2.6: Difference between revisions

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{{DISPLAYTITLE:Asymptotic Approximations - 2.6 Distributional Methods}}
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! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
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| [https://dlmf.nist.gov/2.6.E1 2.6.E1] || [[Item:Q826|<math>S(x) = \int_{0}^{\infty}\frac{1}{(1+t)^{1/3}(x+t)}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>S(x) = \int_{0}^{\infty}\frac{1}{(1+t)^{1/3}(x+t)}\diff{t}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>S(x) = int((1)/((1 + t)^(1/3)*(x + t)), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>S[x] == Integrate[Divide[1,(1 + t)^(1/3)*(x + t)], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [30 / 30]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -1.405894671+.7500000000*I
| [https://dlmf.nist.gov/2.6.E1 2.6.E1] || <math qid="Q826">S(x) = \int_{0}^{\infty}\frac{1}{(1+t)^{1/3}(x+t)}\diff{t}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>S(x) = \int_{0}^{\infty}\frac{1}{(1+t)^{1/3}(x+t)}\diff{t}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>S(x) = int((1)/((1 + t)^(1/3)*(x + t)), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>S[x] == Integrate[Divide[1,(1 + t)^(1/3)*(x + t)], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [30 / 30]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -1.405894671+.7500000000*I
Test Values: {S = 1/2*3^(1/2)+1/2*I, x = 1.5}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -3.111297626+.2500000000*I
Test Values: {S = 1/2*3^(1/2)+1/2*I, x = 1.5}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -3.111297626+.2500000000*I
Test Values: {S = 1/2*3^(1/2)+1/2*I, x = .5}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.774895738+1.*I
Test Values: {S = 1/2*3^(1/2)+1/2*I, x = .5}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.774895738+1.*I
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Test Values: {Rule[S, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
Test Values: {Rule[S, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/2.6.E2 2.6.E2] || [[Item:Q827|<math>(1+t)^{-1/3} = \sum_{s=0}^{\infty}\binom{-\frac{1}{3}}{s}t^{-s-(1/3)}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>(1+t)^{-1/3} = \sum_{s=0}^{\infty}\binom{-\frac{1}{3}}{s}t^{-s-(1/3)}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(1 + t)^(- 1/3) = sum(binomial(-(1)/(3),s)*(t)^(- s -(1/3)), s = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(1 + t)^(- 1/3) == Sum[Binomial[-Divide[1,3],s]*(t)^(- s -(1/3)), {s, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [1 / 6]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Float(infinity)+Float(infinity)*I
| [https://dlmf.nist.gov/2.6.E2 2.6.E2] || <math qid="Q827">(1+t)^{-1/3} = \sum_{s=0}^{\infty}\binom{-\frac{1}{3}}{s}t^{-s-(1/3)}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>(1+t)^{-1/3} = \sum_{s=0}^{\infty}\binom{-\frac{1}{3}}{s}t^{-s-(1/3)}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(1 + t)^(- 1/3) = sum(binomial(-(1)/(3),s)*(t)^(- s -(1/3)), s = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(1 + t)^(- 1/3) == Sum[Binomial[-Divide[1,3],s]*(t)^(- s -(1/3)), {s, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [1 / 6]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Float(infinity)+Float(infinity)*I
Test Values: {t = -.5}</syntaxhighlight><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [1 / 6]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[1.8898815748423095, 1.0911236359717216]
Test Values: {t = -.5}</syntaxhighlight><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [1 / 6]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[1.8898815748423095, 1.0911236359717216]
Test Values: {Rule[t, -0.5]}</syntaxhighlight><br></div></div>
Test Values: {Rule[t, -0.5]}</syntaxhighlight><br></div></div>
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| [https://dlmf.nist.gov/2.6.E4 2.6.E4] || [[Item:Q829|<math>\int_{0}^{\infty}\frac{t^{\alpha-1}}{(x+t)^{\alpha+\beta}}\diff{t} = \frac{\EulerGamma@{\alpha}\EulerGamma@{\beta}}{\EulerGamma@{\alpha+\beta}}\frac{1}{x^{\beta}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\infty}\frac{t^{\alpha-1}}{(x+t)^{\alpha+\beta}}\diff{t} = \frac{\EulerGamma@{\alpha}\EulerGamma@{\beta}}{\EulerGamma@{\alpha+\beta}}\frac{1}{x^{\beta}}</syntaxhighlight> || <math>\realpart@@{\alpha} > 0, \realpart@@{\beta} > 0, \realpart@@{(\alpha)} > 0, \realpart@@{(\beta)} > 0, \realpart@@{(\alpha+\beta)} > 0</math> || <syntaxhighlight lang=mathematica>int(((t)^(alpha - 1))/((x + t)^(alpha + beta)), t = 0..infinity) = (GAMMA(alpha)*GAMMA(beta))/(GAMMA(alpha + beta))*(1)/((x)^(beta))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Divide[(t)^(\[Alpha]- 1),(x + t)^(\[Alpha]+ \[Beta])], {t, 0, Infinity}, GenerateConditions->None] == Divide[Gamma[\[Alpha]]*Gamma[\[Beta]],Gamma[\[Alpha]+ \[Beta]]]*Divide[1,(x)^\[Beta]]</syntaxhighlight> || Failure || Successful || Successful [Tested: 27] || Successful [Tested: 27]
