Jacobian Elliptic Functions - 22.9 Cyclic Identities

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
22.9.E1 s m , p ( 2 ) = sn ⁡ ( z + 2 ⁢ p - 1 ⁢ ( m - 1 ) ⁢ K ⁡ ( k ) , k ) superscript subscript 𝑠 𝑚 𝑝 2 Jacobi-elliptic-sn 𝑧 2 superscript 𝑝 1 𝑚 1 complete-elliptic-integral-first-kind-K 𝑘 𝑘 {\displaystyle{\displaystyle s_{m,p}^{(2)}=\operatorname{sn}\left(z+2p^{-1}(m-% 1)K\left(k\right),k\right)}}
s_{m,p}^{(2)} = \Jacobiellsnk@{z+2p^{-1}(m-1)\compellintKk@{k}}{k}		 

(s[m , p])^(2) = JacobiSN(z + 2*(p)^(- 1)*(m - 1)*EllipticK(k), k)

(Subscript[s, m , p])^(2) == JacobiSN[z + 2*(p)^(- 1)*(m - 1)*EllipticK[(k)^2], (k)^2]
Failure Failure Error
Failed [300 / 300]
Result: Indeterminate
Test Values: {Rule[k, 1], Rule[m, 1], Rule[p, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[s, m, p], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Indeterminate
Test Values: {Rule[k, 1], Rule[m, 2], Rule[p, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[s, m, p], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
22.9.E2 c m , p ( 2 ) = cn ⁡ ( z + 2 ⁢ p - 1 ⁢ ( m - 1 ) ⁢ K ⁡ ( k ) , k ) superscript subscript 𝑐 𝑚 𝑝 2 Jacobi-elliptic-cn 𝑧 2 superscript 𝑝 1 𝑚 1 complete-elliptic-integral-first-kind-K 𝑘 𝑘 {\displaystyle{\displaystyle c_{m,p}^{(2)}=\operatorname{cn}\left(z+2p^{-1}(m-% 1)K\left(k\right),k\right)}}
c_{m,p}^{(2)} = \Jacobiellcnk@{z+2p^{-1}(m-1)\compellintKk@{k}}{k}		 

(c[m , p])^(2) = JacobiCN(z + 2*(p)^(- 1)*(m - 1)*EllipticK(k), k)

(Subscript[c, m , p])^(2) == JacobiCN[z + 2*(p)^(- 1)*(m - 1)*EllipticK[(k)^2], (k)^2]
Failure Failure Error
Failed [300 / 300]
Result: Indeterminate
Test Values: {Rule[k, 1], Rule[m, 1], Rule[p, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[c, m, p], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Indeterminate
Test Values: {Rule[k, 1], Rule[m, 2], Rule[p, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[c, m, p], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
22.9.E3 d m , p ( 2 ) = dn ⁡ ( z + 2 ⁢ p - 1 ⁢ ( m - 1 ) ⁢ K ⁡ ( k ) , k ) superscript subscript 𝑑 𝑚 𝑝 2 Jacobi-elliptic-dn 𝑧 2 superscript 𝑝 1 𝑚 1 complete-elliptic-integral-first-kind-K 𝑘 𝑘 {\displaystyle{\displaystyle d_{m,p}^{(2)}=\operatorname{dn}\left(z+2p^{-1}(m-% 1)K\left(k\right),k\right)}}
d_{m,p}^{(2)} = \Jacobielldnk@{z+2p^{-1}(m-1)\compellintKk@{k}}{k}		 

(d[m , p])^(2) = JacobiDN(z + 2*(p)^(- 1)*(m - 1)*EllipticK(k), k)

(Subscript[d, m , p])^(2) == JacobiDN[z + 2*(p)^(- 1)*(m - 1)*EllipticK[(k)^2], (k)^2]
Failure Failure Error
Failed [300 / 300]
Result: Indeterminate
Test Values: {Rule[k, 1], Rule[m, 1], Rule[p, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[d, m, p], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Indeterminate
Test Values: {Rule[k, 1], Rule[m, 2], Rule[p, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[d, m, p], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
22.9.E4 s m , p ( 4 ) = sn ⁡ ( z + 4 ⁢ p - 1 ⁢ ( m - 1 ) ⁢ K ⁡ ( k ) , k ) superscript subscript 𝑠 𝑚 𝑝 4 Jacobi-elliptic-sn 𝑧 4 superscript 𝑝 1 𝑚 1 complete-elliptic-integral-first-kind-K 𝑘 𝑘 {\displaystyle{\displaystyle s_{m,p}^{(4)}=\operatorname{sn}\left(z+4p^{-1}(m-% 1)K\left(k\right),k\right)}}
s_{m,p}^{(4)} = \Jacobiellsnk@{z+4p^{-1}(m-1)\compellintKk@{k}}{k}		 

(s[m , p])^(4) = JacobiSN(z + 4*(p)^(- 1)*(m - 1)*EllipticK(k), k)

