Jacobian Elliptic Functions - 22.8 Addition Theorems

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
22.8.E1 sn ⁡ ( u + v ) = sn ⁡ u ⁢ cn ⁡ v ⁢ dn ⁡ v + sn ⁡ v ⁢ cn ⁡ u ⁢ dn ⁡ u 1 - k 2 ⁢ sn 2 ⁡ u ⁢ sn 2 ⁡ v Jacobi-elliptic-sn 𝑢 𝑣 𝑘 Jacobi-elliptic-sn 𝑢 𝑘 Jacobi-elliptic-cn 𝑣 𝑘 Jacobi-elliptic-dn 𝑣 𝑘 Jacobi-elliptic-sn 𝑣 𝑘 Jacobi-elliptic-cn 𝑢 𝑘 Jacobi-elliptic-dn 𝑢 𝑘 1 superscript 𝑘 2 Jacobi-elliptic-sn 2 𝑢 𝑘 Jacobi-elliptic-sn 2 𝑣 𝑘 {\displaystyle{\displaystyle\operatorname{sn}(u+v)=\frac{\operatorname{sn}u% \operatorname{cn}v\operatorname{dn}v+\operatorname{sn}v\operatorname{cn}u% \operatorname{dn}u}{1-k^{2}{\operatorname{sn}^{2}}u{\operatorname{sn}^{2}}v}}}
\Jacobiellsnk@@{(u+v)}{k} = \frac{\Jacobiellsnk@@{u}{k}\Jacobiellcnk@@{v}{k}\Jacobielldnk@@{v}{k}+\Jacobiellsnk@@{v}{k}\Jacobiellcnk@@{u}{k}\Jacobielldnk@@{u}{k}}{1-k^{2}\Jacobiellsnk^{2}@@{u}{k}\Jacobiellsnk^{2}@@{v}{k}}		 

JacobiSN(u + v, k) = (JacobiSN(u, k)*JacobiCN(v, k)*JacobiDN(v, k)+ JacobiSN(v, k)*JacobiCN(u, k)*JacobiDN(u, k))/(1 - (k)^(2)* (JacobiSN(u, k))^(2)* (JacobiSN(v, k))^(2))

JacobiSN[u + v, (k)^2] == Divide[JacobiSN[u, (k)^2]*JacobiCN[v, (k)^2]*JacobiDN[v, (k)^2]+ JacobiSN[v, (k)^2]*JacobiCN[u, (k)^2]*JacobiDN[u, (k)^2],1 - (k)^(2)* (JacobiSN[u, (k)^2])^(2)* (JacobiSN[v, (k)^2])^(2)]
Successful Aborted - Successful [Tested: 300]
22.8.E2 cn ⁡ ( u + v ) = cn ⁡ u ⁢ cn ⁡ v - sn ⁡ u ⁢ dn ⁡ u ⁢ sn ⁡ v ⁢ dn ⁡ v 1 - k 2 ⁢ sn 2 ⁡ u ⁢ sn 2 ⁡ v Jacobi-elliptic-cn 𝑢 𝑣 𝑘 Jacobi-elliptic-cn 𝑢 𝑘 Jacobi-elliptic-cn 𝑣 𝑘 Jacobi-elliptic-sn 𝑢 𝑘 Jacobi-elliptic-dn 𝑢 𝑘 Jacobi-elliptic-sn 𝑣 𝑘 Jacobi-elliptic-dn 𝑣 𝑘 1 superscript 𝑘 2 Jacobi-elliptic-sn 2 𝑢 𝑘 Jacobi-elliptic-sn 2 𝑣 𝑘 {\displaystyle{\displaystyle\operatorname{cn}(u+v)=\frac{\operatorname{cn}u% \operatorname{cn}v-\operatorname{sn}u\operatorname{dn}u\operatorname{sn}v% \operatorname{dn}v}{1-k^{2}{\operatorname{sn}^{2}}u{\operatorname{sn}^{2}}v}}}
\Jacobiellcnk@@{(u+v)}{k} = \frac{\Jacobiellcnk@@{u}{k}\Jacobiellcnk@@{v}{k}-\Jacobiellsnk@@{u}{k}\Jacobielldnk@@{u}{k}\Jacobiellsnk@@{v}{k}\Jacobielldnk@@{v}{k}}{1-k^{2}\Jacobiellsnk^{2}@@{u}{k}\Jacobiellsnk^{2}@@{v}{k}}		 

JacobiCN(u + v, k) = (JacobiCN(u, k)*JacobiCN(v, k)- JacobiSN(u, k)*JacobiDN(u, k)*JacobiSN(v, k)*JacobiDN(v, k))/(1 - (k)^(2)* (JacobiSN(u, k))^(2)* (JacobiSN(v, k))^(2))

JacobiCN[u + v, (k)^2] == Divide[JacobiCN[u, (k)^2]*JacobiCN[v, (k)^2]- JacobiSN[u, (k)^2]*JacobiDN[u, (k)^2]*JacobiSN[v, (k)^2]*JacobiDN[v, (k)^2],1 - (k)^(2)* (JacobiSN[u, (k)^2])^(2)* (JacobiSN[v, (k)^2])^(2)]
Successful Aborted - Successful [Tested: 300]
22.8.E3 dn ⁡ ( u + v ) = dn ⁡ u ⁢ dn ⁡ v - k 2 ⁢ sn ⁡ u ⁢ cn ⁡ u ⁢ sn ⁡ v ⁢ cn ⁡ v 1 - k 2 ⁢ sn 2 ⁡ u ⁢ sn 2 ⁡ v Jacobi-elliptic-dn 𝑢 𝑣 𝑘 Jacobi-elliptic-dn 𝑢 𝑘 Jacobi-elliptic-dn 𝑣 𝑘 superscript 𝑘 2 Jacobi-elliptic-sn 𝑢 𝑘 Jacobi-elliptic-cn 𝑢 𝑘 Jacobi-elliptic-sn 𝑣 𝑘 Jacobi-elliptic-cn 𝑣 𝑘 1 superscript 𝑘 2 Jacobi-elliptic-sn 2 𝑢 𝑘 Jacobi-elliptic-sn 2 𝑣 𝑘 {\displaystyle{\displaystyle\operatorname{dn}(u+v)=\frac{\operatorname{dn}u% \operatorname{dn}v-k^{2}\operatorname{sn}u\operatorname{cn}u\operatorname{sn}v% \operatorname{cn}v}{1-k^{2}{\operatorname{sn}^{2}}u{\operatorname{sn}^{2}}v}}}
\Jacobielldnk@@{(u+v)}{k} = \frac{\Jacobielldnk@@{u}{k}\Jacobielldnk@@{v}{k}-k^{2}\Jacobiellsnk@@{u}{k}\Jacobiellcnk@@{u}{k}\Jacobiellsnk@@{v}{k}\Jacobiellcnk@@{v}{k}}{1-k^{2}\Jacobiellsnk^{2}@@{u}{k}\Jacobiellsnk^{2}@@{v}{k}}		 

JacobiDN(u + v, k) = (JacobiDN(u, k)*JacobiDN(v, k)- (k)^(2)* JacobiSN(u, k)*JacobiCN(u, k)*JacobiSN(v, k)*JacobiCN(v, k))/(1 - (k)^(2)* (JacobiSN(u, k))^(2)* (JacobiSN(v, k))^(2))

JacobiDN[u + v, (k)^2] == Divide[JacobiDN[u, (k)^2]*JacobiDN[v, (k)^2]- (k)^(2)* JacobiSN[u, (k)^2]*JacobiCN[u, (k)^2]*JacobiSN[v, (k)^2]*JacobiCN[v, (k)^2],1 - (k)^(2)* (JacobiSN[u, (k)^2])^(2)* (JacobiSN[v, (k)^2])^(2)]
Successful Aborted - Successful [Tested: 300]
22.8.E4 cd ⁡ ( u + v ) = cd ⁡ u ⁢ cd ⁡ v - k ′ 2 ⁢ sd ⁡ u ⁢ nd ⁡ u ⁢ sd ⁡ v ⁢ nd ⁡ v 1 + k 2 ⁢ k ′ 2 ⁢ sd 2 ⁡ u ⁢ sd 2 ⁡ v Jacobi-elliptic-cd 𝑢 𝑣 𝑘 Jacobi-elliptic-cd 𝑢 𝑘 Jacobi-elliptic-cd 𝑣 𝑘 superscript superscript 𝑘 ′ 2 Jacobi-elliptic-sd 𝑢 𝑘 Jacobi-elliptic-nd 𝑢 𝑘 Jacobi-elliptic-sd 𝑣 𝑘 Jacobi-elliptic-nd 𝑣 𝑘 1 superscript 𝑘 2 superscript superscript 𝑘 ′ 2 Jacobi-elliptic-sd 2 𝑢 𝑘 Jacobi-elliptic-sd 2 𝑣 𝑘 {\displaystyle{\displaystyle\operatorname{cd}(u+v)=\frac{\operatorname{cd}u% \operatorname{cd}v-{k^{\prime}}^{2}\operatorname{sd}u\operatorname{nd}u% \operatorname{sd}v\operatorname{nd}v}{1+k^{2}{k^{\prime}}^{2}{\operatorname{sd% }^{2}}u{\operatorname{sd}^{2}}v}}}
\Jacobiellcdk@@{(u+v)}{k} = \frac{\Jacobiellcdk@@{u}{k}\Jacobiellcdk@@{v}{k}-{k^{\prime}}^{2}\Jacobiellsdk@@{u}{k}\Jacobiellndk@@{u}{k}\Jacobiellsdk@@{v}{k}\Jacobiellndk@@{v}{k}}{1+k^{2}{k^{\prime}}^{2}\Jacobiellsdk^{2}@@{u}{k}\Jacobiellsdk^{2}@@{v}{k}}		 

