View source for 22.18

{{DISPLAYTITLE:Jacobian Elliptic Functions - 22.18 Mathematical Applications}}

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! scope="col" style="position: sticky; top: 0;" | Formula 
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! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
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| [https://dlmf.nist.gov/22.18.E1 22.18.E1] || <math qid="Q7164">\left(x^{2}/a^{2}\right)+\left(y^{2}/b^{2}\right) = 1</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\left(x^{2}/a^{2}\right)+\left(y^{2}/b^{2}\right) = 1</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">((x)^(2)/(a)^(2))+((y)^(2)/(b)^(2)) = 1</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">((x)^(2)/(a)^(2))+((y)^(2)/(b)^(2)) == 1</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/22.18#Ex1 22.18#Ex1] || <math qid="Q7165">x = a\Jacobiellsnk@{u}{k}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>x = a\Jacobiellsnk@{u}{k}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>x = a*JacobiSN(u, k)</syntaxhighlight> || <syntaxhighlight lang=mathematica>x == a*JacobiSN[u, (k)^2]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 2.688604135+.3653402056*I
Test Values: {a = -3/2, u = 1/2*3^(1/2)+1/2*I, x = 3/2, k = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 2.519684790-.2348240643e-1*I
Test Values: {a = -3/2, u = 1/2*3^(1/2)+1/2*I, x = 3/2, 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[2.688604134627054, 0.3653402057357006]
Test Values: {Rule[a, -1.5], Rule[k, 1], Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[2.5196847900911203, -0.02348240620400443]
Test Values: {Rule[a, -1.5], Rule[k, 2], Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/22.18#Ex2 22.18#Ex2] || <math qid="Q7166">y = b\Jacobiellcnk@{u}{k}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>y = b\Jacobiellcnk@{u}{k}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>y = b*JacobiCN(u, k)</syntaxhighlight> || <syntaxhighlight lang=mathematica>y == b*JacobiCN[u, (k)^2]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -.433885744-.4073155167*I
Test Values: {b = -3/2, u = 1/2*3^(1/2)+1/2*I, y = -3/2, k = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.399423925+.2175647210e-1*I
Test Values: {b = -3/2, u = 1/2*3^(1/2)+1/2*I, y = -3/2, 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[-0.43388574383527945, -0.40731551667372035]
Test Values: {Rule[b, -1.5], Rule[k, 1], Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[y, -1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-0.39942392524307424, 0.021756471897004394]
Test Values: {Rule[b, -1.5], Rule[k, 2], Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[y, -1.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/22.18.E4 22.18.E4] || <math qid="Q7168">l(r) = (1/\sqrt{2})\aJacobiellcnk@{r}{1/\sqrt{2}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>l(r) = (1/\sqrt{2})\aJacobiellcnk@{r}{1/\sqrt{2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>l(r) = (1/(sqrt(2)))*InverseJacobiCN(r, 1/(sqrt(2)))</syntaxhighlight> || <syntaxhighlight lang=mathematica>l[r] == (1/(Sqrt[2]))*InverseJacobiCN[r, (1/(Sqrt[2]))^2]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [18 / 18]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -4.122057553+.6299669258*I
Test Values: {r = -3/2, l = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -5.622057553+.6299669258*I
Test Values: {r = -3/2, l = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [18 / 18]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-4.12205755429212, 0.629966925905157]
Test Values: {Rule[l, 1], Rule[r, -1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-5.62205755429212, 0.629966925905157]
Test Values: {Rule[l, 2], Rule[r, -1.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/22.18.E5 22.18.E5] || <math qid="Q7169">r = \Jacobiellcnk@{\sqrt{2}l}{1/\sqrt{2}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>r = \Jacobiellcnk@{\sqrt{2}l}{1/\sqrt{2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>r = JacobiCN(sqrt(2)*l, 1/(sqrt(2)))</syntaxhighlight> || <syntaxhighlight lang=mathematica>r == JacobiCN[Sqrt[2]*l, (1/(Sqrt[2]))^2]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [18 / 18]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -1.810737930