(Subscript[s, m , p])^(4) == JacobiSN[z + 4*(p)^(- 1)*(m - 1)*EllipticK[(k)^2], (k)^2]
Failure Failure Error
Failed [300 / 300]
Result: Indeterminate
Test Values: {Rule[k, 1], Rule[m, 1], Rule[p, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[s, m, p], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Indeterminate
Test Values: {Rule[k, 1], Rule[m, 2], Rule[p, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[s, m, p], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
22.9.E5 c m , p ( 4 ) = cn ⁡ ( z + 4 ⁢ p - 1 ⁢ ( m - 1 ) ⁢ K ⁡ ( k ) , k ) superscript subscript 𝑐 𝑚 𝑝 4 Jacobi-elliptic-cn 𝑧 4 superscript 𝑝 1 𝑚 1 complete-elliptic-integral-first-kind-K 𝑘 𝑘 {\displaystyle{\displaystyle c_{m,p}^{(4)}=\operatorname{cn}\left(z+4p^{-1}(m-% 1)K\left(k\right),k\right)}}
c_{m,p}^{(4)} = \Jacobiellcnk@{z+4p^{-1}(m-1)\compellintKk@{k}}{k}		 

(c[m , p])^(4) = JacobiCN(z + 4*(p)^(- 1)*(m - 1)*EllipticK(k), k)

(Subscript[c, m , p])^(4) == JacobiCN[z + 4*(p)^(- 1)*(m - 1)*EllipticK[(k)^2], (k)^2]
Failure Failure Error
Failed [300 / 300]
Result: Indeterminate
Test Values: {Rule[k, 1], Rule[m, 1], Rule[p, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[c, m, p], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Indeterminate
Test Values: {Rule[k, 1], Rule[m, 2], Rule[p, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[c, m, p], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
22.9.E6 d m , p ( 4 ) = dn ⁡ ( z + 4 ⁢ p - 1 ⁢ ( m - 1 ) ⁢ K ⁡ ( k ) , k ) superscript subscript 𝑑 𝑚 𝑝 4 Jacobi-elliptic-dn 𝑧 4 superscript 𝑝 1 𝑚 1 complete-elliptic-integral-first-kind-K 𝑘 𝑘 {\displaystyle{\displaystyle d_{m,p}^{(4)}=\operatorname{dn}\left(z+4p^{-1}(m-% 1)K\left(k\right),k\right)}}
d_{m,p}^{(4)} = \Jacobielldnk@{z+4p^{-1}(m-1)\compellintKk@{k}}{k}		 

(d[m , p])^(4) = JacobiDN(z + 4*(p)^(- 1)*(m - 1)*EllipticK(k), k)

(Subscript[d, m , p])^(4) == JacobiDN[z + 4*(p)^(- 1)*(m - 1)*EllipticK[(k)^2], (k)^2]
Failure Failure Error
Failed [300 / 300]
Result: Indeterminate
Test Values: {Rule[k, 1], Rule[m, 1], Rule[p, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[d, m, p], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Indeterminate
Test Values: {Rule[k, 1], Rule[m, 2], Rule[p, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[d, m, p], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
22.9.E8 s 1 , 3 ( 4 ) ⁢ s 2 , 3 ( 4 ) + s 2 , 3 ( 4 ) ⁢ s 3 , 3 ( 4 ) + s 3 , 3 ( 4 ) ⁢ s 1 , 3 ( 4 ) = κ 2 - 1 k 2 superscript subscript 𝑠 1 3 4 superscript subscript 𝑠 2 3 4 superscript subscript 𝑠 2 3 4 superscript subscript 𝑠 3 3 4 superscript subscript 𝑠 3 3 4 superscript subscript 𝑠 1 3 4 superscript 𝜅 2 1 superscript 𝑘 2 {\displaystyle{\displaystyle 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}}}}
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}}		 

(s[1 , 3])^(4)*(s[2 , 3])^(4)+ (s[2 , 3])^(4)*(s[3 , 3])^(4)+ (s[3 , 3])^(4)*(s[1 , 3])^(4) = ((JacobiDN(2*EllipticK(k)/3, k))^(2)- 1)/((k)^(2))
 
(Subscript[s, 1 , 3])^(4)*(Subscript[s, 2 , 3])^(4)+ (Subscript[s, 2 , 3])^(4)*(Subscript[s, 3 , 3])^(4)+ (Subscript[s, 3 , 3])^(4)*(Subscript[s, 1 , 3])^(4) == Divide[(JacobiDN[2*EllipticK[(k)^2]/3, (k)^2])^(2)- 1,(k)^(2)]
 Skipped - no semantic math Skipped - no semantic math - -
22.9.E9 c 1 , 3 ( 4 ) ⁢ c 2 , 3 ( 4 ) + c 2 , 3 ( 4 ) ⁢ c 3 , 3 ( 4 ) + c 3 , 3 ( 4 ) ⁢ c 1 , 3 ( 4 ) = - κ ⁢ ( κ + 2 ) ( 1 + κ ) 2 superscript subscript 𝑐 1 3 4 superscript subscript 𝑐 2 3 4 superscript subscript 𝑐 2 3 4 superscript subscript 𝑐 3 3 4 superscript subscript 𝑐 3 3 4 superscript subscript 𝑐 1 3 4 𝜅 𝜅 2 superscript 1 𝜅 2 {\displaystyle{\displaystyle 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}}}}
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}}		 

