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

22.9.1 $s_{m,p}^{(2)}=\operatorname{sn}\left(z+2p^{-1}(m-1)K\left(k\right),k\right),$

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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, : complex, : modulus, : positive integer, : 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 $c_{m,p}^{(2)}=\operatorname{cn}\left(z+2p^{-1}(m-1)K\left(k\right),k\right),$

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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, : complex, : modulus, : positive integer, : 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 $d_{m,p}^{(2)}=\operatorname{dn}\left(z+2p^{-1}(m-1)K\left(k\right),k\right),$

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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, : complex, : modulus, : positive integer, : 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 $s_{m,p}^{(4)}=\operatorname{sn}\left(z+4p^{-1}(m-1)K\left(k\right),k\right),$

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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, : complex, : modulus, : positive integer, : 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 $c_{m,p}^{(4)}=\operatorname{cn}\left(z+4p^{-1}(m-1)K\left(k\right),k\right),$

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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, : complex, : modulus, : positive integer, : 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 $d_{m,p}^{(4)}=\operatorname{dn}\left(z+4p^{-1}(m-1)K\left(k\right),k\right).$

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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, : complex, : modulus, : positive integer, : 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 determines the number of points in the identity. The argument is suppressed in the above notation, as all cyclic identities are independent of .

§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 $\kappa=\operatorname{dn}\left(2K\left(k\right)/3,k\right),$

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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 : 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 $s_{1,3}^{(4)}s_{2,3}^{(4)}+s_{2,3}^{(4)}s_{3,3}^{(4)}+s_{3,3}^{(4)}s_{1,3}^{(4% )}=\frac{\kappa^{2}-1}{k^{2}},$

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Symbols:: : 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 . It has been corrected by writing 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 $c_{1,3}^{(4)}c_{2,3}^{(4)}+c_{2,3}^{(4)}c_{3,3}^{(4)}+c_{3,3}^{(4)}c_{1,3}^{(4% )}=-\frac{\kappa(\kappa+2)}{(1+\kappa)^{2}},$

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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 . It has been corrected by writing 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 $d_{1,3}^{(2)}d_{2,3}^{(2)}+d_{2,3}^{(2)}d_{3,3}^{(2)}+d_{3,3}^{(2)}d_{1,3}^{(2% )}=d_{1,3}^{(4)}d_{2,3}^{(4)}+d_{2,3}^{(4)}d_{3,3}^{(4)}+d_{3,3}^{(4)}d_{1,3}^% {(4)}=\kappa(\kappa+2).$

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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 . It has been corrected by writing 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 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 $\left(d_{1,2}^{(2)}\right)^{2}d_{2,2}^{(2)}\pm\left(d_{2,2}^{(2)}\right)^{2}d_% {1,2}^{(2)}=k^{\prime}\left(d_{1,2}^{(2)}\pm d_{2,2}^{(2)}\right),$

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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 $c_{1,2}^{(2)}s_{1,2}^{(2)}d_{2,2}^{(2)}+c_{2,2}^{(2)}s_{2,2}^{(2)}d_{1,2}^{(2)% }=0.$

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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 $s_{1,3}^{(4)}s_{2,3}^{(4)}s_{3,3}^{(4)}=-\frac{1}{1-\kappa^{2}}\left(s_{1,3}^{% (4)}+s_{2,3}^{(4)}+s_{3,3}^{(4)}\right),$

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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 $c_{1,3}^{(4)}c_{2,3}^{(4)}c_{3,3}^{(4)}=\frac{\kappa^{2}}{1-\kappa^{2}}\left(c% _{1,3}^{(4)}+c_{2,3}^{(4)}+c_{3,3}^{(4)}\right),$

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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 $d_{1,3}^{(2)}d_{2,3}^{(2)}d_{3,3}^{(2)}=\frac{\kappa^{2}+k^{2}-1}{1-\kappa^{2}% }\left(d_{1,3}^{(2)}+d_{2,3}^{(2)}+d_{3,3}^{(2)}\right),$