Test Values: {r = -3/2, l = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.8262668012
Test Values: {r = -3/2, l = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [18 / 18]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -1.810737930333856
Test Values: {Rule[l, 1], Rule[r, -1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -0.8262668010254658
Test Values: {Rule[l, 2], Rule[r, -1.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/22.18#Ex3 22.18#Ex3] || <math qid="Q7170">x = \Jacobiellcnk@{\sqrt{2}l}{1/\sqrt{2}}\Jacobielldnk@{\sqrt{2}l}{1/\sqrt{2}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>x = \Jacobiellcnk@{\sqrt{2}l}{1/\sqrt{2}}\Jacobielldnk@{\sqrt{2}l}{1/\sqrt{2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>x = JacobiCN(sqrt(2)*l, 1/(sqrt(2)))*JacobiDN(sqrt(2)*l, 1/(sqrt(2)))</syntaxhighlight> || <syntaxhighlight lang=mathematica>x == JacobiCN[Sqrt[2]*l, (1/(Sqrt[2]))^2]*JacobiDN[Sqrt[2]*l, (1/(Sqrt[2]))^2]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [9 / 9]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 1.269911408
Test Values: {x = 3/2, l = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 2.074437352
Test Values: {x = 3/2, l = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [9 / 9]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 1.2699114077583538
Test Values: {Rule[l, 1], Rule[x, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 2.0744373520381156
Test Values: {Rule[l, 2], Rule[x, 1.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/22.18.E7 22.18.E7] || <math qid="Q7172">ax^{2}y^{2}+b(x^{2}y+xy^{2})+c(x^{2}+y^{2})+2dxy+e(x+y)+f = 0</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>ax^{2}y^{2}+b(x^{2}y+xy^{2})+c(x^{2}+y^{2})+2dxy+e(x+y)+f = 0</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">a*(x)^(2)* (y)^(2)+ b*((x)^(2)* y + x*(y)^(2))+ c*((x)^(2)+ (y)^(2))+ 2*d*x*y + exp(1)*(x + y)+ f = 0</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">a*(x)^(2)* (y)^(2)+ b*((x)^(2)* y + x*(y)^(2))+ c*((x)^(2)+ (y)^(2))+ 2*d*x*y + E*(x + y)+ f == 0</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/22.18#Ex5 22.18#Ex5] || <math qid="Q7173">x_{3} = \frac{x_{1}y_{2}+x_{2}y_{1}}{1-k^{2}x_{1}^{2}x_{2}^{2}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>x_{3} = \frac{x_{1}y_{2}+x_{2}y_{1}}{1-k^{2}x_{1}^{2}x_{2}^{2}}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">x[3] = (x[1]*y[2]+ x[2]*y[1])/(1 - (k)^(2)* (x[1])^(2)*(x[2])^(2))</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[x, 3] == Divide[Subscript[x, 1]*Subscript[y, 2]+ Subscript[x, 2]*Subscript[y, 1],1 - (k)^(2)* (Subscript[x, 1])^(2)*(Subscript[x, 2])^(2)]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/22.18#Ex6 22.18#Ex6] || <math qid="Q7174">y_{3} = \frac{y_{1}y_{2}+x_{2}(-(1+k^{2})x_{1}+2k^{2}x_{1}^{3})}{1-k^{2}x_{1}^{2}x_{2}^{2}}+x_{3}\frac{2k^{2}x_{1}y_{1}x_{2}^{2}}{1-k^{2}x_{1}^{2}x_{2}^{2}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>y_{3} = \frac{y_{1}y_{2}+x_{2}(-(1+k^{2})x_{1}+2k^{2}x_{1}^{3})}{1-k^{2}x_{1}^{2}x_{2}^{2}}+x_{3}\frac{2k^{2}x_{1}y_{1}x_{2}^{2}}{1-k^{2}x_{1}^{2}x_{2}^{2}}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">y[3] = (y[1]*y[2]+ x[2]*(-(1 + (k)^(2))*x[1]+ 2*(k)^(2)* (x[1])^(3)))/(1 - (k)^(2)* (x[1])^(2)*(x[2])^(2))+ x[3]*(2*(k)^(2)* x[1]*y[1]*(x[2])^(2))/(1 - (k)^(2)* (x[1])^(2)*(x[2])^(2))</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[y, 3] == Divide[Subscript[y, 1]*Subscript[y, 2]+ Subscript[x, 2]*(-(1 + (k)^(2))*Subscript[x, 1]+ 2*(k)^(2)* (Subscript[x, 1])^(3)),1 - (k)^(2)* (Subscript[x, 1])^(2)*(Subscript[x, 2])^(2)]+ Subscript[x, 3]*Divide[2*(k)^(2)* Subscript[x, 1]*Subscript[y, 1]*(Subscript[x, 2])^(2),1 - (k)^(2)* (Subscript[x, 1])^(2)*(Subscript[x, 2])^(2)]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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