(c[1 , 3])^(4)*(c[2 , 3])^(4)+ (c[2 , 3])^(4)*(c[3 , 3])^(4)+ (c[3 , 3])^(4)*(c[1 , 3])^(4) = -((JacobiDN(2*EllipticK(k)/3, k))*((JacobiDN(2*EllipticK(k)/3, k))+ 2))/((1 +(JacobiDN(2*EllipticK(k)/3, k)))^(2))
 
(Subscript[c, 1 , 3])^(4)*(Subscript[c, 2 , 3])^(4)+ (Subscript[c, 2 , 3])^(4)*(Subscript[c, 3 , 3])^(4)+ (Subscript[c, 3 , 3])^(4)*(Subscript[c, 1 , 3])^(4) == -Divide[(JacobiDN[2*EllipticK[(k)^2]/3, (k)^2])*((JacobiDN[2*EllipticK[(k)^2]/3, (k)^2])+ 2),(1 +(JacobiDN[2*EllipticK[(k)^2]/3, (k)^2]))^(2)]
 Skipped - no semantic math Skipped - no semantic math - -
22.9.E10 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 ) superscript subscript 𝑑 1 3 2 superscript subscript 𝑑 2 3 2 superscript subscript 𝑑 2 3 2 superscript subscript 𝑑 3 3 2 superscript subscript 𝑑 3 3 2 superscript subscript 𝑑 1 3 2 superscript subscript 𝑑 1 3 4 superscript subscript 𝑑 2 3 4 superscript subscript 𝑑 2 3 4 superscript subscript 𝑑 3 3 4 superscript subscript 𝑑 3 3 4 superscript subscript 𝑑 1 3 4 {\displaystyle{\displaystyle 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)}}}
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)}		 

(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)
 
(Subscript[d, 1 , 3])^(2)*(Subscript[d, 2 , 3])^(2)+ (Subscript[d, 2 , 3])^(2)*(Subscript[d, 3 , 3])^(2)+ (Subscript[d, 3 , 3])^(2)*(Subscript[d, 1 , 3])^(2) == (Subscript[d, 1 , 3])^(4)*(Subscript[d, 2 , 3])^(4)+ (Subscript[d, 2 , 3])^(4)*(Subscript[d, 3 , 3])^(4)+ (Subscript[d, 3 , 3])^(4)*(Subscript[d, 1 , 3])^(4)
 Skipped - no semantic math Skipped - no semantic math - -
22.9.E11 ( d 1 , 2 ( 2 ) ) 2 ⁢ d 2 , 2 ( 2 ) + ( d 2 , 2 ( 2 ) ) 2 ⁢ d 1 , 2 ( 2 ) = k ′ ⁢ ( d 1 , 2 ( 2 ) + d 2 , 2 ( 2 ) ) superscript superscript subscript 𝑑 1 2 2 2 superscript subscript 𝑑 2 2 2 superscript superscript subscript 𝑑 2 2 2 2 superscript subscript 𝑑 1 2 2 superscript 𝑘 ′ superscript subscript 𝑑 1 2 2 superscript subscript 𝑑 2 2 2 {\displaystyle{\displaystyle\left(d_{1,2}^{(2)}\right)^{2}d_{2,2}^{(2)}+\left(% d_{2,2}^{(2)}\right)^{2}d_{1,2}^{(2)}=k^{\prime}\left(d_{1,2}^{(2)}+d_{2,2}^{(% 2)}\right)}}
\left(d_{1,2}^{(2)}\right)^{2}d_{2,2}^{(2)}+\left(d_{2,2}^{(2)}\right)^{2}d_{1,2}^{(2)} = k^{\prime}\left(d_{1,2}^{(2)}+ d_{2,2}^{(2)}\right)		 

((d[1 , 2])^(2))^(2)* (d[2 , 2])^(2)+((d[2 , 2])^(2))^(2)* (d[1 , 2])^(2) = sqrt(1 - (k)^(2))*((d[1 , 2])^(2)+ (d[2 , 2])^(2))
 
((Subscript[d, 1 , 2])^(2))^(2)* (Subscript[d, 2 , 2])^(2)+((Subscript[d, 2 , 2])^(2))^(2)* (Subscript[d, 1 , 2])^(2) == Sqrt[1 - (k)^(2)]*((Subscript[d, 1 , 2])^(2)+ (Subscript[d, 2 , 2])^(2))
 Skipped - no semantic math Skipped - no semantic math - -
22.9.E12 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 superscript subscript 𝑐 1 2 2 superscript subscript 𝑠 1 2 2 superscript subscript 𝑑 2 2 2 superscript subscript 𝑐 2 2 2 superscript subscript 𝑠 2 2 2 superscript subscript 𝑑 1 2 2 0 {\displaystyle{\displaystyle 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}}
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		 