| [https://dlmf.nist.gov/2.6.E4 2.6.E4] || <math qid="Q829">\int_{0}^{\infty}\frac{t^{\alpha-1}}{(x+t)^{\alpha+\beta}}\diff{t} = \frac{\EulerGamma@{\alpha}\EulerGamma@{\beta}}{\EulerGamma@{\alpha+\beta}}\frac{1}{x^{\beta}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\infty}\frac{t^{\alpha-1}}{(x+t)^{\alpha+\beta}}\diff{t} = \frac{\EulerGamma@{\alpha}\EulerGamma@{\beta}}{\EulerGamma@{\alpha+\beta}}\frac{1}{x^{\beta}}</syntaxhighlight> || <math>\realpart@@{\alpha} > 0, \realpart@@{\beta} > 0, \realpart@@{(\alpha)} > 0, \realpart@@{(\beta)} > 0, \realpart@@{(\alpha+\beta)} > 0</math> || <syntaxhighlight lang=mathematica>int(((t)^(alpha - 1))/((x + t)^(alpha + beta)), t = 0..infinity) = (GAMMA(alpha)*GAMMA(beta))/(GAMMA(alpha + beta))*(1)/((x)^(beta))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Divide[(t)^(\[Alpha]- 1),(x + t)^(\[Alpha]+ \[Beta])], {t, 0, Infinity}, GenerateConditions->None] == Divide[Gamma[\[Alpha]]*Gamma[\[Beta]],Gamma[\[Alpha]+ \[Beta]]]*Divide[1,(x)^\[Beta]]</syntaxhighlight> || Failure || Successful || Successful [Tested: 27] || Successful [Tested: 27]
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| [https://dlmf.nist.gov/2.6.E5 2.6.E5] || [[Item:Q830|<math>\int_{0}^{\infty}\frac{t^{-s-(1/3)}}{x+t}\diff{t} = \frac{2\pi}{\sqrt{3}}\frac{(-1)^{s}}{x^{s+(1/3)}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\infty}\frac{t^{-s-(1/3)}}{x+t}\diff{t} = \frac{2\pi}{\sqrt{3}}\frac{(-1)^{s}}{x^{s+(1/3)}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>int(((t)^(- s -(1/3)))/(x + t), t = 0..infinity) = (2*Pi)/(sqrt(3))*((- 1)^(s))/((x)^(s +(1/3)))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Divide[(t)^(- s -(1/3)),x + t], {t, 0, Infinity}, GenerateConditions->None] == Divide[2*Pi,Sqrt[3]]*Divide[(- 1)^(s),(x)^(s +(1/3))]</syntaxhighlight> || Failure || Failure || Skipped - Because timed out || Successful [Tested: 3]
| [https://dlmf.nist.gov/2.6.E5 2.6.E5] || <math qid="Q830">\int_{0}^{\infty}\frac{t^{-s-(1/3)}}{x+t}\diff{t} = \frac{2\pi}{\sqrt{3}}\frac{(-1)^{s}}{x^{s+(1/3)}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\infty}\frac{t^{-s-(1/3)}}{x+t}\diff{t} = \frac{2\pi}{\sqrt{3}}\frac{(-1)^{s}}{x^{s+(1/3)}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>int(((t)^(- s -(1/3)))/(x + t), t = 0..infinity) = (2*Pi)/(sqrt(3))*((- 1)^(s))/((x)^(s +(1/3)))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Divide[(t)^(- s -(1/3)),x + t], {t, 0, Infinity}, GenerateConditions->None] == Divide[2*Pi,Sqrt[3]]*Divide[(- 1)^(s),(x)^(s +(1/3))]</syntaxhighlight> || Failure || Failure || Skipped - Because timed out || Successful [Tested: 3]
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| [https://dlmf.nist.gov/2.6.E22 2.6.E22] || [[Item:Q847|<math>\phi_{\varepsilon}(t) = \frac{e^{-\varepsilon t}}{t+z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\phi_{\varepsilon}(t) = \frac{e^{-\varepsilon t}}{t+z}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">phi[varepsilon](t) = (exp(- varepsilon*t))/(t + z)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[\[Phi], \[CurlyEpsilon]][t] == Divide[Exp[- \[CurlyEpsilon]*t],t + z]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| [https://dlmf.nist.gov/2.6.E22 2.6.E22] || <math qid="Q847">\phi_{\varepsilon}(t) = \frac{e^{-\varepsilon t}}{t+z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\phi_{\varepsilon}(t) = \frac{e^{-\varepsilon t}}{t+z}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">phi[varepsilon](t) = (exp(- varepsilon*t))/(t + z)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[\[Phi], \[CurlyEpsilon]][t] == Divide[Exp[- \[CurlyEpsilon]*t],t + z]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/2.6.E30 2.6.E30] || [[Item:Q856|<math>R_{n}(z) = \frac{(-1)^{n}}{z^{n}}\int_{0}^{\infty}\frac{\tau^{n}f_{n}(\tau)}{\tau+z}\diff{\tau}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>R_{n}(z) = \frac{(-1)^{n}}{z^{n}}\int_{0}^{\infty}\frac{\tau^{n}f_{n}(\tau)}{\tau+z}\diff{\tau}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>R[n](z) = ((- 1)^(n))/((z)^(n))*int(((tau)^(n)* f[n](tau))/(tau + z), tau = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[R, n][z] == Divide[(- 1)^(n),(z)^(n)]*Integrate[Divide[\[Tau]^(n)* Subscript[f, n][\[Tau]],\[Tau]+ z], {\[Tau], 0, Infinity}, GenerateConditions->None]</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
| [https://dlmf.nist.gov/2.6.E30 2.6.E30] || <math qid="Q856">R_{n}(z) = \frac{(-1)^{n}}{z^{n}}\int_{0}^{\infty}\frac{\tau^{n}f_{n}(\tau)}{\tau+z}\diff{\tau}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>R_{n}(z) = \frac{(-1)^{n}}{z^{n}}\int_{0}^{\infty}\frac{\tau^{n}f_{n}(\tau)}{\tau+z}\diff{\tau}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>R[n](z) = ((- 1)^(n))/((z)^(n))*int(((tau)^(n)* f[n](tau))/(tau + z), tau = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[R, n][z] == Divide[(- 1)^(n),(z)^(n)]*Integrate[Divide[\[Tau]^(n)* Subscript[f, n][\[Tau]],\[Tau]+ z], {\[Tau], 0, Infinity}, GenerateConditions->None]</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: {z = 1/2*3^(1/2)+1/2*I, R[n] = 1/2*3^(1/2)+1/2*I, f[n] = 1/2*3^(1/2)+1/2*I, n = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Float(infinity)+Float(infinity)*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, R[n] = 1/2*3^(1/2)+1/2*I, f[n] = 1/2*3^(1/2)+1/2*I, n = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Float(infinity)+Float(infinity)*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, R[n] = 1/2*3^(1/2)+1/2*I, f[n] = 1/2*3^(1/2)+1/2*I, n = 2}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Float(infinity)+Float(infinity)*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, R[n] = 1/2*3^(1/2)+1/2*I, f[n] = 1/2*3^(1/2)+1/2*I, n = 2}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Float(infinity)+Float(infinity)*I
