22 Jacobian Elliptic Functions Properties 22.8 Addition Theorems 22.10 Maclaurin Series

§22.9 Cyclic Identities

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Keywords:: Jacobian elliptic functions, cyclic identities
Referenced by:: §22.8(iii)
Permalink:: http://dlmf.nist.gov/22.9
See also:: Annotations for Ch.22

§22.9(i) Notation
§22.9(ii) Typical Identities of Rank 2
§22.9(iii) Typical Identities of Rank 3
§22.9(iv) Typical Identities of Rank 4
§22.9(v) Identities of Higher Rank

§22.9(i) Notation

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Keywords:: Jacobian elliptic functions, cyclic identities, notation, points, rank
Referenced by:: Erratum (V1.0.3) for References
Permalink:: http://dlmf.nist.gov/22.9.i
Clarification (effective with 1.0.3):: The paragraph after (22.9.6) was revised to state more precisely the definition of rank.
See also:: Annotations for §22.9 and Ch.22

The following notation is a generalization of that of Khare and Sukhatme (2002).

Throughout this subsection $m$ and $p$ are positive integers with $1\leq m\leq p$ .

22.9.1		$s_{m,p}^{(2)}=\operatorname{sn}\left(z+2p^{-1}(m-1)K\left(k\right),k\right),$
ⓘ Symbols: $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $z$ : complex, $k$ : modulus, $m$ : positive integer, $p$ : positive integer and $s_{m,p}^{(r)}$ : cyclic quantity Permalink: http://dlmf.nist.gov/22.9.E1 Encodings: TeX, pMML, png See also: Annotations for §22.9(i), §22.9 and Ch.22

22.9.2		$c_{m,p}^{(2)}=\operatorname{cn}\left(z+2p^{-1}(m-1)K\left(k\right),k\right),$
ⓘ Symbols: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $z$ : complex, $k$ : modulus, $m$ : positive integer, $p$ : positive integer and $c_{m,p}^{(r)}$ : cyclic quantity Permalink: http://dlmf.nist.gov/22.9.E2 Encodings: TeX, pMML, png See also: Annotations for §22.9(i), §22.9 and Ch.22

22.9.3		$d_{m,p}^{(2)}=\operatorname{dn}\left(z+2p^{-1}(m-1)K\left(k\right),k\right),$
ⓘ Symbols: $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $z$ : complex, $k$ : modulus, $m$ : positive integer, $p$ : positive integer and $d_{m,p}^{(r)}$ : cyclic quantity Permalink: http://dlmf.nist.gov/22.9.E3 Encodings: TeX, pMML, png See also: Annotations for §22.9(i), §22.9 and Ch.22

22.9.4		$s_{m,p}^{(4)}=\operatorname{sn}\left(z+4p^{-1}(m-1)K\left(k\right),k\right),$
ⓘ Symbols: $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $z$ : complex, $k$ : modulus, $m$ : positive integer, $p$ : positive integer and $s_{m,p}^{(r)}$ : cyclic quantity Permalink: http://dlmf.nist.gov/22.9.E4 Encodings: TeX, pMML, png See also: Annotations for §22.9(i), §22.9 and Ch.22

22.9.5		$c_{m,p}^{(4)}=\operatorname{cn}\left(z+4p^{-1}(m-1)K\left(k\right),k\right),$
ⓘ Symbols: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $z$ : complex, $k$ : modulus, $m$ : positive integer, $p$ : positive integer and $c_{m,p}^{(r)}$ : cyclic quantity Permalink: http://dlmf.nist.gov/22.9.E5 Encodings: TeX, pMML, png See also: Annotations for §22.9(i), §22.9 and Ch.22

22.9.6		$d_{m,p}^{(4)}=\operatorname{dn}\left(z+4p^{-1}(m-1)K\left(k\right),k\right).$
ⓘ Symbols: $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $z$ : complex, $k$ : modulus, $m$ : positive integer, $p$ : positive integer and $d_{m,p}^{(r)}$ : cyclic quantity Referenced by: §22.9(i) Permalink: http://dlmf.nist.gov/22.9.E6 Encodings: TeX, pMML, png See also: Annotations for §22.9(i), §22.9 and Ch.22

In the remainder of this section the rank of an identity is the largest number of elliptic function factors in any term of the identity. The value of $p$ determines the number of points in the identity. The argument $z$ is suppressed in the above notation, as all cyclic identities are independent of $z$ .