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Symbols:: : 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 $s_{1,3}^{(4)}c_{2,3}^{(4)}c_{3,3}^{(4)}+s_{2,3}^{(4)}c_{3,3}^{(4)}c_{1,3}^{(4)% }+s_{3,3}^{(4)}c_{1,3}^{(4)}c_{2,3}^{(4)}=\frac{\kappa(\kappa+2)}{1-\kappa^{2}% }\left(s_{1,3}^{(4)}+s_{2,3}^{(4)}+s_{3,3}^{(4)}\right).$

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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 $d_{1,4}^{(2)}d_{2,4}^{(2)}d_{3,4}^{(2)}\pm d_{2,4}^{(2)}d_{3,4}^{(2)}d_{4,4}^{% (2)}+d_{3,4}^{(2)}d_{4,4}^{(2)}d_{1,4}^{(2)}\pm d_{4,4}^{(2)}d_{1,4}^{(2)}d_{2% ,4}^{(2)}=k^{\prime}{\left(\pm d_{1,4}^{(2)}+d_{2,4}^{(2)}\pm d_{3,4}^{(2)}+d_% {4,4}^{(2)}\right)},$

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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 $\left(d_{1,4}^{(2)}\right)^{2}d_{3,4}^{(2)}\pm\left(d_{2,4}^{(2)}\right)^{2}d_% {4,4}^{(2)}+\left(d_{3,4}^{(2)}\right)^{2}d_{1,4}^{(2)}\pm\left(d_{4,4}^{(2)}% \right)^{2}d_{2,4}^{(2)}=k^{\prime}{\left(d_{1,4}^{(2)}\pm d_{2,4}^{(2)}+d_{3,% 4}^{(2)}\pm d_{4,4}^{(2)}\right)},$

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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 $c_{1,4}^{(2)}s_{1,4}^{(2)}d_{3,4}^{(2)}+c_{3,4}^{(2)}s_{3,4}^{(2)}d_{1,4}^{(2)% }=c_{2,4}^{(2)}s_{2,4}^{(2)}d_{4,4}^{(2)}+c_{4,4}^{(2)}s_{4,4}^{(2)}d_{2,4}^{(% 2)}=0.$

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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 $\left(d_{1,2}^{(2)}\right)^{3}d_{2,2}^{(2)}\pm\left(d_{2,2}^{(2)}\right)^{3}d_% {1,2}^{(2)}=k^{\prime}\left(\left(d_{1,2}^{(2)}\right)^{2}\pm\left(d_{2,2}^{(2% )}\right)^{2}\right),$

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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 $k^{2}c_{1,2}^{(2)}s_{1,2}^{(2)}c_{2,2}^{(2)}s_{2,2}^{(2)}=k^{\prime}\left(1-% \left(s_{1,2}^{(2)}\right)^{2}-\left(s_{2,2}^{(2)}\right)^{2}\right).$

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Symbols:: : 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 $s_{1,3}^{(2)}c_{1,3}^{(2)}d_{2,3}^{(2)}d_{3,3}^{(2)}+s_{2,3}^{(2)}c_{2,3}^{(2)% }d_{3,3}^{(2)}d_{1,3}^{(2)}+s_{3,3}^{(2)}c_{3,3}^{(2)}d_{1,3}^{(2)}d_{2,3}^{(2% )}=\frac{\kappa^{2}+k^{2}-1}{1-\kappa^{2}}\left(s_{1,3}^{(2)}c_{1,3}^{(2)}+s_{% 2,3}^{(2)}c_{2,3}^{(2)}+s_{3,3}^{(2)}c_{3,3}^{(2)}\right),$

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Symbols:: : 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 $s_{1,3}^{(4)}d_{1,3}^{(4)}c_{2,3}^{(4)}c_{3,3}^{(4)}+s_{2,3}^{(4)}d_{2,3}^{(4)% }c_{3,3}^{(4)}c_{1,3}^{(4)}+s_{3,3}^{(4)}d_{3,3}^{(4)}c_{1,3}^{(4)}c_{2,3}^{(4% )}=\frac{\kappa^{2}}{1-\kappa^{2}}\left(s_{1,3}^{(4)}d_{1,3}^{(4)}+s_{2,3}^{(4% )}d_{2,3}^{(4)}+s_{2,3}^{(4)}d_{2,3}^{(4)}\right).$

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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

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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