JacobiCD(u + v, k) = (JacobiCD(u, k)*JacobiCD(v, k)-1 - (k)^(2)*JacobiSD(u, k)*JacobiND(u, k)*JacobiSD(v, k)*JacobiND(v, k))/(1 + (k)^(2)*1 - (k)^(2)*(JacobiSD(u, k))^(2)* (JacobiSD(v, k))^(2))

JacobiCD[u + v, (k)^2] == Divide[JacobiCD[u, (k)^2]*JacobiCD[v, (k)^2]-1 - (k)^(2)*JacobiSD[u, (k)^2]*JacobiND[u, (k)^2]*JacobiSD[v, (k)^2]*JacobiND[v, (k)^2],1 + (k)^(2)*1 - (k)^(2)*(JacobiSD[u, (k)^2])^(2)* (JacobiSD[v, (k)^2])^(2)]
Failure Aborted
Failed [300 / 300]
Result: .6073373021+.4789879505*I
Test Values: {u = 1/2*3^(1/2)+1/2*I, v = 1/2*3^(1/2)+1/2*I, k = 1}


Result: .5744703200+.1556450229*I
Test Values: {u = 1/2*3^(1/2)+1/2*I, v = 1/2*3^(1/2)+1/2*I, k = 2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[0.6073373022896961, 0.47898795042922426]
Test Values: {Rule[k, 1], Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.5744703197186243, 0.15564502146829437]
Test Values: {Rule[k, 2], Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
22.8.E5 sd ⁡ ( u + v ) = sd ⁡ u ⁢ cd ⁡ v ⁢ nd ⁡ v + sd ⁡ v ⁢ cd ⁡ u ⁢ nd ⁡ u 1 + k 2 ⁢ k ′ 2 ⁢ sd 2 ⁡ u ⁢ sd 2 ⁡ v Jacobi-elliptic-sd 𝑢 𝑣 𝑘 Jacobi-elliptic-sd 𝑢 𝑘 Jacobi-elliptic-cd 𝑣 𝑘 Jacobi-elliptic-nd 𝑣 𝑘 Jacobi-elliptic-sd 𝑣 𝑘 Jacobi-elliptic-cd 𝑢 𝑘 Jacobi-elliptic-nd 𝑢 𝑘 1 superscript 𝑘 2 superscript superscript 𝑘 ′ 2 Jacobi-elliptic-sd 2 𝑢 𝑘 Jacobi-elliptic-sd 2 𝑣 𝑘 {\displaystyle{\displaystyle\operatorname{sd}(u+v)=\frac{\operatorname{sd}u% \operatorname{cd}v\operatorname{nd}v+\operatorname{sd}v\operatorname{cd}u% \operatorname{nd}u}{1+k^{2}{k^{\prime}}^{2}{\operatorname{sd}^{2}}u{% \operatorname{sd}^{2}}v}}}
\Jacobiellsdk@@{(u+v)}{k} = \frac{\Jacobiellsdk@@{u}{k}\Jacobiellcdk@@{v}{k}\Jacobiellndk@@{v}{k}+\Jacobiellsdk@@{v}{k}\Jacobiellcdk@@{u}{k}\Jacobiellndk@@{u}{k}}{1+k^{2}{k^{\prime}}^{2}\Jacobiellsdk^{2}@@{u}{k}\Jacobiellsdk^{2}@@{v}{k}}		 

JacobiSD(u + v, k) = (JacobiSD(u, k)*JacobiCD(v, k)*JacobiND(v, k)+ JacobiSD(v, k)*JacobiCD(u, k)*JacobiND(u, k))/(1 + (k)^(2)*1 - (k)^(2)*(JacobiSD(u, k))^(2)* (JacobiSD(v, k))^(2))

JacobiSD[u + v, (k)^2] == Divide[JacobiSD[u, (k)^2]*JacobiCD[v, (k)^2]*JacobiND[v, (k)^2]+ JacobiSD[v, (k)^2]*JacobiCD[u, (k)^2]*JacobiND[u, (k)^2],1 + (k)^(2)*1 - (k)^(2)*(JacobiSD[u, (k)^2])^(2)* (JacobiSD[v, (k)^2])^(2)]
Failure Aborted
Failed [270 / 300]
Result: 1.189544202+1.637439170*I
Test Values: {u = 1/2*3^(1/2)+1/2*I, v = 1/2*3^(1/2)+1/2*I, k = 1}


Result: -.5756484648e-1+.8251147581*I
Test Values: {u = 1/2*3^(1/2)+1/2*I, v = 1/2*3^(1/2)+1/2*I, k = 2}

... skip entries to safe data
Failed [270 / 300]
Result: Complex[1.189544200468709, 1.6374391687321102]
Test Values: {Rule[k, 1], Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.05756484595277844, 0.825114758131751]
Test Values: {Rule[k, 2], Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
22.8.E6 nd ⁡ ( u + v ) = nd ⁡ u ⁢ nd ⁡ v + k 2 ⁢ sd ⁡ u ⁢ cd ⁡ u ⁢ sd ⁡ v ⁢ cd ⁡ v 1 + k 2 ⁢ k ′ 2 ⁢ sd 2 ⁡ u ⁢ sd 2 ⁡ v Jacobi-elliptic-nd 𝑢 𝑣 𝑘 Jacobi-elliptic-nd 𝑢 𝑘 Jacobi-elliptic-nd 𝑣 𝑘 superscript 𝑘 2 Jacobi-elliptic-sd 𝑢 𝑘 Jacobi-elliptic-cd 𝑢 𝑘 Jacobi-elliptic-sd 𝑣 𝑘 Jacobi-elliptic-cd 𝑣 𝑘 1 superscript 𝑘 2 superscript superscript 𝑘 ′ 2 Jacobi-elliptic-sd 2 𝑢 𝑘 Jacobi-elliptic-sd 2 𝑣 𝑘 {\displaystyle{\displaystyle\operatorname{nd}(u+v)=\frac{\operatorname{nd}u% \operatorname{nd}v+k^{2}\operatorname{sd}u\operatorname{cd}u\operatorname{sd}v% \operatorname{cd}v}{1+k^{2}{k^{\prime}}^{2}{\operatorname{sd}^{2}}u{% \operatorname{sd}^{2}}v}}}
\Jacobiellndk@@{(u+v)}{k} = \frac{\Jacobiellndk@@{u}{k}\Jacobiellndk@@{v}{k}+k^{2}\Jacobiellsdk@@{u}{k}\Jacobiellcdk@@{u}{k}\Jacobiellsdk@@{v}{k}\Jacobiellcdk@@{v}{k}}{1+k^{2}{k^{\prime}}^{2}\Jacobiellsdk^{2}@@{u}{k}\Jacobiellsdk^{2}@@{v}{k}}		 

JacobiND(u + v, k) = (JacobiND(u, k)*JacobiND(v, k)+ (k)^(2)* JacobiSD(u, k)*JacobiCD(u, k)*JacobiSD(v, k)*JacobiCD(v, k))/(1 + (k)^(2)*1 - (k)^(2)*(JacobiSD(u, k))^(2)* (JacobiSD(v, k))^(2))

JacobiND[u + v, (k)^2] == Divide[JacobiND[u, (k)^2]*JacobiND[v, (k)^2]+ (k)^(2)* JacobiSD[u, (k)^2]*JacobiCD[u, (k)^2]*JacobiSD[v, (k)^2]*JacobiCD[v, (k)^2],1 + (k)^(2)*1 - (k)^(2)*(JacobiSD[u, (k)^2])^(2)* (JacobiSD[v, (k)^2])^(2)]
Failure Aborted
Failed [300 / 300]
Result: 1.247856974+1.526848242*I
Test Values: {u = 1/2*3^(1/2)+1/2*I, v = 1/2*3^(1/2)+1/2*I, k = 1}