§22.9(ii) Typical Identities of Rank 2

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Notes:: See Khare and Sukhatme (2002).
Referenced by:: §22.9(v)
Permalink:: http://dlmf.nist.gov/22.9.ii
Correction (effective with 1.0.25):: In the sentence just below (22.9.10), the expression $s_{m,2}^{(4)}s_{n,2}^{(4)}$ has been corrected to read $s_{m,p}^{(4)}s_{n,p}^{(4)}$ .
Suggested 2019-11-07 by Juan Miguel Nieto
See also:: Annotations for §22.9 and Ch.22

In this subsection $1\leq m\leq p$ and $1\leq n\leq p$ .

Three Points

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Keywords:: Jacobian elliptic functions, cyclic identities, simultaneously permuted
See also:: Annotations for §22.9(ii), §22.9 and Ch.22

With

22.9.7		$\kappa=\operatorname{dn}\left(2K\left(k\right)/3,k\right),$
ⓘ Defines: $\kappa$ : change of variable (locally) Symbols: $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind and $k$ : modulus Referenced by: §22.9(iii), §22.9(iv) Permalink: http://dlmf.nist.gov/22.9.E7 Encodings: TeX, pMML, png See also: Annotations for §22.9(ii), §22.9(ii), §22.9 and Ch.22

22.9.8		$s_{1,3}^{(4)}s_{2,3}^{(4)}+s_{2,3}^{(4)}s_{3,3}^{(4)}+s_{3,3}^{(4)}s_{1,3}^{(4% )}=\frac{\kappa^{2}-1}{k^{2}},$
ⓘ Symbols: $k$ : modulus, $s_{m,p}^{(r)}$ : cyclic quantity and $\kappa$ : change of variable Referenced by: Erratum (V1.0.25) for Equations (22.9.8), (22.9.9) and (22.9.10), Erratum (V1.0.25) for Equations (22.9.8), (22.9.9) and (22.9.10) Permalink: http://dlmf.nist.gov/22.9.E8 Encodings: TeX, pMML, png Correction (effective with 1.0.25): Originally this equation was written incorrectly for all $s_{m,p}^{(4)}$ with $p=2$ . It has been corrected by writing $p=3$ throughout. Suggested 2019-11-07 by Juan Miguel Nieto See also: Annotations for §22.9(ii), §22.9(ii), §22.9 and Ch.22

22.9.9		$c_{1,3}^{(4)}c_{2,3}^{(4)}+c_{2,3}^{(4)}c_{3,3}^{(4)}+c_{3,3}^{(4)}c_{1,3}^{(4% )}=-\frac{\kappa(\kappa+2)}{(1+\kappa)^{2}},$
ⓘ Symbols: $c_{m,p}^{(r)}$ : cyclic quantity and $\kappa$ : change of variable Referenced by: Erratum (V1.0.25) for Equations (22.9.8), (22.9.9) and (22.9.10), Erratum (V1.0.25) for Equations (22.9.8), (22.9.9) and (22.9.10) Permalink: http://dlmf.nist.gov/22.9.E9 Encodings: TeX, pMML, png Correction (effective with 1.0.25): Originally this equation was written incorrectly for all $c_{m,p}^{(4)}$ with $p=2$ . It has been corrected by writing $p=3$ throughout. Suggested 2019-11-07 by Juan Miguel Nieto See also: Annotations for §22.9(ii), §22.9(ii), §22.9 and Ch.22

22.9.10		$d_{1,3}^{(2)}d_{2,3}^{(2)}+d_{2,3}^{(2)}d_{3,3}^{(2)}+d_{3,3}^{(2)}d_{1,3}^{(2% )}=d_{1,3}^{(4)}d_{2,3}^{(4)}+d_{2,3}^{(4)}d_{3,3}^{(4)}+d_{3,3}^{(4)}d_{1,3}^% {(4)}=\kappa(\kappa+2).$
ⓘ Symbols: $d_{m,p}^{(r)}$ : cyclic quantity and $\kappa$ : change of variable Referenced by: §22.9(ii), Erratum (V1.0.25) for Equations (22.9.8), (22.9.9) and (22.9.10), Erratum (V1.0.25) for Equations (22.9.8), (22.9.9) and (22.9.10) Permalink: http://dlmf.nist.gov/22.9.E10 Encodings: TeX, pMML, png Correction (effective with 1.0.25): Originally this equation was written incorrectly for all $d_{m,p}^{(2)}$ and $d_{m,p}^{(4)}$ with $p=2$ . It has been corrected by writing $p=3$ throughout. Suggested 2019-11-07 by Juan Miguel Nieto See also: Annotations for §22.9(ii), §22.9(ii), §22.9 and Ch.22