(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
 
(Subscript[c, 1 , 2])^(2)*(Subscript[s, 1 , 2])^(2)*(Subscript[d, 2 , 2])^(2)+ (Subscript[c, 2 , 2])^(2)*(Subscript[s, 2 , 2])^(2)*(Subscript[d, 1 , 2])^(2) == 0
 Skipped - no semantic math Skipped - no semantic math - -
22.9.E13 s 1 , 3 ( 4 ) ⁢ s 2 , 3 ( 4 ) ⁢ s 3 , 3 ( 4 ) = - 1 1 - κ 2 ⁢ ( s 1 , 3 ( 4 ) + s 2 , 3 ( 4 ) + s 3 , 3 ( 4 ) ) superscript subscript 𝑠 1 3 4 superscript subscript 𝑠 2 3 4 superscript subscript 𝑠 3 3 4 1 1 superscript 𝜅 2 superscript subscript 𝑠 1 3 4 superscript subscript 𝑠 2 3 4 superscript subscript 𝑠 3 3 4 {\displaystyle{\displaystyle 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)}}
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)		 

(s[1 , 3])^(4)*(s[2 , 3])^(4)*(s[3 , 3])^(4) = -(1)/(1 -(JacobiDN(2*EllipticK(k)/3, k))^(2))*((s[1 , 3])^(4)+ (s[2 , 3])^(4)+ (s[3 , 3])^(4))
 
(Subscript[s, 1 , 3])^(4)*(Subscript[s, 2 , 3])^(4)*(Subscript[s, 3 , 3])^(4) == -Divide[1,1 -(JacobiDN[2*EllipticK[(k)^2]/3, (k)^2])^(2)]*((Subscript[s, 1 , 3])^(4)+ (Subscript[s, 2 , 3])^(4)+ (Subscript[s, 3 , 3])^(4))
 Skipped - no semantic math Skipped - no semantic math - -
22.9.E14 c 1 , 3 ( 4 ) ⁢ c 2 , 3 ( 4 ) ⁢ c 3 , 3 ( 4 ) = κ 2 1 - κ 2 ⁢ ( c 1 , 3 ( 4 ) + c 2 , 3 ( 4 ) + c 3 , 3 ( 4 ) ) superscript subscript 𝑐 1 3 4 superscript subscript 𝑐 2 3 4 superscript subscript 𝑐 3 3 4 superscript 𝜅 2 1 superscript 𝜅 2 superscript subscript 𝑐 1 3 4 superscript subscript 𝑐 2 3 4 superscript subscript 𝑐 3 3 4 {\displaystyle{\displaystyle 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% )}}
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)		 

(c[1 , 3])^(4)*(c[2 , 3])^(4)*(c[3 , 3])^(4) = ((JacobiDN(2*EllipticK(k)/3, k))^(2))/(1 -(JacobiDN(2*EllipticK(k)/3, k))^(2))*((c[1 , 3])^(4)+ (c[2 , 3])^(4)+ (c[3 , 3])^(4))
 
(Subscript[c, 1 , 3])^(4)*(Subscript[c, 2 , 3])^(4)*(Subscript[c, 3 , 3])^(4) == Divide[(JacobiDN[2*EllipticK[(k)^2]/3, (k)^2])^(2),1 -(JacobiDN[2*EllipticK[(k)^2]/3, (k)^2])^(2)]*((Subscript[c, 1 , 3])^(4)+ (Subscript[c, 2 , 3])^(4)+ (Subscript[c, 3 , 3])^(4))
 Skipped - no semantic math Skipped - no semantic math - -
22.9.E15 d 1 , 3 ( 2 ) ⁢ d 2 , 3 ( 2 ) ⁢ d 3 , 3 ( 2 ) = κ 2 + k 2 - 1 1 - κ 2 ⁢ ( d 1 , 3 ( 2 ) + d 2 , 3 ( 2 ) + d 3 , 3 ( 2 ) ) superscript subscript 𝑑 1 3 2 superscript subscript 𝑑 2 3 2 superscript subscript 𝑑 3 3 2 superscript 𝜅 2 superscript 𝑘 2 1 1 superscript 𝜅 2 superscript subscript 𝑑 1 3 2 superscript subscript 𝑑 2 3 2 superscript subscript 𝑑 3 3 2 {\displaystyle{\displaystyle 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)}}
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)		 

(d[1 , 3])^(2)*(d[2 , 3])^(2)*(d[3 , 3])^(2) = ((JacobiDN(2*EllipticK(k)/3, k))^(2)+ (k)^(2)- 1)/(1 -(JacobiDN(2*EllipticK(k)/3, k))^(2))*((d[1 , 3])^(2)+ (d[2 , 3])^(2)+ (d[3 , 3])^(2))
 
(Subscript[d, 1 , 3])^(2)*(Subscript[d, 2 , 3])^(2)*(Subscript[d, 3 , 3])^(2) == Divide[(JacobiDN[2*EllipticK[(k)^2]/3, (k)^2])^(2)+ (k)^(2)- 1,1 -(JacobiDN[2*EllipticK[(k)^2]/3, (k)^2])^(2)]*((Subscript[d, 1 , 3])^(2)+ (Subscript[d, 2 , 3])^(2)+ (Subscript[d, 3 , 3])^(2))
 Skipped - no semantic math Skipped - no semantic math - -
22.9.E16 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 ) = κ ⁢ ( κ + 2 ) 1 - κ 2 ⁢ ( s 1 , 3 ( 4 ) + s 2 , 3 ( 4 ) + s 3 , 3 ( 4 ) ) superscript subscript 𝑠 1 3 4 superscript subscript 𝑐 2 3 4 superscript subscript 𝑐 3 3 4 superscript subscript 𝑠 2 3 4 superscript subscript 𝑐 3 3 4 superscript subscript 𝑐 1 3 4 superscript subscript 𝑠 3 3 4 superscript subscript 𝑐 1 3 4 superscript subscript 𝑐 2 3 4 𝜅 𝜅 2 1 superscript 𝜅 2 superscript subscript 𝑠 1 3 4 superscript subscript 𝑠 2 3 4 superscript subscript 𝑠 3 3 4 {\displaystyle{\displaystyle 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)}}
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)		 