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Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[R, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[R, 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/2.6.E41 2.6.E41] || [[Item:Q867|<math>f = \sum_{s=0}^{n-1}a_{s}t^{-s-\alpha}-\sum_{s=1}^{n}c_{s}\Diracdelta^{(s-1)}+f_{n}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>f = \sum_{s=0}^{n-1}a_{s}t^{-s-\alpha}-\sum_{s=1}^{n}c_{s}\Diracdelta^{(s-1)}+f_{n}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>f = sum(a[s]*(t)^(- s - alpha), s = 0..n - 1)- sum(c[s]*subs( temp=+, diff( Dirac(temp), temp$(s - 1) ) )*f[n], s = 1..n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>f == Sum[Subscript[a, s]*(t)^(- s - \[Alpha]), {s, 0, n - 1}, GenerateConditions->None]- Sum[Subscript[c, s]*(D[DiracDelta[temp], {temp, s - 1}]/.temp-> +)*Subscript[f, n], {s, 1, n}, GenerateConditions->None]</syntaxhighlight> || Error || Failure || - || Error
| [https://dlmf.nist.gov/2.6.E41 2.6.E41] || <math qid="Q867">f = \sum_{s=0}^{n-1}a_{s}t^{-s-\alpha}-\sum_{s=1}^{n}c_{s}\Diracdelta^{(s-1)}+f_{n}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>f = \sum_{s=0}^{n-1}a_{s}t^{-s-\alpha}-\sum_{s=1}^{n}c_{s}\Diracdelta^{(s-1)}+f_{n}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>f = sum(a[s]*(t)^(- s - alpha), s = 0..n - 1)- sum(c[s]*subs( temp=+, diff( Dirac(temp), temp$(s - 1) ) )*f[n], s = 1..n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>f == Sum[Subscript[a, s]*(t)^(- s - \[Alpha]), {s, 0, n - 1}, GenerateConditions->None]- Sum[Subscript[c, s]*(D[DiracDelta[temp], {temp, s - 1}]/.temp-> +)*Subscript[f, n], {s, 1, n}, GenerateConditions->None]</syntaxhighlight> || Error || Failure || - || Error
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| [https://dlmf.nist.gov/2.6.E42 2.6.E42] || [[Item:Q868|<math>f = \sum_{s=0}^{n-1}a_{s}t^{-s-1}-\sum_{s=1}^{n}d_{s}\Diracdelta^{(s-1)}+f_{n}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>f = \sum_{s=0}^{n-1}a_{s}t^{-s-1}-\sum_{s=1}^{n}d_{s}\Diracdelta^{(s-1)}+f_{n}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>f = sum(a[s]*(t)^(- s - 1), s = 0..n - 1)- sum(d[s]*subs( temp=+, diff( Dirac(temp), temp$(s - 1) ) )*f[n], s = 1..n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>f == Sum[Subscript[a, s]*(t)^(- s - 1), {s, 0, n - 1}, GenerateConditions->None]- Sum[Subscript[d, s]*(D[DiracDelta[temp], {temp, s - 1}]/.temp-> +)*Subscript[f, n], {s, 1, n}, GenerateConditions->None]</syntaxhighlight> || Error || Failure || - || Error
| [https://dlmf.nist.gov/2.6.E42 2.6.E42] || <math qid="Q868">f = \sum_{s=0}^{n-1}a_{s}t^{-s-1}-\sum_{s=1}^{n}d_{s}\Diracdelta^{(s-1)}+f_{n}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>f = \sum_{s=0}^{n-1}a_{s}t^{-s-1}-\sum_{s=1}^{n}d_{s}\Diracdelta^{(s-1)}+f_{n}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>f = sum(a[s]*(t)^(- s - 1), s = 0..n - 1)- sum(d[s]*subs( temp=+, diff( Dirac(temp), temp$(s - 1) ) )*f[n], s = 1..n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>f == Sum[Subscript[a, s]*(t)^(- s - 1), {s, 0, n - 1}, GenerateConditions->None]- Sum[Subscript[d, s]*(D[DiracDelta[temp], {temp, s - 1}]/.temp-> +)*Subscript[f, n], {s, 1, n}, GenerateConditions->None]</syntaxhighlight> || Error || Failure || - || Error
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| [https://dlmf.nist.gov/2.6.E47 2.6.E47] || [[Item:Q873|<math>\delta_{n}(x) = \sum_{j=0}^{n}\binom{n}{j}\frac{\EulerGamma@{\mu+1}}{\EulerGamma@{\mu+1-j}}I^{\mu}\left(t^{n-j}f_{n,j}\right)(x)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\delta_{n}(x) = \sum_{j=0}^{n}\binom{n}{j}\frac{\EulerGamma@{\mu+1}}{\EulerGamma@{\mu+1-j}}I^{\mu}\left(t^{n-j}f_{n,j}\right)(x)</syntaxhighlight> || <math>\realpart@@{(\mu+1)} > 0, \realpart@@{(\mu+1-j)} > 0</math> || <syntaxhighlight lang=mathematica>delta[n](x) = sum(binomial(n,j)*(GAMMA(mu + 1))/(GAMMA(mu + 1 - j))*(I((t)^(n - j)* f[n , j])*)^(mu)*(x), j = 0..n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[\[Delta], n][x] == Sum[Binomial[n,j]*Divide[Gamma[\[Mu]+ 1],Gamma[\[Mu]+ 1 - j]]*(I[(t)^(n - j)* Subscript[f, n , j]]*)^\[Mu]*(x), {j, 0, n}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 2.186980427+1.033699533*I
| [https://dlmf.nist.gov/2.6.E47 2.6.E47] || <math qid="Q873">\delta_{n}(x) = \sum_{j=0}^{n}\binom{n}{j}\frac{\EulerGamma@{\mu+1}}{\EulerGamma@{\mu+1-j}}I^{\mu}\left(t^{n-j}f_{n,j}\right)(x)</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\delta_{n}(x) = \sum_{j=0}^{n}\binom{n}{j}\frac{\EulerGamma@{\mu+1}}{\EulerGamma@{\mu+1-j}}I^{\mu}\left(t^{n-j}f_{n,j}\right)(x)</syntaxhighlight> || <math>\realpart@@{(\mu+1)} > 0, \realpart@@{(\mu+1-j)} > 0</math> || <syntaxhighlight lang=mathematica>delta[n](x) = sum(binomial(n,j)*(GAMMA(mu + 1))/(GAMMA(mu + 1 - j))*(I((t)^(n - j)* f[n , j])*)^(mu)*(x), j = 0..n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[\[Delta], n][x] == Sum[Binomial[n,j]*Divide[Gamma[\[Mu]+ 1],Gamma[\[Mu]+ 1 - j]]*(I[(t)^(n - j)* Subscript[f, n , j]]*)^\[Mu]*(x), {j, 0, n}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 2.186980427+1.033699533*I