Result: -.1237018962-.8644962079e-1*I
Test Values: {u = 1/2*3^(1/2)+1/2*I, v = 1/2*3^(1/2)+1/2*I, k = 2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[1.2478569728519586, 1.5268482411210251]
Test Values: {Rule[k, 1], Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.1237018961558749, -0.0864496199922923]
Test Values: {Rule[k, 2], Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
22.8.E7 dc ⁡ ( u + v ) = dc ⁡ u ⁢ dc ⁡ v + k ′ 2 ⁢ sc ⁡ u ⁢ nc ⁡ u ⁢ sc ⁡ v ⁢ nc ⁡ v 1 - k ′ 2 ⁢ sc 2 ⁡ u ⁢ sc 2 ⁡ v Jacobi-elliptic-dc 𝑢 𝑣 𝑘 Jacobi-elliptic-dc 𝑢 𝑘 Jacobi-elliptic-dc 𝑣 𝑘 superscript superscript 𝑘 ′ 2 Jacobi-elliptic-sc 𝑢 𝑘 Jacobi-elliptic-nc 𝑢 𝑘 Jacobi-elliptic-sc 𝑣 𝑘 Jacobi-elliptic-nc 𝑣 𝑘 1 superscript superscript 𝑘 ′ 2 Jacobi-elliptic-sc 2 𝑢 𝑘 Jacobi-elliptic-sc 2 𝑣 𝑘 {\displaystyle{\displaystyle\operatorname{dc}(u+v)=\frac{\operatorname{dc}u% \operatorname{dc}v+{k^{\prime}}^{2}\operatorname{sc}u\operatorname{nc}u% \operatorname{sc}v\operatorname{nc}v}{1-{k^{\prime}}^{2}{\operatorname{sc}^{2}% }u{\operatorname{sc}^{2}}v}}}
\Jacobielldck@@{(u+v)}{k} = \frac{\Jacobielldck@@{u}{k}\Jacobielldck@@{v}{k}+{k^{\prime}}^{2}\Jacobiellsck@@{u}{k}\Jacobiellnck@@{u}{k}\Jacobiellsck@@{v}{k}\Jacobiellnck@@{v}{k}}{1-{k^{\prime}}^{2}\Jacobiellsck^{2}@@{u}{k}\Jacobiellsck^{2}@@{v}{k}}		 

JacobiDC(u + v, k) = (JacobiDC(u, k)*JacobiDC(v, k)+1 - (k)^(2)*JacobiSC(u, k)*JacobiNC(u, k)*JacobiSC(v, k)*JacobiNC(v, k))/(1 -1 - (k)^(2)*(JacobiSC(u, k))^(2)* (JacobiSC(v, k))^(2))

JacobiDC[u + v, (k)^2] == Divide[JacobiDC[u, (k)^2]*JacobiDC[v, (k)^2]+1 - (k)^(2)*JacobiSC[u, (k)^2]*JacobiNC[u, (k)^2]*JacobiSC[v, (k)^2]*JacobiNC[v, (k)^2],1 -1 - (k)^(2)*(JacobiSC[u, (k)^2])^(2)* (JacobiSC[v, (k)^2])^(2)]
Failure Aborted
Failed [300 / 300]
Result: -1.456738398+.1506627644*I
Test Values: {u = 1/2*3^(1/2)+1/2*I, v = 1/2*3^(1/2)+1/2*I, k = 1}


Result: -4.350355103-.3722352376e-1*I
Test Values: {u = 1/2*3^(1/2)+1/2*I, v = 1/2*3^(1/2)+1/2*I, k = 2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-1.456738400104645, 0.15066276425673586]
Test Values: {Rule[k, 1], Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-4.350355102633989, -0.03722352327899177]
Test Values: {Rule[k, 2], Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
22.8.E8 nc ⁡ ( u + v ) = nc ⁡ u ⁢ nc ⁡ v + sc ⁡ u ⁢ dc ⁡ u ⁢ sc ⁡ v ⁢ dc ⁡ v 1 - k ′ 2 ⁢ sc 2 ⁡ u ⁢ sc 2 ⁡ v Jacobi-elliptic-nc 𝑢 𝑣 𝑘 Jacobi-elliptic-nc 𝑢 𝑘 Jacobi-elliptic-nc 𝑣 𝑘 Jacobi-elliptic-sc 𝑢 𝑘 Jacobi-elliptic-dc 𝑢 𝑘 Jacobi-elliptic-sc 𝑣 𝑘 Jacobi-elliptic-dc 𝑣 𝑘 1 superscript superscript 𝑘 ′ 2 Jacobi-elliptic-sc 2 𝑢 𝑘 Jacobi-elliptic-sc 2 𝑣 𝑘 {\displaystyle{\displaystyle\operatorname{nc}(u+v)=\frac{\operatorname{nc}u% \operatorname{nc}v+\operatorname{sc}u\operatorname{dc}u\operatorname{sc}v% \operatorname{dc}v}{1-{k^{\prime}}^{2}{\operatorname{sc}^{2}}u{\operatorname{% sc}^{2}}v}}}
\Jacobiellnck@@{(u+v)}{k} = \frac{\Jacobiellnck@@{u}{k}\Jacobiellnck@@{v}{k}+\Jacobiellsck@@{u}{k}\Jacobielldck@@{u}{k}\Jacobiellsck@@{v}{k}\Jacobielldck@@{v}{k}}{1-{k^{\prime}}^{2}\Jacobiellsck^{2}@@{u}{k}\Jacobiellsck^{2}@@{v}{k}}		 

JacobiNC(u + v, k) = (JacobiNC(u, k)*JacobiNC(v, k)+ JacobiSC(u, k)*JacobiDC(u, k)*JacobiSC(v, k)*JacobiDC(v, k))/(1 -1 - (k)^(2)*(JacobiSC(u, k))^(2)* (JacobiSC(v, k))^(2))

JacobiNC[u + v, (k)^2] == Divide[JacobiNC[u, (k)^2]*JacobiNC[v, (k)^2]+ JacobiSC[u, (k)^2]*JacobiDC[u, (k)^2]*JacobiSC[v, (k)^2]*JacobiDC[v, (k)^2],1 -1 - (k)^(2)*(JacobiSC[u, (k)^2])^(2)* (JacobiSC[v, (k)^2])^(2)]
Failure Aborted
Failed [300 / 300]
Result: 1.356171111+.335718656*I
Test Values: {u = 1/2*3^(1/2)+1/2*I, v = 1/2*3^(1/2)+1/2*I, k = 1}


Result: .3210452605+.1984107752*I
Test Values: {u = 1/2*3^(1/2)+1/2*I, v = 1/2*3^(1/2)+1/2*I, k = 2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[1.356171110076661, 0.3357186535359711]
Test Values: {Rule[k, 1], Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.3210452604978905, 0.19841077324251138]
Test Values: {Rule[k, 2], Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
22.8.E9 sc ⁡ ( u + v ) = sc ⁡ u ⁢ dc ⁡ v ⁢ nc ⁡ v + sc ⁡ v ⁢ dc ⁡ u ⁢ nc ⁡ u 1 - k ′ 2 ⁢ sc 2 ⁡ u ⁢ sc 2 ⁡ v Jacobi-elliptic-sc 𝑢 𝑣 𝑘 Jacobi-elliptic-sc 𝑢 𝑘 Jacobi-elliptic-dc 𝑣 𝑘 Jacobi-elliptic-nc 𝑣 𝑘 Jacobi-elliptic-sc 𝑣 𝑘 Jacobi-elliptic-dc 𝑢 𝑘 Jacobi-elliptic-nc 𝑢 𝑘 1 superscript superscript 𝑘 ′ 2 Jacobi-elliptic-sc 2 𝑢 𝑘 Jacobi-elliptic-sc 2 𝑣 𝑘 {\displaystyle{\displaystyle\operatorname{sc}(u+v)=\frac{\operatorname{sc}u% \operatorname{dc}v\operatorname{nc}v+\operatorname{sc}v\operatorname{dc}u% \operatorname{nc}u}{1-{k^{\prime}}^{2}{\operatorname{sc}^{2}}u{\operatorname{% sc}^{2}}v}}}
\Jacobiellsck@@{(u+v)}{k} = \frac{\Jacobiellsck@@{u}{k}\Jacobielldck@@{v}{k}\Jacobiellnck@@{v}{k}+\Jacobiellsck@@{v}{k}\Jacobielldck@@{u}{k}\Jacobiellnck@@{u}{k}}{1-{k^{\prime}}^{2}\Jacobiellsck^{2}@@{u}{k}\Jacobiellsck^{2}@@{v}{k}}		 

JacobiSC(u + v, k) = (JacobiSC(u, k)*JacobiDC(v, k)*JacobiNC(v, k)+ JacobiSC(v, k)*JacobiDC(u, k)*JacobiNC(u, k))/(1 -1 - (k)^(2)*(JacobiSC(u, k))^(2)* (JacobiSC(v, k))^(2))