(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) = ((JacobiDN(2*EllipticK(k)/3, k))*((JacobiDN(2*EllipticK(k)/3, k))+ 2))/(1 -(JacobiDN(2*EllipticK(k)/3, k))^(2))*((s[1 , 3])^(4)+ (s[2 , 3])^(4)+ (s[3 , 3])^(4))
 
(Subscript[s, 1 , 3])^(4)*(Subscript[c, 2 , 3])^(4)*(Subscript[c, 3 , 3])^(4)+ (Subscript[s, 2 , 3])^(4)*(Subscript[c, 3 , 3])^(4)*(Subscript[c, 1 , 3])^(4)+ (Subscript[s, 3 , 3])^(4)*(Subscript[c, 1 , 3])^(4)*(Subscript[c, 2 , 3])^(4) == Divide[(JacobiDN[2*EllipticK[(k)^2]/3, (k)^2])*((JacobiDN[2*EllipticK[(k)^2]/3, (k)^2])+ 2),1 -(JacobiDN[2*EllipticK[(k)^2]/3, (k)^2])^(2)]*((Subscript[s, 1 , 3])^(4)+ (Subscript[s, 2 , 3])^(4)+ (Subscript[s, 3 , 3])^(4))
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22.9.E17 d 1 , 4 ( 2 ) ⁢ d 2 , 4 ( 2 ) ⁢ d 3 , 4 ( 2 ) + d 2 , 4 ( 2 ) ⁢ d 3 , 4 ( 2 ) ⁢ d 4 , 4 ( 2 ) + d 3 , 4 ( 2 ) ⁢ d 4 , 4 ( 2 ) ⁢ d 1 , 4 ( 2 ) + d 4 , 4 ( 2 ) ⁢ d 1 , 4 ( 2 ) ⁢ d 2 , 4 ( 2 ) = k ′ ⁢ ( + d 1 , 4 ( 2 ) + d 2 , 4 ( 2 ) + d 3 , 4 ( 2 ) + d 4 , 4 ( 2 ) ) superscript subscript 𝑑 1 4 2 superscript subscript 𝑑 2 4 2 superscript subscript 𝑑 3 4 2 superscript subscript 𝑑 2 4 2 superscript subscript 𝑑 3 4 2 superscript subscript 𝑑 4 4 2 superscript subscript 𝑑 3 4 2 superscript subscript 𝑑 4 4 2 superscript subscript 𝑑 1 4 2 superscript subscript 𝑑 4 4 2 superscript subscript 𝑑 1 4 2 superscript subscript 𝑑 2 4 2 superscript 𝑘 ′ superscript subscript 𝑑 1 4 2 superscript subscript 𝑑 2 4 2 superscript subscript 𝑑 3 4 2 superscript subscript 𝑑 4 4 2 {\displaystyle{\displaystyle d_{1,4}^{(2)}d_{2,4}^{(2)}d_{3,4}^{(2)}+d_{2,4}^{% (2)}d_{3,4}^{(2)}d_{4,4}^{(2)}+d_{3,4}^{(2)}d_{4,4}^{(2)}d_{1,4}^{(2)}+d_{4,4}% ^{(2)}d_{1,4}^{(2)}d_{2,4}^{(2)}=k^{\prime}{\left(+d_{1,4}^{(2)}+d_{2,4}^{(2)}% +d_{3,4}^{(2)}+d_{4,4}^{(2)}\right)}}}
d_{1,4}^{(2)}d_{2,4}^{(2)}d_{3,4}^{(2)}+ d_{2,4}^{(2)}d_{3,4}^{(2)}d_{4,4}^{(2)}+d_{3,4}^{(2)}d_{4,4}^{(2)}d_{1,4}^{(2)}+ d_{4,4}^{(2)}d_{1,4}^{(2)}d_{2,4}^{(2)} = k^{\prime}{\left(+ d_{1,4}^{(2)}+d_{2,4}^{(2)}+ d_{3,4}^{(2)}+d_{4,4}^{(2)}\right)}		 

(d[1 , 4])^(2)*(d[2 , 4])^(2)*(d[3 , 4])^(2)+ (d[2 , 4])^(2)*(d[3 , 4])^(2)*(d[4 , 4])^(2)+ (d[3 , 4])^(2)*(d[4 , 4])^(2)*(d[1 , 4])^(2)+ (d[4 , 4])^(2)*(d[1 , 4])^(2)*(d[2 , 4])^(2) = sqrt(1 - (k)^(2))*(+ (d[1 , 4])^(2)+ (d[2 , 4])^(2)+ (d[3 , 4])^(2)+ (d[4 , 4])^(2))
 
