View source for 3.11

{{DISPLAYTITLE:Numerical Methods - 3.11 Approximation Techniques}}

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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/3.11.E5 3.11.E5] || <math qid="Q1446">\sum_{k=0}^{n}x_{j}^{k}\delta a_{k} = (-1)^{j}(m_{j}-m)</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\sum_{k=0}^{n}x_{j}^{k}\delta a_{k} = (-1)^{j}(m_{j}-m)</syntaxhighlight> || <math>j = 0</math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">sum((x[j])^(k)*delta*a[k], k = 0..n) = (- 1)^(j)*(((- 1)^(j)* epsilon[n](x[j]))- m)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Sum[(Subscript[x, j])^(k)*\[Delta]*Subscript[a, k], {k, 0, n}, GenerateConditions->None] == (- 1)^(j)*(((- 1)^(j)* Subscript[\[Epsilon], n][Subscript[x, j]])- m)</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/3.11.E6 3.11.E6] || <math qid="Q1447">\ChebyshevpolyT{n}@{x} = \cos@{n\acos@@{x}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\ChebyshevpolyT{n}@{x} = \cos@{n\acos@@{x}}</syntaxhighlight> || <math>-1 \leq x, x \leq 1</math> || <syntaxhighlight lang=mathematica>ChebyshevT(n, x) = cos(n*arccos(x))</syntaxhighlight> || <syntaxhighlight lang=mathematica>ChebyshevT[n, x] == Cos[n*ArcCos[x]]</syntaxhighlight> || Failure || Successful || Successful [Tested: 3] || Successful [Tested: 3]
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| [https://dlmf.nist.gov/3.11.E7 3.11.E7] || <math qid="Q1448">\ChebyshevpolyT{n+1}@{x}-2x\ChebyshevpolyT{n}@{x}+\ChebyshevpolyT{n-1}@{x} = 0</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\ChebyshevpolyT{n+1}@{x}-2x\ChebyshevpolyT{n}@{x}+\ChebyshevpolyT{n-1}@{x} = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>ChebyshevT(n + 1, x)- 2*x*ChebyshevT(n, x)+ ChebyshevT(n - 1, x) = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>ChebyshevT[n + 1, x]- 2*x*ChebyshevT[n, x]+ ChebyshevT[n - 1, x] == 0</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 3]
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| [https://dlmf.nist.gov/3.11.E13 3.11.E13] || <math qid="Q1454">\epsilon_{n}(x) = d_{n+1}\ChebyshevpolyT{n+1}@{\frac{2x-a-b}{b-a}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\epsilon_{n}(x) = d_{n+1}\ChebyshevpolyT{n+1}@{\frac{2x-a-b}{b-a}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>epsilon[n](x) = d[n + 1]*ChebyshevT(n + 1, (2*x - a - b)/(b - a))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[\[Epsilon], n][x] == Subscript[d, n + 1]*ChebyshevT[n + 1, Divide[2*x - a - b,b - a]]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Float(-infinity)-Float(infinity)*I
Test Values: {a = -1.5, b = -1.5, x = 1.5, d[n+1] = 1/2*3^(1/2)+1/2*I, epsilon[n] = 1/2*3^(1/2)+1/2*I, epsilon = 1, n = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Float(-infinity)-Float(infinity)*I
Test Values: {a = -1.5, b = -1.5, x = 1.5, d[n+1] = 1/2*3^(1/2)+1/2*I, epsilon[n] = 1/2*3^(1/2)+1/2*I, epsilon = 1, n = 2}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Float(-infinity)-Float(infinity)*I
Test Values: {a = -1.5, b = -1.5, x = 1.5, d[n+1] = 1/2*3^(1/2)+1/2*I, epsilon[n] = 1/2*3^(1/2)+1/2*I, epsilon = 1, n = 3}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Float(-infinity)-Float(infinity)*I
Test Values: {a = -1.5, b = -1.5, x = 1.5, d[n+1] = 1/2*3^(1/2)+1/2*I, epsilon[n] = 1/2*3^(1/2)+1/2*I, epsilon = 2, n = 1}</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: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[n, 1], Rule[x, 1.5], Rule[ϵ, 1], Rule[Subscript[d, Plus[1, n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ϵ, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[n, 1], Rule[x, 1.5], Rule[ϵ, 2], Rule[Subscript[d, Plus[1, n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ϵ, n], 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/3.11.E15 3.11.E15] || <math qid="Q1456">u_{k} = 2xu_{k+1}-u_{k+2}+c_{k}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>u_{k} = 2xu_{k+1}-u_{k+2}+c_{k}</syntaxhighlight> || <math>k = n-1</math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">u[k] = 2*x*u[k + 1]- u[k + 2]+ c[k]</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[u, k] == 2*x*Subscript[u, k + 1]- Subscript[u, k + 2]+ Subscript[c, k]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/3.11.E18 3.11.E18] || <math