JacobiCN", "[", RowBox[List["c", ",", "m"]], "]"]]]], "+", RowBox[List[RowBox[List["JacobiDS", "[", RowBox[List["c", ",", "m"]], "]"]], RowBox[List["(", RowBox[List[RowBox[List["JacobiZeta", "[", RowBox[List[RowBox[List["JacobiAmplitude", "[", RowBox[List[RowBox[List["z", "+", "c"]], ",", "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["c", ",", "m"]], "]"]], ",", "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["m", " ", RowBox[List["JacobiCN", "[", RowBox[List[RowBox[List["c", "-", "a"]], ",", "m"]], "]"]]]], "+", RowBox[List[RowBox[List["JacobiDS", "[", RowBox[List[RowBox[List["c", "-", "a"]], ",", "m"]], "]"]], RowBox[List["(", RowBox[List[RowBox[List["JacobiZeta", "[", RowBox[List[RowBox[List["JacobiAmplitude", "[", RowBox[List[RowBox[List["z", "+", "c"]], ",", "m"]], "]"]], ",", "m"]], "]"]], "-", 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[RowBox[List["c", "-", "a"]], ",", "m"]], "]"]], ",", "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["m", " ", RowBox[List["JacobiCN", "[", RowBox[List[RowBox[List["c", "-", "b"]], ",", "m"]], "]"]]]], "+", RowBox[List[RowBox[List["JacobiDS", "[", RowBox[List[RowBox[List["c", "-", "b"]], ",", "m"]], "]"]], RowBox[List["(", RowBox[List[RowBox[List["JacobiZeta", "[", RowBox[List[RowBox[List["JacobiAmplitude", "[", RowBox[List[RowBox[List["z", "+", "c"]], ",", "m"]], "]"]], ",", "m"]], "]"]], "-", RowBox[List["JacobiZeta", "[", RowBox[List[RowBox[List["JacobiAmplitude", "[", RowBox[List[RowBox[List["z", "+", "b"]], ",", "m"]], "]"]], ",", "m"]], "]"]], "-", RowBox[List["JacobiZeta", "[", RowBox[List[RowBox[List["JacobiAmplitude", "[", RowBox[List[RowBox[List["c", "-", "b"]], ",", "m"]], "]"]], ",", "m"]], "]"]]]], ")"]]]]]], ")"]]]]]]]]]]

MathML Form

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</mo> <mrow> <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> ds </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> b </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> m </mi> <mo> ⁢ </mo> <mrow> <mi> cn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> c </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> <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> c </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> c </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> - 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Rule Form

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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