These identities are cyclic in the sense that each of the indices $m, n$ in the first product of, for example, the form $s_{m,p}^{(4)}s_{n,p}^{(4)}$ are simultaneously permuted in the cyclic order: $m\to m+1\to m+2\to\cdots p\to 1\to 2\to\cdots m-1$ ; $n\to n+1\to n+2\to\cdots p\to 1\to 2\to\cdots n-1$ . Many of the identities that follow also have this property.

§22.9(iii) Typical Identities of Rank 3

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Notes:: See Khare and Sukhatme (2002).
Permalink:: http://dlmf.nist.gov/22.9.iii
See also:: Annotations for §22.9 and Ch.22

Two Points

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See also:: Annotations for §22.9(iii), §22.9 and Ch.22

22.9.11		$\left(d_{1,2}^{(2)}\right)^{2}d_{2,2}^{(2)}\pm\left(d_{2,2}^{(2)}\right)^{2}d_% {1,2}^{(2)}=k^{\prime}\left(d_{1,2}^{(2)}\pm d_{2,2}^{(2)}\right),$
ⓘ Symbols: $k^{\prime}$ : complementary modulus and $d_{m,p}^{(r)}$ : cyclic quantity Permalink: http://dlmf.nist.gov/22.9.E11 Encodings: TeX, pMML, png See also: Annotations for §22.9(iii), §22.9(iii), §22.9 and Ch.22

22.9.12		$c_{1,2}^{(2)}s_{1,2}^{(2)}d_{2,2}^{(2)}+c_{2,2}^{(2)}s_{2,2}^{(2)}d_{1,2}^{(2)% }=0.$
ⓘ Symbols: $s_{m,p}^{(r)}$ : cyclic quantity, $c_{m,p}^{(r)}$ : cyclic quantity and $d_{m,p}^{(r)}$ : cyclic quantity Permalink: http://dlmf.nist.gov/22.9.E12 Encodings: TeX, pMML, png See also: Annotations for §22.9(iii), §22.9(iii), §22.9 and Ch.22

Three Points

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See also:: Annotations for §22.9(iii), §22.9 and Ch.22

With $\kappa$ defined as in (22.9.7),

22.9.13		$s_{1,3}^{(4)}s_{2,3}^{(4)}s_{3,3}^{(4)}=-\frac{1}{1-\kappa^{2}}\left(s_{1,3}^{% (4)}+s_{2,3}^{(4)}+s_{3,3}^{(4)}\right),$
ⓘ Symbols: $s_{m,p}^{(r)}$ : cyclic quantity and $\kappa$ : change of variable Permalink: http://dlmf.nist.gov/22.9.E13 Encodings: TeX, pMML, png See also: Annotations for §22.9(iii), §22.9(iii), §22.9 and Ch.22

22.9.14		$c_{1,3}^{(4)}c_{2,3}^{(4)}c_{3,3}^{(4)}=\frac{\kappa^{2}}{1-\kappa^{2}}\left(c% _{1,3}^{(4)}+c_{2,3}^{(4)}+c_{3,3}^{(4)}\right),$
ⓘ Symbols: $c_{m,p}^{(r)}$ : cyclic quantity and $\kappa$ : change of variable Permalink: http://dlmf.nist.gov/22.9.E14 Encodings: TeX, pMML, png See also: Annotations for §22.9(iii), §22.9(iii), §22.9 and Ch.22

22.9.15		$d_{1,3}^{(2)}d_{2,3}^{(2)}d_{3,3}^{(2)}=\frac{\kappa^{2}+k^{2}-1}{1-\kappa^{2}% }\left(d_{1,3}^{(2)}+d_{2,3}^{(2)}+d_{3,3}^{(2)}\right),$
ⓘ Symbols: $k$ : modulus, $d_{m,p}^{(r)}$ : cyclic quantity and $\kappa$ : change of variable Permalink: http://dlmf.nist.gov/22.9.E15 Encodings: TeX, pMML, png See also: Annotations for §22.9(iii), §22.9(iii), §22.9 and Ch.22