Test Values: {I = 1/2*3^(1/2)+1/2*I, delta = 1/2*3^(1/2)+1/2*I, mu = 1/2*3^(1/2)+1/2*I, t = -1.5, x = 1.5, delta[n] = 1/2*3^(1/2)+1/2*I, f[n,j] = 1/2*3^(1/2)+1/2*I, n = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .6751758732+2.165771578*I
Test Values: {I = 1/2*3^(1/2)+1/2*I, delta = 1/2*3^(1/2)+1/2*I, mu = 1/2*3^(1/2)+1/2*I, t = -1.5, x = 1.5, delta[n] = 1/2*3^(1/2)+1/2*I, f[n,j] = 1/2*3^(1/2)+1/2*I, n = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .6751758732+2.165771578*I
Test Values: {I = 1/2*3^(1/2)+1/2*I, delta = 1/2*3^(1/2)+1/2*I, mu = 1/2*3^(1/2)+1/2*I, t = -1.5, x = 1.5, delta[n] = 1/2*3^(1/2)+1/2*I, f[n,j] = 1/2*3^(1/2)+1/2*I, n = 2}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.429437374-4.142136088*I
Test Values: {I = 1/2*3^(1/2)+1/2*I, delta = 1/2*3^(1/2)+1/2*I, mu = 1/2*3^(1/2)+1/2*I, t = -1.5, x = 1.5, delta[n] = 1/2*3^(1/2)+1/2*I, f[n,j] = 1/2*3^(1/2)+1/2*I, n = 2}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.429437374-4.142136088*I
Line 48: Line 50:
Test Values: {I = 1/2*3^(1/2)+1/2*I, delta = 1/2*3^(1/2)+1/2*I, mu = 1/2*3^(1/2)+1/2*I, t = -1.5, x = 1.5, delta[n] = 1/2*3^(1/2)+1/2*I, f[n,j] = -1/2+1/2*I*3^(1/2), n = 1}</syntaxhighlight><br>... skip entries to safe data</div></div> || Skipped - Because timed out
Test Values: {I = 1/2*3^(1/2)+1/2*I, delta = 1/2*3^(1/2)+1/2*I, mu = 1/2*3^(1/2)+1/2*I, t = -1.5, x = 1.5, delta[n] = 1/2*3^(1/2)+1/2*I, f[n,j] = -1/2+1/2*I*3^(1/2), n = 1}</syntaxhighlight><br>... skip entries to safe data</div></div> || Skipped - Because timed out
|- style="background: #dfe6e9;"
|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/2.6.E50 2.6.E50] || [[Item:Q876|<math>f_{n}(t) = (-1)^{n}\frac{t^{1-n-\alpha}}{1+t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>f_{n}(t) = (-1)^{n}\frac{t^{1-n-\alpha}}{1+t}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">f[n](t) = (- 1)^(n)*((t)^(1 - n - alpha))/(1 + t)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[f, n][t] == (- 1)^(n)*Divide[(t)^(1 - n - \[Alpha]),1 + t]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| [https://dlmf.nist.gov/2.6.E50 2.6.E50] || <math qid="Q876">f_{n}(t) = (-1)^{n}\frac{t^{1-n-\alpha}}{1+t}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>f_{n}(t) = (-1)^{n}\frac{t^{1-n-\alpha}}{1+t}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">f[n](t) = (- 1)^(n)*((t)^(1 - n - alpha))/(1 + t)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[f, n][t] == (- 1)^(n)*Divide[(t)^(1 - n - \[Alpha]),1 + t]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
|-  
|-  
| [https://dlmf.nist.gov/2.6.E53 2.6.E53] || [[Item:Q879|<math>{\left|\delta_{n}(x)\right|} \leq \frac{\EulerGamma@{\mu+1}\EulerGamma@{1-\alpha}}{\EulerGamma@{\mu+1-\alpha}\EulerGamma@{n+\alpha}}\*\sum_{j=0}^{n}\dbinom{n}{j}\frac{\EulerGamma@{n+\alpha-j}}{\left|\EulerGamma@{\mu+1-j}\right|}x^{\mu-\alpha}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>{\left|\delta_{n}(x)\right|} \leq \frac{\EulerGamma@{\mu+1}\EulerGamma@{1-\alpha}}{\EulerGamma@{\mu+1-\alpha}\EulerGamma@{n+\alpha}}\*\sum_{j=0}^{n}\dbinom{n}{j}\frac{\EulerGamma@{n+\alpha-j}}{\left|\EulerGamma@{\mu+1-j}\right|}x^{\mu-\alpha}</syntaxhighlight> || <math>\realpart@@{(\mu+1)} > 0, \realpart@@{(1-\alpha)} > 0, \realpart@@{(\mu+1-\alpha)} > 0, \realpart@@{(n+\alpha)} > 0, \realpart@@{(n+\alpha-j)} > 0, \realpart@@{(\mu+1-j)} > 0</math> || <syntaxhighlight lang=mathematica>abs(delta[n](x)) <= (GAMMA(mu + 1)*GAMMA(1 - alpha))/(GAMMA(mu + 1 - alpha)*GAMMA(n + alpha))* sum(binomial(n,j)*(GAMMA(n + alpha - j))/(abs(GAMMA(mu + 1 - j)))*(x)^(mu - alpha), j = 0..n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Abs[Subscript[\[Delta], n][x]] <= Divide[Gamma[\[Mu]+ 1]*Gamma[1 - \[Alpha]],Gamma[\[Mu]+ 1 - \[Alpha]]*Gamma[n + \[Alpha]]]* Sum[Binomial[n,j]*Divide[Gamma[n + \[Alpha]- j],Abs[Gamma[\[Mu]+ 1 - j]]]*(x)^(\[Mu]- \[Alpha]), {j, 0, n}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Successful [Tested: 300] || Skipped - Because timed out
| [https://dlmf.nist.gov/2.6.E53 2.6.E53] || <math qid="Q879">{\left|\delta_{n}(x)\right|} \leq \frac{\EulerGamma@{\mu+1}\EulerGamma@{1-\alpha}}{\EulerGamma@{\mu+1-\alpha}\EulerGamma@{n+\alpha}}\*\sum_{j=0}^{n}\dbinom{n}{j}\frac{\EulerGamma@{n+\alpha-j}}{\left|\EulerGamma@{\mu+1-j}\right|}x^{\mu-\alpha}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>{\left|\delta_{n}(x)\right|} \leq \frac{\EulerGamma@{\mu+1}\EulerGamma@{1-\alpha}}{\EulerGamma@{\mu+1-\alpha}\EulerGamma@{n+\alpha}}\*\sum_{j=0}^{n}\dbinom{n}{j}\frac{\EulerGamma@{n+\alpha-j}}{\left|\EulerGamma@{\mu+1-j}\right|}x^{\mu-\alpha}</syntaxhighlight> || <math>\realpart@@{(\mu+1)} > 0, \realpart@@{(1-\alpha)} > 0, \realpart@@{(\mu+1-\alpha)} > 0, \realpart@@{(n+\alpha)} > 0, \realpart@@{(n+\alpha-j)} > 0, \realpart@@{(\mu+1-j)} > 0</math> || <syntaxhighlight lang=mathematica>abs(delta[n](x)) <= (GAMMA(mu + 1)*GAMMA(1 - alpha))/(GAMMA(mu + 1 - alpha)*GAMMA(n + alpha))* sum(binomial(n,j)*(GAMMA(n + alpha - j))/(abs(GAMMA(mu + 1 - j)))*(x)^(mu - alpha), j = 0..n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Abs[Subscript[\[Delta], n][x]] <= Divide[Gamma[\[Mu]+ 1]*Gamma[1 - \[Alpha]],Gamma[\[Mu]+ 1 - \[Alpha]]*Gamma[n + \[Alpha]]]* Sum[Binomial[n,j]*Divide[Gamma[n + \[Alpha]- j],Abs[Gamma[\[Mu]+ 1 - j]]]*(x)^(\[Mu]- \[Alpha]), {j, 0, n}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Successful [Tested: 300] || Skipped - Because timed out