JacobiSC[u + v, (k)^2] == Divide[JacobiSC[u, (k)^2]*JacobiDC[v, (k)^2]*JacobiNC[v, (k)^2]+ JacobiSC[v, (k)^2]*JacobiDC[u, (k)^2]*JacobiNC[u, (k)^2],1 -1 - (k)^(2)*(JacobiSC[u, (k)^2])^(2)* (JacobiSC[v, (k)^2])^(2)]
Failure Aborted
Failed [270 / 300]
Result: 1.370082581+.423198902*I
Test Values: {u = 1/2*3^(1/2)+1/2*I, v = 1/2*3^(1/2)+1/2*I, k = 1}


Result: .2742031773e-1-2.068263955*I
Test Values: {u = 1/2*3^(1/2)+1/2*I, v = 1/2*3^(1/2)+1/2*I, k = 2}

... skip entries to safe data
Failed [270 / 300]
Result: Complex[1.3700825790735573, 0.42319889849983916]
Test Values: {Rule[k, 1], Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.027420317388659004, -2.068263954207401]
Test Values: {Rule[k, 2], Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
22.8.E10 ns ⁡ ( u + v ) = ns ⁡ u ⁢ ds ⁡ v ⁢ cs ⁡ v - ns ⁡ v ⁢ ds ⁡ u ⁢ cs ⁡ u cs 2 ⁡ v - cs 2 ⁡ u Jacobi-elliptic-ns 𝑢 𝑣 𝑘 Jacobi-elliptic-ns 𝑢 𝑘 Jacobi-elliptic-ds 𝑣 𝑘 Jacobi-elliptic-cs 𝑣 𝑘 Jacobi-elliptic-ns 𝑣 𝑘 Jacobi-elliptic-ds 𝑢 𝑘 Jacobi-elliptic-cs 𝑢 𝑘 Jacobi-elliptic-cs 2 𝑣 𝑘 Jacobi-elliptic-cs 2 𝑢 𝑘 {\displaystyle{\displaystyle\operatorname{ns}(u+v)=\frac{\operatorname{ns}u% \operatorname{ds}v\operatorname{cs}v-\operatorname{ns}v\operatorname{ds}u% \operatorname{cs}u}{{\operatorname{cs}^{2}}v-{\operatorname{cs}^{2}}u}}}
\Jacobiellnsk@@{(u+v)}{k} = \frac{\Jacobiellnsk@@{u}{k}\Jacobielldsk@@{v}{k}\Jacobiellcsk@@{v}{k}-\Jacobiellnsk@@{v}{k}\Jacobielldsk@@{u}{k}\Jacobiellcsk@@{u}{k}}{\Jacobiellcsk^{2}@@{v}{k}-\Jacobiellcsk^{2}@@{u}{k}}		 

JacobiNS(u + v, k) = (JacobiNS(u, k)*JacobiDS(v, k)*JacobiCS(v, k)- JacobiNS(v, k)*JacobiDS(u, k)*JacobiCS(u, k))/((JacobiCS(v, k))^(2)- (JacobiCS(u, k))^(2))

JacobiNS[u + v, (k)^2] == Divide[JacobiNS[u, (k)^2]*JacobiDS[v, (k)^2]*JacobiCS[v, (k)^2]- JacobiNS[v, (k)^2]*JacobiDS[u, (k)^2]*JacobiCS[u, (k)^2],(JacobiCS[v, (k)^2])^(2)- (JacobiCS[u, (k)^2])^(2)]
Successful Aborted -
Failed [60 / 300]
Result: Indeterminate
Test Values: {Rule[k, 1], Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Indeterminate
Test Values: {Rule[k, 2], Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
22.8.E11 ds ⁡ ( u + v ) = ds ⁡ u ⁢ cs ⁡ v ⁢ ns ⁡ v - ds ⁡ v ⁢ cs ⁡ u ⁢ ns ⁡ u cs 2 ⁡ v - cs 2 ⁡ u Jacobi-elliptic-ds 𝑢 𝑣 𝑘 Jacobi-elliptic-ds 𝑢 𝑘 Jacobi-elliptic-cs 𝑣 𝑘 Jacobi-elliptic-ns 𝑣 𝑘 Jacobi-elliptic-ds 𝑣 𝑘 Jacobi-elliptic-cs 𝑢 𝑘 Jacobi-elliptic-ns 𝑢 𝑘 Jacobi-elliptic-cs 2 𝑣 𝑘 Jacobi-elliptic-cs 2 𝑢 𝑘 {\displaystyle{\displaystyle\operatorname{ds}(u+v)=\frac{\operatorname{ds}u% \operatorname{cs}v\operatorname{ns}v-\operatorname{ds}v\operatorname{cs}u% \operatorname{ns}u}{{\operatorname{cs}^{2}}v-{\operatorname{cs}^{2}}u}}}
\Jacobielldsk@@{(u+v)}{k} = \frac{\Jacobielldsk@@{u}{k}\Jacobiellcsk@@{v}{k}\Jacobiellnsk@@{v}{k}-\Jacobielldsk@@{v}{k}\Jacobiellcsk@@{u}{k}\Jacobiellnsk@@{u}{k}}{\Jacobiellcsk^{2}@@{v}{k}-\Jacobiellcsk^{2}@@{u}{k}}		 

JacobiDS(u + v, k) = (JacobiDS(u, k)*JacobiCS(v, k)*JacobiNS(v, k)- JacobiDS(v, k)*JacobiCS(u, k)*JacobiNS(u, k))/((JacobiCS(v, k))^(2)- (JacobiCS(u, k))^(2))

JacobiDS[u + v, (k)^2] == Divide[JacobiDS[u, (k)^2]*JacobiCS[v, (k)^2]*JacobiNS[v, (k)^2]- JacobiDS[v, (k)^2]*JacobiCS[u, (k)^2]*JacobiNS[u, (k)^2],(JacobiCS[v, (k)^2])^(2)- (JacobiCS[u, (k)^2])^(2)]
Successful Aborted -
Failed [60 / 300]
Result: Indeterminate
Test Values: {Rule[k, 1], Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Indeterminate
Test Values: {Rule[k, 2], Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
22.8.E12 cs ⁡ ( u + v ) = cs ⁡ u ⁢ ds ⁡ v ⁢ ns ⁡ v - cs ⁡ v ⁢ ds ⁡ u ⁢ ns ⁡ u cs 2 ⁡ v - cs 2 ⁡ u Jacobi-elliptic-cs 𝑢 𝑣 𝑘 Jacobi-elliptic-cs 𝑢 𝑘 Jacobi-elliptic-ds 𝑣 𝑘 Jacobi-elliptic-ns 𝑣 𝑘 Jacobi-elliptic-cs 𝑣 𝑘 Jacobi-elliptic-ds 𝑢 𝑘 Jacobi-elliptic-ns 𝑢 𝑘 Jacobi-elliptic-cs 2 𝑣 𝑘 Jacobi-elliptic-cs 2 𝑢 𝑘 {\displaystyle{\displaystyle\operatorname{cs}(u+v)=\frac{\operatorname{cs}u% \operatorname{ds}v\operatorname{ns}v-\operatorname{cs}v\operatorname{ds}u% \operatorname{ns}u}{{\operatorname{cs}^{2}}v-{\operatorname{cs}^{2}}u}}}
\Jacobiellcsk@@{(u+v)}{k} = \frac{\Jacobiellcsk@@{u}{k}\Jacobielldsk@@{v}{k}\Jacobiellnsk@@{v}{k}-\Jacobiellcsk@@{v}{k}\Jacobielldsk@@{u}{k}\Jacobiellnsk@@{u}{k}}{\Jacobiellcsk^{2}@@{v}{k}-\Jacobiellcsk^{2}@@{u}{k}}		 

JacobiCS(u + v, k) = (JacobiCS(u, k)*JacobiDS(v, k)*JacobiNS(v, k)- JacobiCS(v, k)*JacobiDS(u, k)*JacobiNS(u, k))/((JacobiCS(v, k))^(2)- (JacobiCS(u, k))^(2))