22.9.16		$s_{1,3}^{(4)}c_{2,3}^{(4)}c_{3,3}^{(4)}+s_{2,3}^{(4)}c_{3,3}^{(4)}c_{1,3}^{(4)% }+s_{3,3}^{(4)}c_{1,3}^{(4)}c_{2,3}^{(4)}=\frac{\kappa(\kappa+2)}{1-\kappa^{2}% }\left(s_{1,3}^{(4)}+s_{2,3}^{(4)}+s_{3,3}^{(4)}\right).$
ⓘ Symbols: $s_{m,p}^{(r)}$ : cyclic quantity, $c_{m,p}^{(r)}$ : cyclic quantity and $\kappa$ : change of variable Permalink: http://dlmf.nist.gov/22.9.E16 Encodings: TeX, pMML, png See also: Annotations for §22.9(iii), §22.9(iii), §22.9 and Ch.22

Four Points

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See also:: Annotations for §22.9(iii), §22.9 and Ch.22

22.9.17		$d_{1,4}^{(2)}d_{2,4}^{(2)}d_{3,4}^{(2)}\pm d_{2,4}^{(2)}d_{3,4}^{(2)}d_{4,4}^{% (2)}+d_{3,4}^{(2)}d_{4,4}^{(2)}d_{1,4}^{(2)}\pm d_{4,4}^{(2)}d_{1,4}^{(2)}d_{2% ,4}^{(2)}=k^{\prime}{\left(\pm d_{1,4}^{(2)}+d_{2,4}^{(2)}\pm d_{3,4}^{(2)}+d_% {4,4}^{(2)}\right)},$
ⓘ Symbols: $k^{\prime}$ : complementary modulus and $d_{m,p}^{(r)}$ : cyclic quantity Permalink: http://dlmf.nist.gov/22.9.E17 Encodings: TeX, pMML, png See also: Annotations for §22.9(iii), §22.9(iii), §22.9 and Ch.22

22.9.18		$\left(d_{1,4}^{(2)}\right)^{2}d_{3,4}^{(2)}\pm\left(d_{2,4}^{(2)}\right)^{2}d_% {4,4}^{(2)}+\left(d_{3,4}^{(2)}\right)^{2}d_{1,4}^{(2)}\pm\left(d_{4,4}^{(2)}% \right)^{2}d_{2,4}^{(2)}=k^{\prime}{\left(d_{1,4}^{(2)}\pm d_{2,4}^{(2)}+d_{3,% 4}^{(2)}\pm d_{4,4}^{(2)}\right)},$
ⓘ Symbols: $k^{\prime}$ : complementary modulus and $d_{m,p}^{(r)}$ : cyclic quantity Permalink: http://dlmf.nist.gov/22.9.E18 Encodings: TeX, pMML, png See also: Annotations for §22.9(iii), §22.9(iii), §22.9 and Ch.22

22.9.19		$c_{1,4}^{(2)}s_{1,4}^{(2)}d_{3,4}^{(2)}+c_{3,4}^{(2)}s_{3,4}^{(2)}d_{1,4}^{(2)% }=c_{2,4}^{(2)}s_{2,4}^{(2)}d_{4,4}^{(2)}+c_{4,4}^{(2)}s_{4,4}^{(2)}d_{2,4}^{(% 2)}=0.$
ⓘ Symbols: $s_{m,p}^{(r)}$ : cyclic quantity, $c_{m,p}^{(r)}$ : cyclic quantity and $d_{m,p}^{(r)}$ : cyclic quantity Permalink: http://dlmf.nist.gov/22.9.E19 Encodings: TeX, pMML, png See also: Annotations for §22.9(iii), §22.9(iii), §22.9 and Ch.22

§22.9(iv) Typical Identities of Rank 4

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Notes:: See Khare et al. (2003), Khare and Sukhatme (2002).
Referenced by:: §22.9(v)
Permalink:: http://dlmf.nist.gov/22.9.iv
See also:: Annotations for §22.9 and Ch.22

Two Points

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See also:: Annotations for §22.9(iv), §22.9 and Ch.22

