Jacobi elliptic function dn: Identities involving the group of functions (formula 09.29.18.0096)

JacobiDN

$Mathematica Notation$

Elliptic Functions

JacobiDN[z,m]

Identities involving the group of functions

Local identities of rank 4

Rank 4 identities with 3 distinct arguments

http://functions.wolfram.com/09.29.18.0096.01

Input Form

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

Standard Form

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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> <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> <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> <mo> ⁢ </mo> <mrow> <mi> sn </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> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> ns </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> b </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> </mrow> <mo> - </mo> <mrow> <mrow> <mi> cs </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> b </mi> <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> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> cs </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> <mo> ⁢ </mo> <mrow> <mi> cs </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> b </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mrow> <mi> cs </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> b </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> dn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <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> a </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> </mrow> <mo> - </mo> <mrow> <mrow> <mi> cs </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> a </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> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <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> <mo> ⁢ </mo> <mrow> <mi> ns </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> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <times /> <apply> <power /> <apply> <ci> JacobiDN </ci> <ci> z </ci> <ci> m </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <ci> JacobiDN </ci> <apply> <plus /> <ci> a </ci> <ci> z </ci> </apply> <ci> m </ci> </apply> <apply> <ci> JacobiSN </ci> <apply> <plus /> <ci> b </ci> <ci> z </ci> </apply> <ci> m </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <ci> JacobiNS </ci> <ci> b </ci> <ci> m </ci> </apply> <apply> <ci> JacobiCN </ci> <ci> z </ci> <ci> m </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> JacobiCS </ci> <ci> b </ci> <ci> m </ci> </apply> <apply> <ci> JacobiCN </ci> <apply> <plus /> <ci> b </ci> <ci> z </ci> </apply> <ci> m </ci> </apply> </apply> </apply> </apply> <apply> <ci> JacobiCS </ci> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <ci> m </ci> </apply> <apply> <ci> JacobiCS </ci> <ci> b </ci> <ci> m </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> JacobiCS </ci> <ci> a </ci> <ci> m </ci> </apply> <apply> <ci> JacobiNS </ci> <ci> b </ci> <ci> m </ci> </apply> <apply> <ci> JacobiSN </ci> <ci> z </ci> <ci> m </ci> </apply> <apply> <ci> JacobiDN </ci> <ci> z </ci> <ci> m </ci> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <ci> JacobiNS </ci> <ci> a </ci> <ci> m </ci> </apply> <apply> <ci> JacobiCN </ci> <ci> z </ci> <ci> m </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> JacobiCS </ci> <ci> a </ci> <ci> m </ci> </apply> <apply> <ci> JacobiCN </ci> <apply> <plus /> <ci> a </ci> <ci> z </ci> </apply> <ci> m </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> JacobiCS </ci> <ci> a </ci> <ci> m </ci> </apply> </apply> <apply> <ci> JacobiNS </ci> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <ci> m </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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