JacobiCS[u + v, (k)^2] == Divide[JacobiCS[u, (k)^2]*JacobiDS[v, (k)^2]*JacobiNS[v, (k)^2]- JacobiCS[v, (k)^2]*JacobiDS[u, (k)^2]*JacobiNS[u, (k)^2],(JacobiCS[v, (k)^2])^(2)- (JacobiCS[u, (k)^2])^(2)]
Successful Aborted -
Failed [60 / 300]
Result: Indeterminate
Test Values: {Rule[k, 1], Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Indeterminate
Test Values: {Rule[k, 2], Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
22.8.E13 sn ⁡ ( u + v ) = sn 2 ⁡ u - sn 2 ⁡ v sn ⁡ u ⁢ cn ⁡ v ⁢ dn ⁡ v - sn ⁡ v ⁢ cn ⁡ u ⁢ dn ⁡ u Jacobi-elliptic-sn 𝑢 𝑣 𝑘 Jacobi-elliptic-sn 2 𝑢 𝑘 Jacobi-elliptic-sn 2 𝑣 𝑘 Jacobi-elliptic-sn 𝑢 𝑘 Jacobi-elliptic-cn 𝑣 𝑘 Jacobi-elliptic-dn 𝑣 𝑘 Jacobi-elliptic-sn 𝑣 𝑘 Jacobi-elliptic-cn 𝑢 𝑘 Jacobi-elliptic-dn 𝑢 𝑘 {\displaystyle{\displaystyle\operatorname{sn}(u+v)=\frac{{\operatorname{sn}^{2% }}u-{\operatorname{sn}^{2}}v}{\operatorname{sn}u\operatorname{cn}v% \operatorname{dn}v-\operatorname{sn}v\operatorname{cn}u\operatorname{dn}u}}}
\Jacobiellsnk@@{(u+v)}{k} = \frac{\Jacobiellsnk^{2}@@{u}{k}-\Jacobiellsnk^{2}@@{v}{k}}{\Jacobiellsnk@@{u}{k}\Jacobiellcnk@@{v}{k}\Jacobielldnk@@{v}{k}-\Jacobiellsnk@@{v}{k}\Jacobiellcnk@@{u}{k}\Jacobielldnk@@{u}{k}}		 

JacobiSN(u + v, k) = ((JacobiSN(u, k))^(2)- (JacobiSN(v, k))^(2))/(JacobiSN(u, k)*JacobiCN(v, k)*JacobiDN(v, k)- JacobiSN(v, k)*JacobiCN(u, k)*JacobiDN(u, k))

JacobiSN[u + v, (k)^2] == Divide[(JacobiSN[u, (k)^2])^(2)- (JacobiSN[v, (k)^2])^(2),JacobiSN[u, (k)^2]*JacobiCN[v, (k)^2]*JacobiDN[v, (k)^2]- JacobiSN[v, (k)^2]*JacobiCN[u, (k)^2]*JacobiDN[u, (k)^2]]
Successful Aborted -
Failed [30 / 300]
Result: Indeterminate
Test Values: {Rule[k, 1], Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Indeterminate
Test Values: {Rule[k, 2], Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
22.8.E14 sn ⁡ ( u + v ) = sn ⁡ u ⁢ cn ⁡ u ⁢ dn ⁡ v + sn ⁡ v ⁢ cn ⁡ v ⁢ dn ⁡ u cn ⁡ u ⁢ cn ⁡ v + sn ⁡ u ⁢ dn ⁡ u ⁢ sn ⁡ v ⁢ dn ⁡ v Jacobi-elliptic-sn 𝑢 𝑣 𝑘 Jacobi-elliptic-sn 𝑢 𝑘 Jacobi-elliptic-cn 𝑢 𝑘 Jacobi-elliptic-dn 𝑣 𝑘 Jacobi-elliptic-sn 𝑣 𝑘 Jacobi-elliptic-cn 𝑣 𝑘 Jacobi-elliptic-dn 𝑢 𝑘 Jacobi-elliptic-cn 𝑢 𝑘 Jacobi-elliptic-cn 𝑣 𝑘 Jacobi-elliptic-sn 𝑢 𝑘 Jacobi-elliptic-dn 𝑢 𝑘 Jacobi-elliptic-sn 𝑣 𝑘 Jacobi-elliptic-dn 𝑣 𝑘 {\displaystyle{\displaystyle\operatorname{sn}(u+v)=\frac{\operatorname{sn}u% \operatorname{cn}u\operatorname{dn}v+\operatorname{sn}v\operatorname{cn}v% \operatorname{dn}u}{\operatorname{cn}u\operatorname{cn}v+\operatorname{sn}u% \operatorname{dn}u\operatorname{sn}v\operatorname{dn}v}}}
\Jacobiellsnk@@{(u+v)}{k} = \frac{\Jacobiellsnk@@{u}{k}\Jacobiellcnk@@{u}{k}\Jacobielldnk@@{v}{k}+\Jacobiellsnk@@{v}{k}\Jacobiellcnk@@{v}{k}\Jacobielldnk@@{u}{k}}{\Jacobiellcnk@@{u}{k}\Jacobiellcnk@@{v}{k}+\Jacobiellsnk@@{u}{k}\Jacobielldnk@@{u}{k}\Jacobiellsnk@@{v}{k}\Jacobielldnk@@{v}{k}}		 

JacobiSN(u + v, k) = (JacobiSN(u, k)*JacobiCN(u, k)*JacobiDN(v, k)+ JacobiSN(v, k)*JacobiCN(v, k)*JacobiDN(u, k))/(JacobiCN(u, k)*JacobiCN(v, k)+ JacobiSN(u, k)*JacobiDN(u, k)*JacobiSN(v, k)*JacobiDN(v, k))

JacobiSN[u + v, (k)^2] == Divide[JacobiSN[u, (k)^2]*JacobiCN[u, (k)^2]*JacobiDN[v, (k)^2]+ JacobiSN[v, (k)^2]*JacobiCN[v, (k)^2]*JacobiDN[u, (k)^2],JacobiCN[u, (k)^2]*JacobiCN[v, (k)^2]+ JacobiSN[u, (k)^2]*JacobiDN[u, (k)^2]*JacobiSN[v, (k)^2]*JacobiDN[v, (k)^2]]
Successful Aborted - Successful [Tested: 300]
22.8.E15 cn ⁡ ( u + v ) = sn ⁡ u ⁢ cn ⁡ u ⁢ dn ⁡ v - sn ⁡ v ⁢ cn ⁡ v ⁢ dn ⁡ u sn ⁡ u ⁢ cn ⁡ v ⁢ dn ⁡ v - sn ⁡ v ⁢ cn ⁡ u ⁢ dn ⁡ u Jacobi-elliptic-cn 𝑢 𝑣 𝑘 Jacobi-elliptic-sn 𝑢 𝑘 Jacobi-elliptic-cn 𝑢 𝑘 Jacobi-elliptic-dn 𝑣 𝑘 Jacobi-elliptic-sn 𝑣 𝑘 Jacobi-elliptic-cn 𝑣 𝑘 Jacobi-elliptic-dn 𝑢 𝑘 Jacobi-elliptic-sn 𝑢 𝑘 Jacobi-elliptic-cn 𝑣 𝑘 Jacobi-elliptic-dn 𝑣 𝑘 Jacobi-elliptic-sn 𝑣 𝑘 Jacobi-elliptic-cn 𝑢 𝑘 Jacobi-elliptic-dn 𝑢 𝑘 {\displaystyle{\displaystyle\operatorname{cn}(u+v)=\frac{\operatorname{sn}u% \operatorname{cn}u\operatorname{dn}v-\operatorname{sn}v\operatorname{cn}v% \operatorname{dn}u}{\operatorname{sn}u\operatorname{cn}v\operatorname{dn}v-% \operatorname{sn}v\operatorname{cn}u\operatorname{dn}u}}}
\Jacobiellcnk@@{(u+v)}{k} = \frac{\Jacobiellsnk@@{u}{k}\Jacobiellcnk@@{u}{k}\Jacobielldnk@@{v}{k}-\Jacobiellsnk@@{v}{k}\Jacobiellcnk@@{v}{k}\Jacobielldnk@@{u}{k}}{\Jacobiellsnk@@{u}{k}\Jacobiellcnk@@{v}{k}\Jacobielldnk@@{v}{k}-\Jacobiellsnk@@{v}{k}\Jacobiellcnk@@{u}{k}\Jacobielldnk@@{u}{k}}		 

JacobiCN(u + v, k) = (JacobiSN(u, k)*JacobiCN(u, k)*JacobiDN(v, k)- JacobiSN(v, k)*JacobiCN(v, k)*JacobiDN(u, k))/(JacobiSN(u, k)*JacobiCN(v, k)*JacobiDN(v, k)- JacobiSN(v, k)*JacobiCN(u, k)*JacobiDN(u, k))