|- style="background: #dfe6e9;"
|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/2.6.E56 2.6.E56] || [[Item:Q882|<math>h(t) = \sum_{s=0}^{n-1}b_{s}t^{-s-\beta}+h_{n}(t)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>h(t) = \sum_{s=0}^{n-1}b_{s}t^{-s-\beta}+h_{n}(t)</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">h(t) = sum(b[s]*(t)^(- s - beta), s = 0..n - 1)+ h[n](t)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">h[t] == Sum[Subscript[b, s]*(t)^(- s - \[Beta]), {s, 0, n - 1}, GenerateConditions->None]+ Subscript[h, n][t]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| [https://dlmf.nist.gov/2.6.E56 2.6.E56] || <math qid="Q882">h(t) = \sum_{s=0}^{n-1}b_{s}t^{-s-\beta}+h_{n}(t)</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>h(t) = \sum_{s=0}^{n-1}b_{s}t^{-s-\beta}+h_{n}(t)</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">h(t) = sum(b[s]*(t)^(- s - beta), s = 0..n - 1)+ h[n](t)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">h[t] == Sum[Subscript[b, s]*(t)^(- s - \[Beta]), {s, 0, n - 1}, GenerateConditions->None]+ Subscript[h, n][t]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
|-  
|-  
| [https://dlmf.nist.gov/2.6.E59 2.6.E59] || [[Item:Q885|<math>\int_{0}^{\infty}t^{\lambda}\diff{t} = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\infty}t^{\lambda}\diff{t} = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>int((t)^(lambda), t = 0..infinity) = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[(t)^\[Lambda], {t, 0, Infinity}, GenerateConditions->None] == 0</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [10 / 10]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Float(undefined)
| [https://dlmf.nist.gov/2.6.E59 2.6.E59] || <math qid="Q885">\int_{0}^{\infty}t^{\lambda}\diff{t} = 0</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\infty}t^{\lambda}\diff{t} = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>int((t)^(lambda), t = 0..infinity) = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[(t)^\[Lambda], {t, 0, Infinity}, GenerateConditions->None] == 0</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [10 / 10]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Float(undefined)
Test Values: {lambda = 1/2*3^(1/2)+1/2*I, lambda = 1+I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Float(undefined)
Test Values: {lambda = 1/2*3^(1/2)+1/2*I, lambda = 1+I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Float(undefined)
Test Values: {lambda = -1/2+1/2*I*3^(1/2), lambda = 1+I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Float(undefined)
Test Values: {lambda = -1/2+1/2*I*3^(1/2), lambda = 1+I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Float(undefined)
Line 61: Line 63:
Test Values: {Rule[λ, Complex[1, 1]]}</syntaxhighlight><br></div></div>
Test Values: {Rule[λ, Complex[1, 1]]}</syntaxhighlight><br></div></div>
|-  
|-  
| [https://dlmf.nist.gov/2.6#Ex2 2.6#Ex2] || [[Item:Q889|<math>\delta_{n}(x) = \int_{0}^{\infty}f_{n}(t)h_{n}(xt)\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\delta_{n}(x) = \int_{0}^{\infty}f_{n}(t)h_{n}(xt)\diff{t}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>delta[n](x) = int(f[n](t)* h[n](x*t), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[\[Delta], n][x] == Integrate[Subscript[f, n][t]* Subscript[h, n][x*t], {t, 0, Infinity}, GenerateConditions->None]</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
| [https://dlmf.nist.gov/2.6#Ex2 2.6#Ex2] || <math qid="Q889">\delta_{n}(x) = \int_{0}^{\infty}f_{n}(t)h_{n}(xt)\diff{t}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\delta_{n}(x) = \int_{0}^{\infty}f_{n}(t)h_{n}(xt)\diff{t}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>delta[n](x) = int(f[n](t)* h[n](x*t), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[\[Delta], n][x] == Integrate[Subscript[f, n][t]* Subscript[h, n][x*t], {t, 0, Infinity}, GenerateConditions->None]</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: {delta = 1/2*3^(1/2)+1/2*I, x = 1.5, delta[n] = 1/2*3^(1/2)+1/2*I, f[n] = 1/2*3^(1/2)+1/2*I, h[n] = 1/2*3^(1/2)+1/2*I, n = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Float(infinity)+Float(infinity)*I
Test Values: {delta = 1/2*3^(1/2)+1/2*I, x = 1.5, delta[n] = 1/2*3^(1/2)+1/2*I, f[n] = 1/2*3^(1/2)+1/2*I, h[n] = 1/2*3^(1/2)+1/2*I, n = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Float(infinity)+Float(infinity)*I
Test Values: {delta = 1/2*3^(1/2)+1/2*I, x = 1.5, delta[n] = 1/2*3^(1/2)+1/2*I, f[n] = 1/2*3^(1/2)+1/2*I, h[n] = 1/2*3^(1/2)+1/2*I, n = 2}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Float(infinity)+Float(infinity)*I
Test Values: {delta = 1/2*3^(1/2)+1/2*I, x = 1.5, delta[n] = 1/2*3^(1/2)+1/2*I, f[n] = 1/2*3^(1/2)+1/2*I, h[n] = 1/2*3^(1/2)+1/2*I, n = 2}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Float(infinity)+Float(infinity)*I

