Jacobi elliptic function dn: Transformations (formula 09.29.16.0123)

JacobiDN

$Mathematica Notation$

Elliptic Functions

JacobiDN[z,m]

Transformations

Sums over products of four Jacobi functions

http://functions.wolfram.com/09.29.16.0123.01

Input Form

Sum[JacobiCN[z + 2 k (EllipticK[m]/p), m]^3 (JacobiCN[z + 2 (k + r) (EllipticK[m]/p), m] + JacobiCN[z + 2 (k - r) (EllipticK[m]/p), m]), {k, 0, p - 1}] == (2/m^2) JacobiNS[2 r (EllipticK[m]/p), m] JacobiCS[2 r (EllipticK[m]/p), m] Sum[JacobiDN[z + 2 k (EllipticK[m]/p), m]^2, {k, 0, p - 1}] + (p/(2 EllipticK[m])) (Integrate[JacobiCN[t, m]^3 (JacobiCN[t + 2 r (EllipticK[m]/p), m] + JacobiCN[t - 2 r (EllipticK[m]/p), m]), {t, 0, 2 EllipticK[m]}] - (4/m^2) JacobiNS[2 r (EllipticK[m]/p), m] JacobiCS[2 r (EllipticK[m]/p), m] EllipticE[m]) /; Element[p, Integers] && p >= 1 && Element[r, Integers] && Inequality[1, LessEqual, r, Less, p - 1]

Standard Form

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MathML Form

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</mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> p </mi> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> <mo> ∧ </mo> <mrow> <mi> r </mi> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> <mo> ∧ </mo> <mrow> <mi> r </mi> <mo> < </mo> <mrow> <mi> p </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> p </ci> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <times /> <apply> <power /> <apply> <ci> JacobiCN </ci> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> <apply> <power /> <ci> p </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <ci> m </ci> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <plus /> <apply> <ci> JacobiCN </ci> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> r </ci> </apply> </apply> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> <apply> <power /> <ci> p </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <ci> m </ci> </apply> <apply> <ci> JacobiCN </ci> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> k </ci> <ci> r </ci> </apply> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> <apply> <power /> <ci> p </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <ci> m </ci> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <ci> p </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> </apply> </uplimit> <apply> <times /> <apply> <power /> <apply> <ci> JacobiCN </ci> <ci> t </ci> <ci> m </ci> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <plus /> <apply> <ci> JacobiCN </ci> <apply> <plus /> <ci> t </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> r </ci> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> <apply> <power /> <ci> p </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <ci> m </ci> </apply> <apply> <ci> JacobiCN </ci> <apply> <plus /> <ci> t </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> r </ci> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> <apply> <power /> <ci> p </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <ci> m </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <apply> <power /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> JacobiNS </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> r </ci> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> <apply> <power /> <ci> p </ci> <cn type='integer'> -1 </cn> </apply> </apply> <ci> m </ci> </apply> <apply> <ci> JacobiCS </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> r </ci> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> <apply> <power /> <ci> p </ci> <cn type='integer'> -1 </cn> </apply> </apply> <ci> m </ci> </apply> <apply> <ci> EllipticE </ci> <ci> m </ci> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <power /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> JacobiCS </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> r </ci> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> <apply> <power /> <ci> p </ci> <cn type='integer'> -1 </cn> </apply> </apply> <ci> m </ci> </apply> <apply> <ci> JacobiNS </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> r </ci> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> <apply> <power /> <ci> p </ci> <cn type='integer'> -1 </cn> </apply> </apply> <ci> m </ci> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> p </ci> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <power /> <apply> <ci> JacobiDN </ci> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> <apply> <power /> <ci> p </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <ci> m </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> p </ci> <apply> <ci> SuperPlus </ci> <ci> ℕ </ci> </apply> </apply> <apply> <in /> <ci> r </ci> <apply> <ci> SuperPlus </ci> <ci> ℕ </ci> </apply> </apply> <apply> <lt /> <ci> r </ci> <apply> <plus /> <ci> p </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

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References

A. Khare, A. Lakshminarayan, U. Sukhatme, "Cyclic Identities Involving Jacobi Elliptic Functions. II", math-ph/0207019, (2002) http://arXiv.org/abs/math-ph/0207019

A. Khare, A. Lakshminarayan, U. Sukhatme, "Cyclic Identities Involving Jacobi Elliptic Functions", Journal of Mathematical Physics, v. 44, issue 4, pp. 1822-1841 (2003)

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

2002-12-18