JacobiCN[u + v, (k)^2] == Divide[JacobiSN[u, (k)^2]*JacobiCN[u, (k)^2]*JacobiDN[v, (k)^2]- JacobiSN[v, (k)^2]*JacobiCN[v, (k)^2]*JacobiDN[u, (k)^2],JacobiSN[u, (k)^2]*JacobiCN[v, (k)^2]*JacobiDN[v, (k)^2]- JacobiSN[v, (k)^2]*JacobiCN[u, (k)^2]*JacobiDN[u, (k)^2]]
Successful Aborted -
Failed [30 / 300]
Result: Indeterminate
Test Values: {Rule[k, 1], Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Indeterminate
Test Values: {Rule[k, 2], Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
22.8.E16 cn ⁡ ( u + v ) = 1 - sn 2 ⁡ u - sn 2 ⁡ v + k 2 ⁢ sn 2 ⁡ u ⁢ sn 2 ⁡ v cn ⁡ u ⁢ cn ⁡ v + sn ⁡ u ⁢ dn ⁡ u ⁢ sn ⁡ v ⁢ dn ⁡ v Jacobi-elliptic-cn 𝑢 𝑣 𝑘 1 Jacobi-elliptic-sn 2 𝑢 𝑘 Jacobi-elliptic-sn 2 𝑣 𝑘 superscript 𝑘 2 Jacobi-elliptic-sn 2 𝑢 𝑘 Jacobi-elliptic-sn 2 𝑣 𝑘 Jacobi-elliptic-cn 𝑢 𝑘 Jacobi-elliptic-cn 𝑣 𝑘 Jacobi-elliptic-sn 𝑢 𝑘 Jacobi-elliptic-dn 𝑢 𝑘 Jacobi-elliptic-sn 𝑣 𝑘 Jacobi-elliptic-dn 𝑣 𝑘 {\displaystyle{\displaystyle\operatorname{cn}(u+v)=\frac{1-{\operatorname{sn}^% {2}}u-{\operatorname{sn}^{2}}v+k^{2}{\operatorname{sn}^{2}}u{\operatorname{sn}% ^{2}}v}{\operatorname{cn}u\operatorname{cn}v+\operatorname{sn}u\operatorname{% dn}u\operatorname{sn}v\operatorname{dn}v}}}
\Jacobiellcnk@@{(u+v)}{k} = \frac{1-\Jacobiellsnk^{2}@@{u}{k}-\Jacobiellsnk^{2}@@{v}{k}+k^{2}\Jacobiellsnk^{2}@@{u}{k}\Jacobiellsnk^{2}@@{v}{k}}{\Jacobiellcnk@@{u}{k}\Jacobiellcnk@@{v}{k}+\Jacobiellsnk@@{u}{k}\Jacobielldnk@@{u}{k}\Jacobiellsnk@@{v}{k}\Jacobielldnk@@{v}{k}}		 

JacobiCN(u + v, k) = (1 - (JacobiSN(u, k))^(2)- (JacobiSN(v, k))^(2)+ (k)^(2)* (JacobiSN(u, k))^(2)* (JacobiSN(v, k))^(2))/(JacobiCN(u, k)*JacobiCN(v, k)+ JacobiSN(u, k)*JacobiDN(u, k)*JacobiSN(v, k)*JacobiDN(v, k))

JacobiCN[u + v, (k)^2] == Divide[1 - (JacobiSN[u, (k)^2])^(2)- (JacobiSN[v, (k)^2])^(2)+ (k)^(2)* (JacobiSN[u, (k)^2])^(2)* (JacobiSN[v, (k)^2])^(2),JacobiCN[u, (k)^2]*JacobiCN[v, (k)^2]+ JacobiSN[u, (k)^2]*JacobiDN[u, (k)^2]*JacobiSN[v, (k)^2]*JacobiDN[v, (k)^2]]
Successful Aborted - Successful [Tested: 300]
22.8.E17 dn ⁡ ( u + v ) = sn ⁡ u ⁢ cn ⁡ v ⁢ dn ⁡ u - sn ⁡ v ⁢ cn ⁡ u ⁢ dn ⁡ v sn ⁡ u ⁢ cn ⁡ v ⁢ dn ⁡ v - sn ⁡ v ⁢ cn ⁡ u ⁢ dn ⁡ u Jacobi-elliptic-dn 𝑢 𝑣 𝑘 Jacobi-elliptic-sn 𝑢 𝑘 Jacobi-elliptic-cn 𝑣 𝑘 Jacobi-elliptic-dn 𝑢 𝑘 Jacobi-elliptic-sn 𝑣 𝑘 Jacobi-elliptic-cn 𝑢 𝑘 Jacobi-elliptic-dn 𝑣 𝑘 Jacobi-elliptic-sn 𝑢 𝑘 Jacobi-elliptic-cn 𝑣 𝑘 Jacobi-elliptic-dn 𝑣 𝑘 Jacobi-elliptic-sn 𝑣 𝑘 Jacobi-elliptic-cn 𝑢 𝑘 Jacobi-elliptic-dn 𝑢 𝑘 {\displaystyle{\displaystyle\operatorname{dn}(u+v)=\frac{\operatorname{sn}u% \operatorname{cn}v\operatorname{dn}u-\operatorname{sn}v\operatorname{cn}u% \operatorname{dn}v}{\operatorname{sn}u\operatorname{cn}v\operatorname{dn}v-% \operatorname{sn}v\operatorname{cn}u\operatorname{dn}u}}}
\Jacobielldnk@@{(u+v)}{k} = \frac{\Jacobiellsnk@@{u}{k}\Jacobiellcnk@@{v}{k}\Jacobielldnk@@{u}{k}-\Jacobiellsnk@@{v}{k}\Jacobiellcnk@@{u}{k}\Jacobielldnk@@{v}{k}}{\Jacobiellsnk@@{u}{k}\Jacobiellcnk@@{v}{k}\Jacobielldnk@@{v}{k}-\Jacobiellsnk@@{v}{k}\Jacobiellcnk@@{u}{k}\Jacobielldnk@@{u}{k}}		 

JacobiDN(u + v, k) = (JacobiSN(u, k)*JacobiCN(v, k)*JacobiDN(u, k)- JacobiSN(v, k)*JacobiCN(u, k)*JacobiDN(v, k))/(JacobiSN(u, k)*JacobiCN(v, k)*JacobiDN(v, k)- JacobiSN(v, k)*JacobiCN(u, k)*JacobiDN(u, k))

JacobiDN[u + v, (k)^2] == Divide[JacobiSN[u, (k)^2]*JacobiCN[v, (k)^2]*JacobiDN[u, (k)^2]- JacobiSN[v, (k)^2]*JacobiCN[u, (k)^2]*JacobiDN[v, (k)^2],JacobiSN[u, (k)^2]*JacobiCN[v, (k)^2]*JacobiDN[v, (k)^2]- JacobiSN[v, (k)^2]*JacobiCN[u, (k)^2]*JacobiDN[u, (k)^2]]
Successful Aborted -
Failed [30 / 300]
Result: Indeterminate
Test Values: {Rule[k, 1], Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Indeterminate
Test Values: {Rule[k, 2], Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
22.8.E18 dn ⁡ ( u + v ) = cn ⁡ u ⁢ dn ⁡ u ⁢ cn ⁡ v ⁢ dn ⁡ v + k ′ 2 ⁢ sn ⁡ u ⁢ sn ⁡ v cn ⁡ u ⁢ cn ⁡ v + sn ⁡ u ⁢ dn ⁡ u ⁢ sn ⁡ v ⁢ dn ⁡ v Jacobi-elliptic-dn 𝑢 𝑣 𝑘 Jacobi-elliptic-cn 𝑢 𝑘 Jacobi-elliptic-dn 𝑢 𝑘 Jacobi-elliptic-cn 𝑣 𝑘 Jacobi-elliptic-dn 𝑣 𝑘 superscript superscript 𝑘 ′ 2 Jacobi-elliptic-sn 𝑢 𝑘 Jacobi-elliptic-sn 𝑣 𝑘 Jacobi-elliptic-cn 𝑢 𝑘 Jacobi-elliptic-cn 𝑣 𝑘 Jacobi-elliptic-sn 𝑢 𝑘 Jacobi-elliptic-dn 𝑢 𝑘 Jacobi-elliptic-sn 𝑣 𝑘 Jacobi-elliptic-dn 𝑣 𝑘 {\displaystyle{\displaystyle\operatorname{dn}(u+v)=\frac{\operatorname{cn}u% \operatorname{dn}u\operatorname{cn}v\operatorname{dn}v+{k^{\prime}}^{2}% \operatorname{sn}u\operatorname{sn}v}{\operatorname{cn}u\operatorname{cn}v+% \operatorname{sn}u\operatorname{dn}u\operatorname{sn}v\operatorname{dn}v}}}
\Jacobielldnk@@{(u+v)}{k} = \frac{\Jacobiellcnk@@{u}{k}\Jacobielldnk@@{u}{k}\Jacobiellcnk@@{v}{k}\Jacobielldnk@@{v}{k}+{k^{\prime}}^{2}\Jacobiellsnk@@{u}{k}\Jacobiellsnk@@{v}{k}}{\Jacobiellcnk@@{u}{k}\Jacobiellcnk@@{v}{k}+\Jacobiellsnk@@{u}{k}\Jacobielldnk@@{u}{k}\Jacobiellsnk@@{v}{k}\Jacobielldnk@@{v}{k}}		 

JacobiDN(u + v, k) = (JacobiCN(u, k)*JacobiDN(u, k)*JacobiCN(v, k)*JacobiDN(v, k)+1 - (k)^(2)*JacobiSN(u, k)*JacobiSN(v, k))/(JacobiCN(u, k)*JacobiCN(v, k)+ JacobiSN(u, k)*JacobiDN(u, k)*JacobiSN(v, k)*JacobiDN(v, k))