(Subscript[d, 1 , 4])^(2)*(Subscript[d, 2 , 4])^(2)*(Subscript[d, 3 , 4])^(2)+ (Subscript[d, 2 , 4])^(2)*(Subscript[d, 3 , 4])^(2)*(Subscript[d, 4 , 4])^(2)+ (Subscript[d, 3 , 4])^(2)*(Subscript[d, 4 , 4])^(2)*(Subscript[d, 1 , 4])^(2)+ (Subscript[d, 4 , 4])^(2)*(Subscript[d, 1 , 4])^(2)*(Subscript[d, 2 , 4])^(2) == Sqrt[1 - (k)^(2)]*(+ (Subscript[d, 1 , 4])^(2)+ (Subscript[d, 2 , 4])^(2)+ (Subscript[d, 3 , 4])^(2)+ (Subscript[d, 4 , 4])^(2))
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22.9.E18 ( d 1 , 4 ( 2 ) ) 2 ⁢ d 3 , 4 ( 2 ) + ( d 2 , 4 ( 2 ) ) 2 ⁢ d 4 , 4 ( 2 ) + ( d 3 , 4 ( 2 ) ) 2 ⁢ d 1 , 4 ( 2 ) + ( d 4 , 4 ( 2 ) ) 2 ⁢ d 2 , 4 ( 2 ) = k ′ ⁢ ( d 1 , 4 ( 2 ) + d 2 , 4 ( 2 ) + d 3 , 4 ( 2 ) + d 4 , 4 ( 2 ) ) superscript superscript subscript 𝑑 1 4 2 2 superscript subscript 𝑑 3 4 2 superscript superscript subscript 𝑑 2 4 2 2 superscript subscript 𝑑 4 4 2 superscript superscript subscript 𝑑 3 4 2 2 superscript subscript 𝑑 1 4 2 superscript superscript subscript 𝑑 4 4 2 2 superscript subscript 𝑑 2 4 2 superscript 𝑘 ′ superscript subscript 𝑑 1 4 2 superscript subscript 𝑑 2 4 2 superscript subscript 𝑑 3 4 2 superscript subscript 𝑑 4 4 2 {\displaystyle{\displaystyle\left(d_{1,4}^{(2)}\right)^{2}d_{3,4}^{(2)}+\left(% d_{2,4}^{(2)}\right)^{2}d_{4,4}^{(2)}+\left(d_{3,4}^{(2)}\right)^{2}d_{1,4}^{(% 2)}+\left(d_{4,4}^{(2)}\right)^{2}d_{2,4}^{(2)}=k^{\prime}{\left(d_{1,4}^{(2)}% +d_{2,4}^{(2)}+d_{3,4}^{(2)}+d_{4,4}^{(2)}\right)}}}
\left(d_{1,4}^{(2)}\right)^{2}d_{3,4}^{(2)}+\left(d_{2,4}^{(2)}\right)^{2}d_{4,4}^{(2)}+\left(d_{3,4}^{(2)}\right)^{2}d_{1,4}^{(2)}+\left(d_{4,4}^{(2)}\right)^{2}d_{2,4}^{(2)} = k^{\prime}{\left(d_{1,4}^{(2)}+ d_{2,4}^{(2)}+d_{3,4}^{(2)}+ d_{4,4}^{(2)}\right)}		 

((d[1 , 4])^(2))^(2)* (d[3 , 4])^(2)+((d[2 , 4])^(2))^(2)* (d[4 , 4])^(2)+((d[3 , 4])^(2))^(2)* (d[1 , 4])^(2)+((d[4 , 4])^(2))^(2)* (d[2 , 4])^(2) = sqrt(1 - (k)^(2))*((d[1 , 4])^(2)+ (d[2 , 4])^(2)+ (d[3 , 4])^(2)+ (d[4 , 4])^(2))
 
((Subscript[d, 1 , 4])^(2))^(2)* (Subscript[d, 3 , 4])^(2)+((Subscript[d, 2 , 4])^(2))^(2)* (Subscript[d, 4 , 4])^(2)+((Subscript[d, 3 , 4])^(2))^(2)* (Subscript[d, 1 , 4])^(2)+((Subscript[d, 4 , 4])^(2))^(2)* (Subscript[d, 2 , 4])^(2) == Sqrt[1 - (k)^(2)]*((Subscript[d, 1 , 4])^(2)+ (Subscript[d, 2 , 4])^(2)+ (Subscript[d, 3 , 4])^(2)+ (Subscript[d, 4 , 4])^(2))
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22.9.E19 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 ) superscript subscript 𝑐 1 4 2 superscript subscript 𝑠 1 4 2 superscript subscript 𝑑 3 4 2 superscript subscript 𝑐 3 4 2 superscript subscript 𝑠 3 4 2 superscript subscript 𝑑 1 4 2 superscript subscript 𝑐 2 4 2 superscript subscript 𝑠 2 4 2 superscript subscript 𝑑 4 4 2 superscript subscript 𝑐 4 4 2 superscript subscript 𝑠 4 4 2 superscript subscript 𝑑 2 4 2 {\displaystyle{\displaystyle 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)}}}
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)}		 