Latest revision as of 11:01, 28 June 2021


DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
2.6.E1 S ⁢ ( x ) = ∫ 0 ∞ 1 ( 1 + t ) 1 / 3 ⁢ ( x + t ) ⁢ d t 𝑆 𝑥 superscript subscript 0 1 superscript 1 𝑡 1 3 𝑥 𝑡 𝑡 {\displaystyle{\displaystyle S(x)=\int_{0}^{\infty}\frac{1}{(1+t)^{1/3}(x+t)}% \mathrm{d}t}}
S(x) = \int_{0}^{\infty}\frac{1}{(1+t)^{1/3}(x+t)}\diff{t}		 

S(x) = int((1)/((1 + t)^(1/3)*(x + t)), t = 0..infinity)

S[x] == Integrate[Divide[1,(1 + t)^(1/3)*(x + t)], {t, 0, Infinity}, GenerateConditions->None]
Failure Failure
Failed [30 / 30]
Result: -1.405894671+.7500000000*I
Test Values: {S = 1/2*3^(1/2)+1/2*I, x = 1.5}


Result: -3.111297626+.2500000000*I
Test Values: {S = 1/2*3^(1/2)+1/2*I, x = .5}


Result: -.774895738+1.*I
Test Values: {S = 1/2*3^(1/2)+1/2*I, x = 2}


Result: -3.454932777+1.299038106*I
Test Values: {S = -1/2+1/2*I*3^(1/2), x = 1.5}

... skip entries to safe data
Failed [30 / 30]
Result: Complex[-1.4058946699058708, 0.7499999999999999]
Test Values: {Rule[S, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5]}


Result: Complex[-3.1112976262861083, 0.24999999999999997]
Test Values: {Rule[S, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 0.5]}

... skip entries to safe data
2.6.E2 ( 1 + t ) - 1 / 3 = ∑ s = 0 ∞ ( - 1 3 s ) ⁢ t - s - ( 1 / 3 ) superscript 1 𝑡 1 3 superscript subscript 𝑠 0 binomial 1 3 𝑠 superscript 𝑡 𝑠 1 3 {\displaystyle{\displaystyle(1+t)^{-1/3}=\sum_{s=0}^{\infty}\genfrac{(}{)}{0.0% pt}{}{-\frac{1}{3}}{s}t^{-s-(1/3)}}}
(1+t)^{-1/3} = \sum_{s=0}^{\infty}\binom{-\frac{1}{3}}{s}t^{-s-(1/3)}		 

(1 + t)^(- 1/3) = sum(binomial(-(1)/(3),s)*(t)^(- s -(1/3)), s = 0..infinity)

