View source for 22.8

{{DISPLAYTITLE:Jacobian Elliptic Functions - 22.8 Addition Theorems}}

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! scope="col" style="position: sticky; top: 0;" | DLMF 
! scope="col" style="position: sticky; top: 0;" | Formula 
! scope="col" style="position: sticky; top: 0;" | Constraints 
! scope="col" style="position: sticky; top: 0;" | Maple
! scope="col" style="position: sticky; top: 0;" | Mathematica
! scope="col" style="position: sticky; top: 0;" | Symbolic<br>Maple
! scope="col" style="position: sticky; top: 0;" | Symbolic<br>Mathematica
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Maple
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
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| [https://dlmf.nist.gov/22.8.E1 22.8.E1] || <math qid="Q6966">\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}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>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))</syntaxhighlight> || <syntaxhighlight lang=mathematica>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)]</syntaxhighlight> || Successful || Aborted || - || Successful [Tested: 300]
|- 
| [https://dlmf.nist.gov/22.8.E2 22.8.E2] || <math qid="Q6967">\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}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>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))</syntaxhighlight> || <syntaxhighlight lang=mathematica>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)]</syntaxhighlight> || Successful || Aborted || - || Successful [Tested: 300]
|- 
| [https://dlmf.nist.gov/22.8.E3 22.8.E3] || <math qid="Q6968">\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}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>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))</syntaxhighlight> || <syntaxhighlight lang=mathematica>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)]</syntaxhighlight> || Successful || Aborted || - || Successful [Tested: 300]
|- 
| [https://dlmf.nist.gov/22.8.E4 22.8.E4] || <math qid="Q6969">\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}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>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))</syntaxhighlight> || <syntaxhighlight lang=mathematica>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)]</syntaxhighlight> || Failure || Aborted || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/22.8.E5 22.8.E5] || <math qid="Q6970">\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}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>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))</syntaxhighlight> || <syntaxhighlight lang=mathematica>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)]</syntaxhighlight> || Failure || Aborted || <div class="toccolours mw-collapsible mw-collapsed">Failed [270 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [270 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/22.8.E6 22.8.E6] || <math qid="Q6971">\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}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>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))</syntaxhighlight> || <syntaxhighlight lang=mathematica>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)]</syntaxhighlight> || Failure || Aborted || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/22.8.E7 22.8.E7] || <math qid="Q6972">\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}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>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))</syntaxhighlight> || <syntaxhighlight lang=mathematica>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)]</syntaxhighlight> || Failure || Aborted || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/22.8.E8 22.8.E8] || <math qid="Q6973">\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}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>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))</syntaxhighlight> || <syntaxhighlight lang=mathematica>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)]</syntaxhighlight> || Failure || Aborted || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/22.8.E9 22.8.E9] || <math qid="Q6974">\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}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>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))</syntaxhighlight> || <syntaxhighlight lang=mathematica>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)]</syntaxhighlight> || Failure || Aborted || <div class="toccolours mw-collapsible mw-collapsed">Failed [270 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [270 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/22.8.E10 22.8.E10] || <math qid="Q6975">\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}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>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))</syntaxhighlight> || <syntaxhighlight lang=mathematica>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)]</syntaxhighlight> || Successful || Aborted || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [60 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/22.8.E11 22.8.E11] || <math qid="Q6976">\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}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>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))</syntaxhighlight> || <syntaxhighlight lang=mathematica>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)]</syntaxhighlight> || Successful || Aborted || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [60 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/22.8.E12 22.8.E12] || <math qid="Q6977">\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}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>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))</syntaxhighlight> || <syntaxhighlight lang=mathematica>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)]</syntaxhighlight> || Successful || Aborted || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [60 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/22.8.E13 22.8.E13] || <math qid="Q6978">\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}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>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))</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]]</syntaxhighlight> || Successful || Aborted || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [30 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/22.8.E14 22.8.E14] || <math qid="Q6979">\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}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>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))</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]]</syntaxhighlight> || Successful || Aborted || - || Successful [Tested: 300]
|- 
