View source for 22.20

{{DISPLAYTITLE:Jacobian Elliptic Functions - 22.20 Methods of Computation}}

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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/22.20#Ex1 22.20#Ex1] || [[Item:Q7184|<math>a_{n} = \tfrac{1}{2}\left(a_{n-1}+b_{n-1}\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>a_{n} = \tfrac{1}{2}\left(a_{n-1}+b_{n-1}\right)</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">a[n] = (1)/(2)*(a[n - 1]+ b[n - 1])</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[a, n] == Divide[1,2]*(Subscript[a, n - 1]+ Subscript[b, n - 1])</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/22.20#Ex2 22.20#Ex2] || [[Item:Q7185|<math>b_{n} = \left(a_{n-1}b_{n-1}\right)^{1/2}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>b_{n} = \left(a_{n-1}b_{n-1}\right)^{1/2}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">b[n] = (a[n - 1]*b[n - 1])^(1/2)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[b, n] == (Subscript[a, n - 1]*Subscript[b, n - 1])^(1/2)</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/22.20#Ex3 22.20#Ex3] || [[Item:Q7186|<math>c_{n} = \tfrac{1}{2}\left(a_{n-1}-b_{n-1}\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>c_{n} = \tfrac{1}{2}\left(a_{n-1}-b_{n-1}\right)</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">c[n] = (1)/(2)*(a[n - 1]- b[n - 1])</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[c, n] == Divide[1,2]*(Subscript[a, n - 1]- Subscript[b, n - 1])</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/22.20.E3 22.20.E3] || [[Item:Q7188|<math>\phi_{N} = 2^{N}a_{N}x</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\phi_{N} = 2^{N}a_{N}x</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">phi[N] = (2)^(N)* a[N]*x</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[\[Phi], N] == (2)^(N)* Subscript[a, N]*x</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/22.20.E4 22.20.E4] || [[Item:Q7189|<math>\phi_{n-1} = \frac{1}{2}\left(\phi_{n}+\asin@{\frac{c_{n}}{a_{n}}\sin@@{\phi_{n}}}\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\phi_{n-1} = \frac{1}{2}\left(\phi_{n}+\asin@{\frac{c_{n}}{a_{n}}\sin@@{\phi_{n}}}\right)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>phi[n - 1] = (1)/(2)*(phi[n]+ arcsin((c[n])/(a[n])*sin(phi[n])))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[\[Phi], n - 1] == Divide[1,2]*(Subscript[\[Phi], n]+ ArcSin[Divide[Subscript[c, n],Subscript[a, n]]*Sin[Subscript[\[Phi], n]]])</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [276 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -1.366025404+.3660254040*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, a[n] = 1/2*3^(1/2)+1/2*I, c[n] = 1/2*3^(1/2)+1/2*I, phi[n] = 1/2*3^(1/2)+1/2*I, phi[-1+n] = -1/2+1/2*I*3^(1/2), n = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -1.366025404+.3660254040*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, a[n] = 1/2*3^(1/2)+1/2*I, c[n] = 1/2*3^(1/2)+1/2*I, phi[n] = 1/2*3^(1/2)+1/2*I, phi[-1+n] = -1/2+1/2*I*3^(1/2), n = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [276 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[1.3660254037844384, -0.36602540378443876]
Test Values: {Rule[n, 1], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[a, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[c, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ϕ, Plus[-1, n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ϕ, n], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[1.3660254037844384, -0.36602540378443876]
Test Values: {Rule[n, 2], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[a, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[c, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ϕ, Plus[-1, n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ϕ, n], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/22.20#Ex4 22.20#Ex4] || [[Item:Q7190|<math>\Jacobiellsnk@{x}{k} = \sin@@{\phi_{0}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Jacobiellsnk@{x}{k} = \sin@@{\phi_{0}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>JacobiSN(x, k) = sin(phi[0])</syntaxhighlight> || <syntaxhighlight lang=mathematica>JacobiSN[x, (k)^2] == Sin[Subscript[\[Phi], 0]]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .461679191e-1-.3375964631*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, x = 3/2, phi[0] = 1/2*3^(1/2)+1/2*I, k = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.6784403409-.3375964631*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, x = 3/2, phi[0] = 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[0.046167919344728525, -0.33759646322287]
Test Values: {Rule[k, 1], Rule[x, 1.5], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ϕ, 0], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-0.678440340667692, -0.33759646322287]
Test Values: {Rule[k, 2], Rule[x, 1.5], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ϕ, 0], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/22.20#Ex5 22.20#Ex5] || [[Item:Q7191|<math>\Jacobiellcnk@{x}{k} = \cos@@{\phi_{0}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Jacobiellcnk@{x}{k} = \cos@@{\phi_{0}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>JacobiCN(x, k) = cos(phi[0])</syntaxhighlight> || <syntaxhighlight lang=mathematica>JacobiCN[x, (k)^2] == Cos[Subscript[\[Phi], 0]]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -.3054469840+.3969495503*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, x = 3/2, phi[0] = 1/2*3^(1/2)+1/2*I, k = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .2530246253+.3969495503*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, x = 3/2, phi[0] = 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[-0.3054469841149447, 0.3969495502290325]
Test Values: {Rule[k, 1], Rule[x, 1.5], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ϕ, 0], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[0.2530246251336542, 0.3969495502290325]
Test Values: {Rule[k, 2], Rule[x, 1.5], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ϕ, 0], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/22.20#Ex6 22.20#Ex6] || [[Item:Q7192|<math>\Jacobielldnk@{x}{k} = \frac{\cos@@{\phi_{0}}}{\cos@{\phi_{1}-\phi_{0}}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Jacobielldnk@{x}{k} = \frac{\cos@@{\phi_{0}}}{\cos@{\phi_{1}-\phi_{0}}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>JacobiDN(x, k) = (cos(phi[0]))/(cos(phi[1]- phi[0]))</syntaxhighlight> || <syntaxhighlight lang=mathematica>JacobiDN[x, (k)^2] == Divide[Cos[Subscript[\[Phi], 0]],Cos[Subscript[\[Phi], 1]- Subscript[\[Phi], 0]]]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -.3054469840+.3969495503*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, x = 3/2, phi[0] = 1/2*3^(1/2)+1/2*I, phi[1] = 1/2*3^(1/2)+1/2*I, k = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -1.663077867+.3969495503*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, x = 3/2, phi[0] = 1/2*3^(1/2)+1/2*I, phi[1] = 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[-0.3054469841149447, 0.3969495502290325]
Test Values: {Rule[k, 1], Rule[x, 1.5], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ϕ, 0], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ϕ, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-1.6630778670906836, 0.3969495502290325]
Test Values: {Rule[k, 2], Rule[x, 1.5], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ϕ, 0], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ϕ, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/22.20#Ex7 22.20#Ex7] || [[Item:Q7193|<math>K = \frac{\pi}{2M(1,k^{\prime})}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>K = \frac{\pi}{2M(1,k^{\prime})}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">EllipticK(k) = (Pi)/(2*M(1 ,sqrt(1 - (k)^(2))))</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">EllipticK[(k)^2] == Divide[Pi,2*M[1 ,Sqrt[1 - (k)^(2)]]]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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