(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)
 
(Subscript[c, 1 , 4])^(2)*(Subscript[s, 1 , 4])^(2)*(Subscript[d, 3 , 4])^(2)+ (Subscript[c, 3 , 4])^(2)*(Subscript[s, 3 , 4])^(2)*(Subscript[d, 1 , 4])^(2) == (Subscript[c, 2 , 4])^(2)*(Subscript[s, 2 , 4])^(2)*(Subscript[d, 4 , 4])^(2)+ (Subscript[c, 4 , 4])^(2)*(Subscript[s, 4 , 4])^(2)*(Subscript[d, 2 , 4])^(2)
 Skipped - no semantic math Skipped - no semantic math - -
22.9.E20 ( d 1 , 2 ( 2 ) ) 3 ⁢ d 2 , 2 ( 2 ) + ( d 2 , 2 ( 2 ) ) 3 ⁢ d 1 , 2 ( 2 ) = k ′ ⁢ ( ( d 1 , 2 ( 2 ) ) 2 + ( d 2 , 2 ( 2 ) ) 2 ) superscript superscript subscript 𝑑 1 2 2 3 superscript subscript 𝑑 2 2 2 superscript superscript subscript 𝑑 2 2 2 3 superscript subscript 𝑑 1 2 2 superscript 𝑘 ′ superscript superscript subscript 𝑑 1 2 2 2 superscript superscript subscript 𝑑 2 2 2 2 {\displaystyle{\displaystyle\left(d_{1,2}^{(2)}\right)^{3}d_{2,2}^{(2)}+\left(% d_{2,2}^{(2)}\right)^{3}d_{1,2}^{(2)}=k^{\prime}\left(\left(d_{1,2}^{(2)}% \right)^{2}+\left(d_{2,2}^{(2)}\right)^{2}\right)}}
\left(d_{1,2}^{(2)}\right)^{3}d_{2,2}^{(2)}+\left(d_{2,2}^{(2)}\right)^{3}d_{1,2}^{(2)} = k^{\prime}\left(\left(d_{1,2}^{(2)}\right)^{2}+\left(d_{2,2}^{(2)}\right)^{2}\right)		 

((d[1 , 2])^(2))^(3)* (d[2 , 2])^(2)+((d[2 , 2])^(2))^(3)* (d[1 , 2])^(2) = sqrt(1 - (k)^(2))*(((d[1 , 2])^(2))^(2)+((d[2 , 2])^(2))^(2))
 
((Subscript[d, 1 , 2])^(2))^(3)* (Subscript[d, 2 , 2])^(2)+((Subscript[d, 2 , 2])^(2))^(3)* (Subscript[d, 1 , 2])^(2) == Sqrt[1 - (k)^(2)]*(((Subscript[d, 1 , 2])^(2))^(2)+((Subscript[d, 2 , 2])^(2))^(2))
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22.9.E21 k 2 ⁢ c 1 , 2 ( 2 ) ⁢ s 1 , 2 ( 2 ) ⁢ c 2 , 2 ( 2 ) ⁢ s 2 , 2 ( 2 ) = k ′ ⁢ ( 1 - ( s 1 , 2 ( 2 ) ) 2 - ( s 2 , 2 ( 2 ) ) 2 ) superscript 𝑘 2 superscript subscript 𝑐 1 2 2 superscript subscript 𝑠 1 2 2 superscript subscript 𝑐 2 2 2 superscript subscript 𝑠 2 2 2 superscript 𝑘 ′ 1 superscript superscript subscript 𝑠 1 2 2 2 superscript superscript subscript 𝑠 2 2 2 2 {\displaystyle{\displaystyle 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)}}
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)		 

(k)^(2)* (c[1 , 2])^(2)*(s[1 , 2])^(2)*(c[2 , 2])^(2)*(s[2 , 2])^(2) = sqrt(1 - (k)^(2))*(1 -((s[1 , 2])^(2))^(2)-((s[2 , 2])^(2))^(2))
 
(k)^(2)* (Subscript[c, 1 , 2])^(2)*(Subscript[s, 1 , 2])^(2)*(Subscript[c, 2 , 2])^(2)*(Subscript[s, 2 , 2])^(2) == Sqrt[1 - (k)^(2)]*(1 -((Subscript[s, 1 , 2])^(2))^(2)-((Subscript[s, 2 , 2])^(2))^(2))
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22.9.E22 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 ) = κ 2 + k 2 - 1 1 - κ 2 ⁢ ( s 1 , 3 ( 2 ) ⁢ c 1 , 3 ( 2 ) + s 2 , 3 ( 2 ) ⁢ c 2 , 3 ( 2 ) + s 3 , 3 ( 2 ) ⁢ c 3 , 3 ( 2 ) ) superscript subscript 𝑠 1 3 2 superscript subscript 𝑐 1 3 2 superscript subscript 𝑑 2 3 2 superscript subscript 𝑑 3 3 2 superscript subscript 𝑠 2 3 2 superscript subscript 𝑐 2 3 2 superscript subscript 𝑑 3 3 2 superscript subscript 𝑑 1 3 2 superscript subscript 𝑠 3 3 2 superscript subscript 𝑐 3 3 2 superscript subscript 𝑑 1 3 2 superscript subscript 𝑑 2 3 2 superscript 𝜅 2 superscript 𝑘 2 1 1 superscript 𝜅 2 superscript subscript 𝑠 1 3 2 superscript subscript 𝑐 1 3 2 superscript subscript 𝑠 2 3 2 superscript subscript 𝑐 2 3 2 superscript subscript 𝑠 3 3 2 superscript subscript 𝑐 3 3 2 {\displaystyle{\displaystyle 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)}}
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)		 