JacobiDN[u + v, (k)^2] == Divide[JacobiCN[u, (k)^2]*JacobiDN[u, (k)^2]*JacobiCN[v, (k)^2]*JacobiDN[v, (k)^2]+1 - (k)^(2)*JacobiSN[u, (k)^2]*JacobiSN[v, (k)^2],JacobiCN[u, (k)^2]*JacobiCN[v, (k)^2]+ JacobiSN[u, (k)^2]*JacobiDN[u, (k)^2]*JacobiSN[v, (k)^2]*JacobiDN[v, (k)^2]]
Failure Aborted
Failed [300 / 300]
Result: -.6011182715+.1479228534*I
Test Values: {u = 1/2*3^(1/2)+1/2*I, v = 1/2*3^(1/2)+1/2*I, k = 1}


Result: -2.889107517+1.386528377*I
Test Values: {u = 1/2*3^(1/2)+1/2*I, v = 1/2*3^(1/2)+1/2*I, k = 2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-0.6011182715762831, 0.14792285354183748]
Test Values: {Rule[k, 1], Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-2.8891075280231666, 1.3865283695917823]
Test Values: {Rule[k, 2], Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
22.8.E19 z 1 + z 2 + z 3 + z 4 = 0 subscript 𝑧 1 subscript 𝑧 2 subscript 𝑧 3 subscript 𝑧 4 0 {\displaystyle{\displaystyle z_{1}+z_{2}+z_{3}+z_{4}=0}}
z_{1}+z_{2}+z_{3}+z_{4} = 0		 

z[1]+ z[2]+ z[3]+ z[4] = 0
 
Subscript[z, 1]+ Subscript[z, 2]+ Subscript[z, 3]+ Subscript[z, 4] == 0
 Skipped - no semantic math Skipped - no semantic math - -
22.8.E21 k ′ 2 - k ′ 2 ⁢ k 2 ⁢ sn ⁡ z 1 ⁢ sn ⁡ z 2 ⁢ sn ⁡ z 3 ⁢ sn ⁡ z 4 + k 2 ⁢ cn ⁡ z 1 ⁢ cn ⁡ z 2 ⁢ cn ⁡ z 3 ⁢ cn ⁡ z 4 - dn ⁡ z 1 ⁢ dn ⁡ z 2 ⁢ dn ⁡ z 3 ⁢ dn ⁡ z 4 = 0 superscript superscript 𝑘 ′ 2 superscript superscript 𝑘 ′ 2 superscript 𝑘 2 Jacobi-elliptic-sn subscript 𝑧 1 𝑘 Jacobi-elliptic-sn subscript 𝑧 2 𝑘 Jacobi-elliptic-sn subscript 𝑧 3 𝑘 Jacobi-elliptic-sn subscript 𝑧 4 𝑘 superscript 𝑘 2 Jacobi-elliptic-cn subscript 𝑧 1 𝑘 Jacobi-elliptic-cn subscript 𝑧 2 𝑘 Jacobi-elliptic-cn subscript 𝑧 3 𝑘 Jacobi-elliptic-cn subscript 𝑧 4 𝑘 Jacobi-elliptic-dn subscript 𝑧 1 𝑘 Jacobi-elliptic-dn subscript 𝑧 2 𝑘 Jacobi-elliptic-dn subscript 𝑧 3 𝑘 Jacobi-elliptic-dn subscript 𝑧 4 𝑘 0 {\displaystyle{\displaystyle{k^{\prime}}^{2}-{k^{\prime}}^{2}k^{2}% \operatorname{sn}z_{1}\operatorname{sn}z_{2}\operatorname{sn}z_{3}% \operatorname{sn}z_{4}+k^{2}\operatorname{cn}z_{1}\operatorname{cn}z_{2}% \operatorname{cn}z_{3}\operatorname{cn}z_{4}-\operatorname{dn}z_{1}% \operatorname{dn}z_{2}\operatorname{dn}z_{3}\operatorname{dn}z_{4}=0}}
{k^{\prime}}^{2}-{k^{\prime}}^{2}k^{2}\Jacobiellsnk@@{z_{1}}{k}\Jacobiellsnk@@{z_{2}}{k}\Jacobiellsnk@@{z_{3}}{k}\Jacobiellsnk@@{z_{4}}{k}+k^{2}\Jacobiellcnk@@{z_{1}}{k}\Jacobiellcnk@@{z_{2}}{k}\Jacobiellcnk@@{z_{3}}{k}\Jacobiellcnk@@{z_{4}}{k}-\Jacobielldnk@@{z_{1}}{k}\Jacobielldnk@@{z_{2}}{k}\Jacobielldnk@@{z_{3}}{k}\Jacobielldnk@@{z_{4}}{k} = 0		 

1 - (k)^(2)-1 - (k)^(2)*(k)^(2)* JacobiSN(z[1], k)*JacobiSN(z[2], k)*JacobiSN(z[3], k)*JacobiSN(z[4], k)+ (k)^(2)* JacobiCN(z[1], k)*JacobiCN(z[2], k)*JacobiCN(z[3], k)*JacobiCN(z[4], k)- JacobiDN(z[1], k)*JacobiDN(z[2], k)*JacobiDN(z[3], k)*JacobiDN(z[4], k) = 0

1 - (k)^(2)-1 - (k)^(2)*(k)^(2)* JacobiSN[Subscript[z, 1], (k)^2]*JacobiSN[Subscript[z, 2], (k)^2]*JacobiSN[Subscript[z, 3], (k)^2]*JacobiSN[Subscript[z, 4], (k)^2]+ (k)^(2)* JacobiCN[Subscript[z, 1], (k)^2]*JacobiCN[Subscript[z, 2], (k)^2]*JacobiCN[Subscript[z, 3], (k)^2]*JacobiCN[Subscript[z, 4], (k)^2]- JacobiDN[Subscript[z, 1], (k)^2]*JacobiDN[Subscript[z, 2], (k)^2]*JacobiDN[Subscript[z, 3], (k)^2]*JacobiDN[Subscript[z, 4], (k)^2] == 0
Failure Failure
Failed [300 / 300]
Result: -1.174291399-.4389390377*I
Test Values: {z[1] = 1/2*3^(1/2)+1/2*I, z[2] = 1/2*3^(1/2)+1/2*I, z[3] = 1/2*3^(1/2)+1/2*I, z[4] = 1/2*3^(1/2)+1/2*I, k = 1}


Result: -6.960363418+.5505072293*I
Test Values: {z[1] = 1/2*3^(1/2)+1/2*I, z[2] = 1/2*3^(1/2)+1/2*I, z[3] = 1/2*3^(1/2)+1/2*I, z[4] = 1/2*3^(1/2)+1/2*I, k = 2}

... skip entries to safe data
Failed [2 / 3]
Result: -3.0
Test Values: {Rule[k, 2]}

Result: -8.0
Test Values: {Rule[k, 3]}

22.8.E22 z 1 + z 2 + z 3 + z 4 = 2 ⁢ K ⁡ ( k ) subscript 𝑧 1 subscript 𝑧 2 subscript 𝑧 3 subscript 𝑧 4 2 complete-elliptic-integral-first-kind-K 𝑘 {\displaystyle{\displaystyle z_{1}+z_{2}+z_{3}+z_{4}=2K\left(k\right)}}
z_{1}+z_{2}+z_{3}+z_{4} = 2\compellintKk@{k}		 

z[1]+ z[2]+ z[3]+ z[4] = 2*EllipticK(k)
 
Subscript[z, 1]+ Subscript[z, 2]+ Subscript[z, 3]+ Subscript[z, 4] == 2*EllipticK[(k)^2]
 Failure Failure Error
Failed [300 / 300]
Result: DirectedInfinity[]
Test Values: {Rule[k, 1], Rule[Subscript[z, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[z, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[z, 3], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[z, 4], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Complex[1.7783512603251586, 4.156515647499643]
Test Values: {Rule[k, 2], Rule[Subscript[z, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[z, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[z, 3], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[z, 4], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
22.8.E24 z 1 - z 2 = z 2 - z 3 subscript 𝑧 1 subscript 𝑧 2 subscript 𝑧 2 subscript 𝑧 3 {\displaystyle{\displaystyle z_{1}-z_{2}=z_{2}-z_{3}}}
z_{1}-z_{2} = z_{2}-z_{3}		 

z[1]- z[2] = z[2]- z[3]
 
Subscript[z, 1]- Subscript[z, 2] == Subscript[z, 2]- Subscript[z, 3]
 Failure Failure
Failed [297 / 300]
Result: -1.366025404+.3660254040*I
Test Values: {z[1] = 1/2*3^(1/2)+1/2*I, z[2] = 1/2*3^(1/2)+1/2*I, z[3] = -1/2+1/2*I*3^(1/2)}

Result: -.3660254040-1.366025404*I
Test Values: {z[1] = 1/2*3^(1/2)+1/2*I, z[2] = 1/2*3^(1/2)+1/2*I, z[3] = 1/2-1/2*I*3^(1/2)}