qid="Q1459">m_{j} = (-1)^{j}\epsilon_{k,\ell}(x_{j})</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>m_{j} = (-1)^{j}\epsilon_{k,\ell}(x_{j})</syntaxhighlight> || <math>j = 0</math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">((- 1)^(j)* epsilon[n](x[j])) = (- 1)^(j)* epsilon[k , ell](x[j])</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">((- 1)^(j)* Subscript[\[Epsilon], n][Subscript[x, j]]) == (- 1)^(j)* Subscript[\[Epsilon], k , \[ScriptL]][Subscript[x, j]]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/3.11.E19 3.11.E19] || <math qid="Q1460">R_{3,3}(x) = \frac{p_{0}+p_{1}x+p_{2}x^{2}+p_{3}x^{3}}{1+q_{1}x+q_{2}x^{2}+q_{3}x^{3}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>R_{3,3}(x) = \frac{p_{0}+p_{1}x+p_{2}x^{2}+p_{3}x^{3}}{1+q_{1}x+q_{2}x^{2}+q_{3}x^{3}}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">R[3 , 3](x) = (p[0]+ p[1]*x + p[2]*(x)^(2)+ p[3]*(x)^(3))/(1 + q[1]*x + q[2]*(x)^(2)+ q[3]*(x)^(3))</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[R, 3 , 3][x] == Divide[Subscript[p, 0]+ Subscript[p, 1]*x + Subscript[p, 2]*(x)^(2)+ Subscript[p, 3]*(x)^(3),1 + Subscript[q, 1]*x + Subscript[q, 2]*(x)^(2)+ Subscript[q, 3]*(x)^(3)]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/3.11#E23X 3.11#E23X] || <math qid="Q1464">\displaystyle a_{0} = c_{0}b_{0}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\displaystyle a_{0} = c_{0}b_{0}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">a[0] = c[0]*b[0]</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[a, 0] == Subscript[c, 0]*Subscript[b, 0]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/3.11#E23Xa 3.11#E23Xa] || <math qid="Q1465">\displaystyle a_{1} = c_{1}b_{0}+c_{0}b_{1}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\displaystyle a_{1} = c_{1}b_{0}+c_{0}b_{1}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">a[1] = c[1]*b[0]+ c[0]*b[1]</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[a, 1] == Subscript[c, 1]*Subscript[b, 0]+ Subscript[c, 0]*Subscript[b, 1]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/3.11.E25 3.11.E25] || <math qid="Q1472">(N-C)^{-1}+(S-C)^{-1} = (W-C)^{-1}+(E-C)^{-1}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>(N-C)^{-1}+(S-C)^{-1} = (W-C)^{-1}+(E-C)^{-1}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(N - C)^(- 1)+(S - C)^(- 1) = (W - C)^(- 1)+(E - C)^(- 1)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(N - C)^(- 1)+(S - C)^(- 1) == (W - C)^(- 1)+(E - C)^(- 1)</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/3.11.E34 3.11.E34] || <math qid="Q1482">X_{k\ell} = \sum_{j=1}^{J}w(x_{j})\phi_{k}(x_{j})\phi_{\ell}(x_{j})</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>X_{k\ell} = \sum_{j=1}^{J}w(x_{j})\phi_{k}(x_{j})\phi_{\ell}(x_{j})</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">X[k*ell] = sum(w(x[j])* phi[k](x[j])* phi[ell](x[j]), j = 1..J)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[X, k*\[ScriptL]] == Sum[w[Subscript[x, j]]* Subscript[\[Phi], k][Subscript[x, j]]* Subscript[\[Phi], \[ScriptL]][Subscript[x, j]], {j, 1, J}, GenerateConditions->None]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/3.11.E37 3.11.E37] || <math qid="Q1485">\sum_{j=0}^{n-1}\phi_{k}(x_{j})\conj{\phi_{\ell}(x_{j})} = n\Kroneckerdelta{k}{\ell}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{j=0}^{n-1}\phi_{k}(x_{j})\conj{\phi_{\ell}(x_{j})} = n\Kroneckerdelta{k}{\ell}</syntaxhighlight> || <math>k = 0, \ell = 0</math> || <syntaxhighlight lang=mathematica>sum(phi[k](x[j])* conjugate(phi[ell](x[j])), j = 0..n - 1) = n*KroneckerDelta[k, ell]</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[Subscript[\[Phi], k][Subscript[x, j]]* Conjugate[Subscript[\[Phi], \[ScriptL]][Subscript[x, j]]], {j, 0, n - 1}, GenerateConditions->None] == n*KroneckerDelta[k, \[ScriptL]]</syntaxhighlight> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[1.0, Times[-1.0, KroneckerDelta[1.0, ℓ]]]
Test Values: {Rule[k, 1], Rule[n, 1], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[x, j], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ϕ, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ϕ, ℓ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[2.0, Times[-2.0, KroneckerDelta[1.0, ℓ]]]