(1 + t)^(- 1/3) == Sum[Binomial[-Divide[1,3],s]*(t)^(- s -(1/3)), {s, 0, Infinity}, GenerateConditions->None]
Failure Failure
Failed [1 / 6]
Result: Float(infinity)+Float(infinity)*I
Test Values: {t = -.5}

Failed [1 / 6]
Result: Complex[1.8898815748423095, 1.0911236359717216]
Test Values: {Rule[t, -0.5]}

2.6.E4 ∫ 0 ∞ t α - 1 ( x + t ) α + β ⁢ d t = Γ ⁡ ( α ) ⁢ Γ ⁡ ( β ) Γ ⁡ ( α + β ) ⁢ 1 x β superscript subscript 0 superscript 𝑡 𝛼 1 superscript 𝑥 𝑡 𝛼 𝛽 𝑡 Euler-Gamma 𝛼 Euler-Gamma 𝛽 Euler-Gamma 𝛼 𝛽 1 superscript 𝑥 𝛽 {\displaystyle{\displaystyle\int_{0}^{\infty}\frac{t^{\alpha-1}}{(x+t)^{\alpha% +\beta}}\mathrm{d}t=\frac{\Gamma\left(\alpha\right)\Gamma\left(\beta\right)}{% \Gamma\left(\alpha+\beta\right)}\frac{1}{x^{\beta}}}}
\int_{0}^{\infty}\frac{t^{\alpha-1}}{(x+t)^{\alpha+\beta}}\diff{t} = \frac{\EulerGamma@{\alpha}\EulerGamma@{\beta}}{\EulerGamma@{\alpha+\beta}}\frac{1}{x^{\beta}}		 ℜ ⁡ α > 0 , ℜ ⁡ β > 0 , ℜ ⁡ ( α ) > 0 , ℜ ⁡ ( β ) > 0 , ℜ ⁡ ( α + β ) > 0 formulae-sequence 𝛼 0 formulae-sequence 𝛽 0 formulae-sequence 𝛼 0 formulae-sequence 𝛽 0 𝛼 𝛽 0 {\displaystyle{\displaystyle\Re\alpha>0,\Re\beta>0,\Re(\alpha)>0,\Re(\beta)>0,% \Re(\alpha+\beta)>0}} 
int(((t)^(alpha - 1))/((x + t)^(alpha + beta)), t = 0..infinity) = (GAMMA(alpha)*GAMMA(beta))/(GAMMA(alpha + beta))*(1)/((x)^(beta))

Integrate[Divide[(t)^(\[Alpha]- 1),(x + t)^(\[Alpha]+ \[Beta])], {t, 0, Infinity}, GenerateConditions->None] == Divide[Gamma[\[Alpha]]*Gamma[\[Beta]],Gamma[\[Alpha]+ \[Beta]]]*Divide[1,(x)^\[Beta]]
Failure Successful Successful [Tested: 27] Successful [Tested: 27]
2.6.E5 ∫ 0 ∞ t - s - ( 1 / 3 ) x + t ⁢ d t = 2 ⁢ π 3 ⁢ ( - 1 ) s x s + ( 1 / 3 ) superscript subscript 0 superscript 𝑡 𝑠 1 3 𝑥 𝑡 𝑡 2 𝜋 3 superscript 1 𝑠 superscript 𝑥 𝑠 1 3 {\displaystyle{\displaystyle\int_{0}^{\infty}\frac{t^{-s-(1/3)}}{x+t}\mathrm{d% }t=\frac{2\pi}{\sqrt{3}}\frac{(-1)^{s}}{x^{s+(1/3)}}}}
\int_{0}^{\infty}\frac{t^{-s-(1/3)}}{x+t}\diff{t} = \frac{2\pi}{\sqrt{3}}\frac{(-1)^{s}}{x^{s+(1/3)}}		 

int(((t)^(- s -(1/3)))/(x + t), t = 0..infinity) = (2*Pi)/(sqrt(3))*((- 1)^(s))/((x)^(s +(1/3)))

Integrate[Divide[(t)^(- s -(1/3)),x + t], {t, 0, Infinity}, GenerateConditions->None] == Divide[2*Pi,Sqrt[3]]*Divide[(- 1)^(s),(x)^(s +(1/3))]
Failure Failure Skipped - Because timed out Successful [Tested: 3]
2.6.E22 ϕ ε ⁢ ( t ) = e - ε ⁢ t t + z subscript italic-ϕ 𝜀 𝑡 superscript 𝑒 𝜀 𝑡 𝑡 𝑧 {\displaystyle{\displaystyle\phi_{\varepsilon}(t)=\frac{e^{-\varepsilon t}}{t+% z}}}
\phi_{\varepsilon}(t) = \frac{e^{-\varepsilon t}}{t+z}		 

phi[varepsilon](t) = (exp(- varepsilon*t))/(t + z)
 
Subscript[\[Phi], \[CurlyEpsilon]][t] == Divide[Exp[- \[CurlyEpsilon]*t],t + z]
 Skipped - no semantic math Skipped - no semantic math - -
2.6.E30 R n ⁢ ( z ) = ( - 1 ) n z n ⁢ ∫ 0 ∞ τ n ⁢ f n ⁢ ( τ ) τ + z ⁢ d τ subscript 𝑅 𝑛 𝑧 superscript 1 𝑛 superscript 𝑧 𝑛 superscript subscript 0 superscript 𝜏 𝑛 subscript 𝑓 𝑛 𝜏 𝜏 𝑧 𝜏 {\displaystyle{\displaystyle R_{n}(z)=\frac{(-1)^{n}}{z^{n}}\int_{0}^{\infty}% \frac{\tau^{n}f_{n}(\tau)}{\tau+z}\mathrm{d}\tau}}
R_{n}(z) = \frac{(-1)^{n}}{z^{n}}\int_{0}^{\infty}\frac{\tau^{n}f_{n}(\tau)}{\tau+z}\diff{\tau}		 