... skip entries to safe data
Failed [297 / 300]
Result: Complex[-1.3660254037844384, 0.36602540378443876]
Test Values: {Rule[Subscript[z, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[z, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[z, 3], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

Result: Complex[-0.3660254037844386, -1.3660254037844386]
Test Values: {Rule[Subscript[z, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[z, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[z, 3], Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]}

... skip entries to safe data
22.8.E24 z 2 - z 3 = 2 3 ⁢ K ⁡ ( k ) subscript 𝑧 2 subscript 𝑧 3 2 3 complete-elliptic-integral-first-kind-K 𝑘 {\displaystyle{\displaystyle z_{2}-z_{3}=\tfrac{2}{3}K\left(k\right)}}
z_{2}-z_{3} = \tfrac{2}{3}\compellintKk@{k}		 

z[2]- z[3] = (2)/(3)*EllipticK(k)
 
Subscript[z, 2]- Subscript[z, 3] == Divide[2,3]*EllipticK[(k)^2]
 Failure Failure Error
Failed [300 / 300]
Result: DirectedInfinity[]
Test Values: {Rule[k, 1], Rule[Subscript[z, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[z, 3], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Complex[-0.561916784937532, 0.7188385491665478]
Test Values: {Rule[k, 2], Rule[Subscript[z, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[z, 3], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
22.8.E26 z 1 - z 2 = z 2 - z 3 subscript 𝑧 1 subscript 𝑧 2 subscript 𝑧 2 subscript 𝑧 3 {\displaystyle{\displaystyle z_{1}-z_{2}=z_{2}-z_{3}}}
z_{1}-z_{2} = z_{2}-z_{3}		 

z[1]- z[2] = z[2]- z[3]
 
Subscript[z, 1]- Subscript[z, 2] == Subscript[z, 2]- Subscript[z, 3]
 Failure Failure
Failed [297 / 300]
Result: -1.366025404+.3660254040*I
Test Values: {z[1] = 1/2*3^(1/2)+1/2*I, z[2] = 1/2*3^(1/2)+1/2*I, z[3] = -1/2+1/2*I*3^(1/2)}

Result: -.3660254040-1.366025404*I
Test Values: {z[1] = 1/2*3^(1/2)+1/2*I, z[2] = 1/2*3^(1/2)+1/2*I, z[3] = 1/2-1/2*I*3^(1/2)}

... skip entries to safe data
Failed [297 / 300]
Result: Complex[-1.3660254037844384, 0.36602540378443876]
Test Values: {Rule[Subscript[z, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[z, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[z, 3], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

Result: Complex[-0.3660254037844386, -1.3660254037844386]
Test Values: {Rule[Subscript[z, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[z, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[z, 3], Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]}

... skip entries to safe data
22.8.E26 z 2 - z 3 = z 3 - z 4 subscript 𝑧 2 subscript 𝑧 3 subscript 𝑧 3 subscript 𝑧 4 {\displaystyle{\displaystyle z_{2}-z_{3}=z_{3}-z_{4}}}
z_{2}-z_{3} = z_{3}-z_{4}		 

z[2]- z[3] = z[3]- z[4]
 
Subscript[z, 2]- Subscript[z, 3] == Subscript[z, 3]- Subscript[z, 4]
 Failure Failure
Failed [297 / 300]
Result: -1.366025404+.3660254040*I
Test Values: {z[2] = 1/2*3^(1/2)+1/2*I, z[3] = 1/2*3^(1/2)+1/2*I, z[4] = -1/2+1/2*I*3^(1/2)}

Result: -.3660254040-1.366025404*I
Test Values: {z[2] = 1/2*3^(1/2)+1/2*I, z[3] = 1/2*3^(1/2)+1/2*I, z[4] = 1/2-1/2*I*3^(1/2)}

... skip entries to safe data
Failed [297 / 300]
Result: Complex[-1.3660254037844384, 0.36602540378443876]
Test Values: {Rule[Subscript[z, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[z, 3], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[z, 4], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

Result: Complex[-0.3660254037844386, -1.3660254037844386]
Test Values: {Rule[Subscript[z, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[z, 3], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[z, 4], Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]}

... skip entries to safe data
22.8.E26 z 3 - z 4 = 1 2 ⁢ K ⁡ ( k ) subscript 𝑧 3 subscript 𝑧 4 1 2 complete-elliptic-integral-first-kind-K 𝑘 {\displaystyle{\displaystyle z_{3}-z_{4}=\tfrac{1}{2}K\left(k\right)}}
z_{3}-z_{4} = \tfrac{1}{2}\compellintKk@{k}		 

z[3]- z[4] = (1)/(2)*EllipticK(k)
 
Subscript[z, 3]- Subscript[z, 4] == Divide[1,2]*EllipticK[(k)^2]
 Failure Failure Error
Failed [300 / 300]
Result: DirectedInfinity[]
Test Values: {Rule[k, 1], Rule[Subscript[z, 3], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[z, 4], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Complex[-0.42143758870314907, 0.5391289118749109]
Test Values: {Rule[k, 2], Rule[Subscript[z, 3], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[z, 4], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
22.8.E27 dn ⁡ z 1 ⁢ dn ⁡ z 3 = dn ⁡ z 2 ⁢ dn ⁡ z 4 Jacobi-elliptic-dn subscript 𝑧 1 𝑘 Jacobi-elliptic-dn subscript 𝑧 3 𝑘 Jacobi-elliptic-dn subscript 𝑧 2 𝑘 Jacobi-elliptic-dn subscript 𝑧 4 𝑘 {\displaystyle{\displaystyle\operatorname{dn}z_{1}\operatorname{dn}z_{3}=% \operatorname{dn}z_{2}\operatorname{dn}z_{4}}}
\Jacobielldnk@@{z_{1}}{k}\Jacobielldnk@@{z_{3}}{k} = \Jacobielldnk@@{z_{2}}{k}\Jacobielldnk@@{z_{4}}{k}		 

JacobiDN(z[1], k)*JacobiDN(z[3], k) = JacobiDN(z[2], k)*JacobiDN(z[4], k)
 
JacobiDN[Subscript[z, 1], (k)^2]*JacobiDN[Subscript[z, 3], (k)^2] == JacobiDN[Subscript[z, 2], (k)^2]*JacobiDN[Subscript[z, 4], (k)^2]
 Failure Failure
Failed [240 / 300]
Result: -.4756423320-.5071574760*I
Test Values: {z[1] = 1/2*3^(1/2)+1/2*I, z[2] = 1/2*3^(1/2)+1/2*I, z[3] = 1/2*3^(1/2)+1/2*I, z[4] = -1/2+1/2*I*3^(1/2), k = 1}

Result: -2.554390475+.7152903744*I
Test Values: {z[1] = 1/2*3^(1/2)+1/2*I, z[2] = 1/2*3^(1/2)+1/2*I, z[3] = 1/2*3^(1/2)+1/2*I, z[4] = -1/2+1/2*I*3^(1/2), k = 2}

... skip entries to safe data
Failed [240 / 300]
Result: Complex[-0.475642332072964, -0.5071574758549496]
Test Values: {Rule[k, 1], Rule[Subscript[z, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[z, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[z, 3], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[z, 4], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

Result: Complex[-2.5543904750381285, 0.715290373196519]
Test Values: {Rule[k, 2], Rule[Subscript[z, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[z, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[z, 3], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[z, 4], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
22.8.E27 dn ⁡ z 2 ⁢ dn ⁡ z 4 = k ′ Jacobi-elliptic-dn subscript 𝑧 2 𝑘 Jacobi-elliptic-dn subscript 𝑧 4 𝑘 superscript 𝑘 ′ {\displaystyle{\displaystyle\operatorname{dn}z_{2}\operatorname{dn}z_{4}=k^{% \prime}}}
\Jacobielldnk@@{z_{2}}{k}\Jacobielldnk@@{z_{4}}{k} = k^{\prime}		 

JacobiDN(z[2], k)*JacobiDN(z[4], k) = sqrt(1 - (k)^(2))
 
JacobiDN[Subscript[z, 2], (k)^2]*JacobiDN[Subscript[z, 4], (k)^2] == Sqrt[1 - (k)^(2)]
 Failure Failure
Failed [300 / 300]
Result: .4314194118-.3859954480*I
Test Values: {z[2] = 1/2*3^(1/2)+1/2*I, z[4] = 1/2*3^(1/2)+1/2*I, k = 1}

Result: -.8474767071-1.646914265*I
Test Values: {z[2] = 1/2*3^(1/2)+1/2*I, z[4] = 1/2*3^(1/2)+1/2*I, k = 2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[0.4314194120331003, -0.3859954480737353]
Test Values: {Rule[k, 1], Rule[Subscript[z, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[z, 4], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Complex[-0.8474767070969642, -1.6469142655565594]
Test Values: {Rule[k, 2], Rule[Subscript[z, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[z, 4], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
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