Test Values: {Rule[k, 1], Rule[n, 2], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[x, j], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ϕ, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ϕ, ℓ], 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/3.11.E38 3.11.E38] || <math qid="Q1486">f_{j} = \sum_{k=0}^{n-1}a_{k}\phi_{k}(x_{j})</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>f_{j} = \sum_{k=0}^{n-1}a_{k}\phi_{k}(x_{j})</syntaxhighlight> || <math>j = 0</math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">f[j] = sum(a[k]*phi[k](x[j]), k = 0..n - 1)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[f, j] == Sum[Subscript[a, k]*Subscript[\[Phi], k][Subscript[x, j]], {k, 0, n - 1}, GenerateConditions->None]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/3.11.E39 3.11.E39] || <math qid="Q1487">a_{k} = \frac{1}{n}\sum_{j=0}^{n-1}f_{j}\conj{\phi_{k}(x_{j})}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>a_{k} = \frac{1}{n}\sum_{j=0}^{n-1}f_{j}\conj{\phi_{k}(x_{j})}</syntaxhighlight> || <math>k = 0</math> || <syntaxhighlight lang=mathematica>a[k] = (1)/(n)*sum(f[j]*conjugate(phi[k](x[j])), j = 0..n - 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[a, k] == Divide[1,n]*Sum[Subscript[f, j]*Conjugate[Subscript[\[Phi], k][Subscript[x, j]]], {j, 0, n - 1}, GenerateConditions->None]</syntaxhighlight> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.0, 0.9999999999999999]
Test Values: {Rule[k, 1], Rule[n, 1], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[a, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, j], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[x, j], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ϕ, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[0.0, 0.9999999999999999]
Test Values: {Rule[k, 1], Rule[n, 2], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[a, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, j], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[x, j], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ϕ, k], 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/3.11#Ex5 3.11#Ex5] || <math qid="Q1490">f_{j} = \sum_{k=0}^{n-1}a_{k}\omega_{n}^{jk}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>f_{j} = \sum_{k=0}^{n-1}a_{k}\omega_{n}^{jk}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">f[j] = sum(a[k]*(omega[n])^(j*k), k = 0..n - 1)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[f, j] == Sum[Subscript[a, k]*(Subscript[\[Omega], n])^(j*k), {k, 0, n - 1}, GenerateConditions->None]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/3.11#Ex6 3.11#Ex6] || <math qid="Q1491">\omega_{n} = e^{2\cpi i/n}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\omega_{n} = e^{2\cpi i/n}</syntaxhighlight> || <math>j = 0</math> || <syntaxhighlight lang=mathematica>omega[n] = exp(2*Pi*I/n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[\[Omega], n] == Exp[2*Pi*I/n]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [290 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -.1339745960+.4999999992*I
Test Values: {omega = 1/2*3^(1/2)+1/2*I, omega[n] = 1/2*3^(1/2)+1/2*I, n = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 1.866025404+.5000000004*I
Test Values: {omega = 1/2*3^(1/2)+1/2*I, omega[n] = 1/2*3^(1/2)+1/2*I, n = 2}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 1.366025404-.3660254040*I
Test Values: {omega = 1/2*3^(1/2)+1/2*I, omega[n] = 1/2*3^(1/2)+1/2*I, n = 3}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -1.500000000+.8660254032*I
Test Values: {omega = 1/2*3^(1/2)+1/2*I, omega[n] = -1/2+1/2*I*3^(1/2), n = 1}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [290 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-0.1339745962155613, 0.49999999999999994]
Test Values: {Rule[n, 1], Rule[ω, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ω, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[1.8660254037844388, 0.49999999999999994]
Test Values: {Rule[n, 2], Rule[ω, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ω, n], 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/3.11.E42 3.11.E42] || <math qid="Q1492">\omega_{n}^{2(k-(n/2))} = \omega_{n/2}^{k}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\omega_{n}^{2(k-(n/2))} = \omega_{n/2}^{k}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(omega[n])^(2*(k -(n/2))) = (omega[n/2])^(k)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(Subscript[\[Omega], n])^(2*(k -(n/2))) == (Subscript[\[Omega], n/2])^(k)</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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