(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) = ((JacobiDN(2*EllipticK(k)/3, k))^(2)+ (k)^(2)- 1)/(1 -(JacobiDN(2*EllipticK(k)/3, k))^(2))*((s[1 , 3])^(2)*(c[1 , 3])^(2)+ (s[2 , 3])^(2)*(c[2 , 3])^(2)+ (s[3 , 3])^(2)*(c[3 , 3])^(2))
 
(Subscript[s, 1 , 3])^(2)*(Subscript[c, 1 , 3])^(2)*(Subscript[d, 2 , 3])^(2)*(Subscript[d, 3 , 3])^(2)+ (Subscript[s, 2 , 3])^(2)*(Subscript[c, 2 , 3])^(2)*(Subscript[d, 3 , 3])^(2)*(Subscript[d, 1 , 3])^(2)+ (Subscript[s, 3 , 3])^(2)*(Subscript[c, 3 , 3])^(2)*(Subscript[d, 1 , 3])^(2)*(Subscript[d, 2 , 3])^(2) == Divide[(JacobiDN[2*EllipticK[(k)^2]/3, (k)^2])^(2)+ (k)^(2)- 1,1 -(JacobiDN[2*EllipticK[(k)^2]/3, (k)^2])^(2)]*((Subscript[s, 1 , 3])^(2)*(Subscript[c, 1 , 3])^(2)+ (Subscript[s, 2 , 3])^(2)*(Subscript[c, 2 , 3])^(2)+ (Subscript[s, 3 , 3])^(2)*(Subscript[c, 3 , 3])^(2))
 Skipped - no semantic math Skipped - no semantic math - -
22.9.E23 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 ) = κ 2 1 - κ 2 ⁢ ( s 1 , 3 ( 4 ) ⁢ d 1 , 3 ( 4 ) + s 2 , 3 ( 4 ) ⁢ d 2 , 3 ( 4 ) + s 2 , 3 ( 4 ) ⁢ d 2 , 3 ( 4 ) ) superscript subscript 𝑠 1 3 4 superscript subscript 𝑑 1 3 4 superscript subscript 𝑐 2 3 4 superscript subscript 𝑐 3 3 4 superscript subscript 𝑠 2 3 4 superscript subscript 𝑑 2 3 4 superscript subscript 𝑐 3 3 4 superscript subscript 𝑐 1 3 4 superscript subscript 𝑠 3 3 4 superscript subscript 𝑑 3 3 4 superscript subscript 𝑐 1 3 4 superscript subscript 𝑐 2 3 4 superscript 𝜅 2 1 superscript 𝜅 2 superscript subscript 𝑠 1 3 4 superscript subscript 𝑑 1 3 4 superscript subscript 𝑠 2 3 4 superscript subscript 𝑑 2 3 4 superscript subscript 𝑠 2 3 4 superscript subscript 𝑑 2 3 4 {\displaystyle{\displaystyle 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)}}
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)		 

(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) = ((JacobiDN(2*EllipticK(k)/3, k))^(2))/(1 -(JacobiDN(2*EllipticK(k)/3, k))^(2))*((s[1 , 3])^(4)*(d[1 , 3])^(4)+ (s[2 , 3])^(4)*(d[2 , 3])^(4)+ (s[2 , 3])^(4)*(d[2 , 3])^(4))
 
(Subscript[s, 1 , 3])^(4)*(Subscript[d, 1 , 3])^(4)*(Subscript[c, 2 , 3])^(4)*(Subscript[c, 3 , 3])^(4)+ (Subscript[s, 2 , 3])^(4)*(Subscript[d, 2 , 3])^(4)*(Subscript[c, 3 , 3])^(4)*(Subscript[c, 1 , 3])^(4)+ (Subscript[s, 3 , 3])^(4)*(Subscript[d, 3 , 3])^(4)*(Subscript[c, 1 , 3])^(4)*(Subscript[c, 2 , 3])^(4) == Divide[(JacobiDN[2*EllipticK[(k)^2]/3, (k)^2])^(2),1 -(JacobiDN[2*EllipticK[(k)^2]/3, (k)^2])^(2)]*((Subscript[s, 1 , 3])^(4)*(Subscript[d, 1 , 3])^(4)+ (Subscript[s, 2 , 3])^(4)*(Subscript[d, 2 , 3])^(4)+ (Subscript[s, 2 , 3])^(4)*(Subscript[d, 2 , 3])^(4))
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