22.9.20		$\left(d_{1,2}^{(2)}\right)^{3}d_{2,2}^{(2)}\pm\left(d_{2,2}^{(2)}\right)^{3}d_% {1,2}^{(2)}=k^{\prime}\left(\left(d_{1,2}^{(2)}\right)^{2}\pm\left(d_{2,2}^{(2% )}\right)^{2}\right),$
ⓘ Symbols: $k^{\prime}$ : complementary modulus and $d_{m,p}^{(r)}$ : cyclic quantity Permalink: http://dlmf.nist.gov/22.9.E20 Encodings: TeX, pMML, png See also: Annotations for §22.9(iv), §22.9(iv), §22.9 and Ch.22

22.9.21		$k^{2}c_{1,2}^{(2)}s_{1,2}^{(2)}c_{2,2}^{(2)}s_{2,2}^{(2)}=k^{\prime}\left(1-% \left(s_{1,2}^{(2)}\right)^{2}-\left(s_{2,2}^{(2)}\right)^{2}\right).$
ⓘ Symbols: $k$ : modulus, $k^{\prime}$ : complementary modulus, $s_{m,p}^{(r)}$ : cyclic quantity and $c_{m,p}^{(r)}$ : cyclic quantity Permalink: http://dlmf.nist.gov/22.9.E21 Encodings: TeX, pMML, png See also: Annotations for §22.9(iv), §22.9(iv), §22.9 and Ch.22

Three Points

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See also:: Annotations for §22.9(iv), §22.9 and Ch.22

Again with $\kappa$ defined as in (22.9.7),

22.9.22		$s_{1,3}^{(2)}c_{1,3}^{(2)}d_{2,3}^{(2)}d_{3,3}^{(2)}+s_{2,3}^{(2)}c_{2,3}^{(2)% }d_{3,3}^{(2)}d_{1,3}^{(2)}+s_{3,3}^{(2)}c_{3,3}^{(2)}d_{1,3}^{(2)}d_{2,3}^{(2% )}=\frac{\kappa^{2}+k^{2}-1}{1-\kappa^{2}}\left(s_{1,3}^{(2)}c_{1,3}^{(2)}+s_{% 2,3}^{(2)}c_{2,3}^{(2)}+s_{3,3}^{(2)}c_{3,3}^{(2)}\right),$
ⓘ Symbols: $k$ : modulus, $s_{m,p}^{(r)}$ : cyclic quantity, $c_{m,p}^{(r)}$ : cyclic quantity, $d_{m,p}^{(r)}$ : cyclic quantity and $\kappa$ : change of variable Permalink: http://dlmf.nist.gov/22.9.E22 Encodings: TeX, pMML, png See also: Annotations for §22.9(iv), §22.9(iv), §22.9 and Ch.22

22.9.23		$s_{1,3}^{(4)}d_{1,3}^{(4)}c_{2,3}^{(4)}c_{3,3}^{(4)}+s_{2,3}^{(4)}d_{2,3}^{(4)% }c_{3,3}^{(4)}c_{1,3}^{(4)}+s_{3,3}^{(4)}d_{3,3}^{(4)}c_{1,3}^{(4)}c_{2,3}^{(4% )}=\frac{\kappa^{2}}{1-\kappa^{2}}\left(s_{1,3}^{(4)}d_{1,3}^{(4)}+s_{2,3}^{(4% )}d_{2,3}^{(4)}+s_{2,3}^{(4)}d_{2,3}^{(4)}\right).$
ⓘ Symbols: $s_{m,p}^{(r)}$ : cyclic quantity, $c_{m,p}^{(r)}$ : cyclic quantity, $d_{m,p}^{(r)}$ : cyclic quantity and $\kappa$ : change of variable Permalink: http://dlmf.nist.gov/22.9.E23 Encodings: TeX, pMML, png See also: Annotations for §22.9(iv), §22.9(iv), §22.9 and Ch.22

§22.9(v) Identities of Higher Rank

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Keywords:: Jacobian elliptic functions, cyclic identities
Permalink:: http://dlmf.nist.gov/22.9.v
See also:: Annotations for §22.9 and Ch.22

For extensions of the identities given in §§22.9(ii)–22.9(iv), and also to related elliptic functions, see Khare and Sukhatme (2002), Khare et al. (2003).

22.8 Addition Theorems 22.10 Maclaurin Series

NIST

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§22.9 Cyclic Identities

Contents

§22.9(i) Notation

§22.9(ii) Typical Identities of Rank 2

Three Points

§22.9(iii) Typical Identities of Rank 3

Two Points

Three Points

Four Points

§22.9(iv) Typical Identities of Rank 4

Two Points

Three Points

§22.9(v) Identities of Higher Rank