| [https://dlmf.nist.gov/22.8.E15 22.8.E15] || <math qid="Q6980">\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}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>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))</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]]</syntaxhighlight> || Successful || Aborted || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [30 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/22.8.E16 22.8.E16] || <math qid="Q6981">\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}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>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))</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]]</syntaxhighlight> || Successful || Aborted || - || Successful [Tested: 300]
|- 
| [https://dlmf.nist.gov/22.8.E17 22.8.E17] || <math qid="Q6982">\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}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>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))</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]]</syntaxhighlight> || Successful || Aborted || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [30 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/22.8.E18 22.8.E18] || <math qid="Q6983">\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}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>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))</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]]</syntaxhighlight> || Failure || Aborted || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/22.8.E19 22.8.E19] || <math qid="Q6984">z_{1}+z_{2}+z_{3}+z_{4} = 0</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>z_{1}+z_{2}+z_{3}+z_{4} = 0</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">z[1]+ z[2]+ z[3]+ z[4] = 0</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[z, 1]+ Subscript[z, 2]+ Subscript[z, 3]+ Subscript[z, 4] == 0</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
|- 
| [https://dlmf.nist.gov/22.8.E21 22.8.E21] || <math qid="Q6986">{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</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>{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</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>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</syntaxhighlight> || <syntaxhighlight lang=mathematica>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</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -3.0
Test Values: {Rule[k, 2]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -8.0
Test Values: {Rule[k, 3]}</syntaxhighlight><br></div></div>
|- 
| [https://dlmf.nist.gov/22.8.E22 22.8.E22] || <math qid="Q6987">z_{1}+z_{2}+z_{3}+z_{4} = 2\compellintKk@{k}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>z_{1}+z_{2}+z_{3}+z_{4} = 2\compellintKk@{k}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>z[1]+ z[2]+ z[3]+ z[4] = 2*EllipticK(k)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[z, 1]+ Subscript[z, 2]+ Subscript[z, 3]+ Subscript[z, 4] == 2*EllipticK[(k)^2]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/22.8.E24 22.8.E24] || <math qid="Q6989">z_{1}-z_{2} = z_{2}-z_{3}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>z_{1}-z_{2} = z_{2}-z_{3}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>z[1]- z[2] = z[2]- z[3]</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[z, 1]- Subscript[z, 2] == Subscript[z, 2]- Subscript[z, 3]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [297 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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)}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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)}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [297 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/22.8.E24 22.8.E24] || <math qid="Q6989">z_{2}-z_{3} = \tfrac{2}{3}\compellintKk@{k}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>z_{2}-z_{3} = \tfrac{2}{3}\compellintKk@{k}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>z[2]- z[3] = (2)/(3)*EllipticK(k)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[z, 2]- Subscript[z, 3] == Divide[2,3]*EllipticK[(k)^2]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/22.8.E26 22.8.E26] || <math qid="Q6991">z_{1}-z_{2} = z_{2}-z_{3}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>z_{1}-z_{2} = z_{2}-z_{3}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>z[1]- z[2] = z[2]- z[3]</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[z, 1]- Subscript[z, 2] == Subscript[z, 2]- Subscript[z, 3]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [297 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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)}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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)}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [297 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/22.8.E26 22.8.E26] || <math qid="Q6991">z_{2}-z_{3} = z_{3}-z_{4}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>z_{2}-z_{3} = z_{3}-z_{4}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>z[2]- z[3] = z[3]- z[4]</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[z, 2]- Subscript[z, 3] == Subscript[z, 3]- Subscript[z, 4]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [297 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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)}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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)}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [297 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/22.8.E26 22.8.E26] || <math qid="Q6991">z_{3}-z_{4} = \tfrac{1}{2}\compellintKk@{k}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>z_{3}-z_{4} = \tfrac{1}{2}\compellintKk@{k}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>z[3]- z[4] = (1)/(2)*EllipticK(k)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[z, 3]- Subscript[z, 4] == Divide[1,2]*EllipticK[(k)^2]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/22.8.E27 22.8.E27] || <math qid="Q6992">\Jacobielldnk@@{z_{1}}{k}\Jacobielldnk@@{z_{3}}{k} = \Jacobielldnk@@{z_{2}}{k}\Jacobielldnk@@{z_{4}}{k}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Jacobielldnk@@{z_{1}}{k}\Jacobielldnk@@{z_{3}}{k} = \Jacobielldnk@@{z_{2}}{k}\Jacobielldnk@@{z_{4}}{k}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>JacobiDN(z[1], k)*JacobiDN(z[3], k) = JacobiDN(z[2], k)*JacobiDN(z[4], k)</syntaxhighlight> || <syntaxhighlight lang=mathematica>JacobiDN[Subscript[z, 1], (k)^2]*JacobiDN[Subscript[z, 3], (k)^2] == JacobiDN[Subscript[z, 2], (k)^2]*JacobiDN[Subscript[z, 4], (k)^2]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [240 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [240 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- 
| [https://dlmf.nist.gov/22.8.E27 22.8.E27] || <math qid="Q6992">\Jacobielldnk@@{z_{2}}{k}\Jacobielldnk@@{z_{4}}{k} = k^{\prime}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Jacobielldnk@@{z_{2}}{k}\Jacobielldnk@@{z_{4}}{k} = k^{\prime}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>JacobiDN(z[2], k)*JacobiDN(z[4], k) = sqrt(1 - (k)^(2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>JacobiDN[Subscript[z, 2], (k)^2]*JacobiDN[Subscript[z, 4], (k)^2] == Sqrt[1 - (k)^(2)]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|}
</div>