R[n](z) = ((- 1)^(n))/((z)^(n))*int(((tau)^(n)* f[n](tau))/(tau + z), tau = 0..infinity)

Subscript[R, n][z] == Divide[(- 1)^(n),(z)^(n)]*Integrate[Divide[\[Tau]^(n)* Subscript[f, n][\[Tau]],\[Tau]+ z], {\[Tau], 0, Infinity}, GenerateConditions->None]
Failure Failure
Failed [300 / 300]
Result: Float(infinity)+Float(infinity)*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, R[n] = 1/2*3^(1/2)+1/2*I, f[n] = 1/2*3^(1/2)+1/2*I, n = 1}


Result: Float(infinity)+Float(infinity)*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, R[n] = 1/2*3^(1/2)+1/2*I, f[n] = 1/2*3^(1/2)+1/2*I, n = 2}


Result: Float(infinity)+Float(infinity)*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, R[n] = 1/2*3^(1/2)+1/2*I, f[n] = 1/2*3^(1/2)+1/2*I, n = 3}


Result: Float(infinity)+Float(infinity)*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, R[n] = 1/2*3^(1/2)+1/2*I, f[n] = -1/2+1/2*I*3^(1/2), n = 1}

... skip entries to safe data
Failed [300 / 300]
Result: DirectedInfinity[]
Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[R, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: DirectedInfinity[]
Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[R, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
2.6.E41 f = ∑ s = 0 n - 1 a s ⁢ t - s - α - ∑ s = 1 n c s ⁢ δ ( s - 1 ) + f n 𝑓 superscript subscript 𝑠 0 𝑛 1 subscript 𝑎 𝑠 superscript 𝑡 𝑠 𝛼 superscript subscript 𝑠 1 𝑛 subscript 𝑐 𝑠 Dirac-delta 𝑠 1 subscript 𝑓 𝑛 {\displaystyle{\displaystyle f=\sum_{s=0}^{n-1}a_{s}t^{-s-\alpha}-\sum_{s=1}^{% n}c_{s}{\delta^{(s-1)}}+f_{n}}}
f = \sum_{s=0}^{n-1}a_{s}t^{-s-\alpha}-\sum_{s=1}^{n}c_{s}\Diracdelta^{(s-1)}+f_{n}		 

f = sum(a[s]*(t)^(- s - alpha), s = 0..n - 1)- sum(c[s]*subs( temp=+, diff( Dirac(temp), temp$(s - 1) ) )*f[n], s = 1..n)

f == Sum[Subscript[a, s]*(t)^(- s - \[Alpha]), {s, 0, n - 1}, GenerateConditions->None]- Sum[Subscript[c, s]*(D[DiracDelta[temp], {temp, s - 1}]/.temp-> +)*Subscript[f, n], {s, 1, n}, GenerateConditions->None]
Error Failure - Error
2.6.E42 f = ∑ s = 0 n - 1 a s ⁢ t - s - 1 - ∑ s = 1 n d s ⁢ δ ( s - 1 ) + f n 𝑓 superscript subscript 𝑠 0 𝑛 1 subscript 𝑎 𝑠 superscript 𝑡 𝑠 1 superscript subscript 𝑠 1 𝑛 subscript 𝑑 𝑠 Dirac-delta 𝑠 1 subscript 𝑓 𝑛 {\displaystyle{\displaystyle f=\sum_{s=0}^{n-1}a_{s}t^{-s-1}-\sum_{s=1}^{n}d_{% s}{\delta^{(s-1)}}+f_{n}}}
f = \sum_{s=0}^{n-1}a_{s}t^{-s-1}-\sum_{s=1}^{n}d_{s}\Diracdelta^{(s-1)}+f_{n}		 

f = sum(a[s]*(t)^(- s - 1), s = 0..n - 1)- sum(d[s]*subs( temp=+, diff( Dirac(temp), temp$(s - 1) ) )*f[n], s = 1..n)

f == Sum[Subscript[a, s]*(t)^(- s - 1), {s, 0, n - 1}, GenerateConditions->None]- Sum[Subscript[d, s]*(D[DiracDelta[temp], {temp, s - 1}]/.temp-> +)*Subscript[f, n], {s, 1, n}, GenerateConditions->None]
Error Failure - Error
2.6.E47 δ n ⁢ ( x ) = ∑ j = 0 n ( n j ) ⁢ Γ ⁡ ( μ + 1 ) Γ ⁡ ( μ + 1 - j ) ⁢ I μ ⁢ ( t n - j ⁢ f n , j ) ⁢ ( x ) subscript 𝛿 𝑛 𝑥 superscript subscript 𝑗 0 𝑛 binomial 𝑛 𝑗 Euler-Gamma 𝜇 1 Euler-Gamma 𝜇 1 𝑗 superscript 𝐼 𝜇 superscript 𝑡 𝑛 𝑗 subscript 𝑓 𝑛 𝑗 𝑥 {\displaystyle{\displaystyle\delta_{n}(x)=\sum_{j=0}^{n}\genfrac{(}{)}{0.0pt}{% }{n}{j}\frac{\Gamma\left(\mu+1\right)}{\Gamma\left(\mu+1-j\right)}I^{\mu}\left% (t^{n-j}f_{n,j}\right)(x)}}
\delta_{n}(x) = \sum_{j=0}^{n}\binom{n}{j}\frac{\EulerGamma@{\mu+1}}{\EulerGamma@{\mu+1-j}}I^{\mu}\left(t^{n-j}f_{n,j}\right)(x)		 ℜ ⁡ ( μ + 1 ) > 0 , ℜ ⁡ ( μ + 1 - j ) > 0 formulae-sequence 𝜇 1 0 𝜇 1 𝑗 0 {\displaystyle{\displaystyle\Re(\mu+1)>0,\Re(\mu+1-j)>0}} 
delta[n](x) = sum(binomial(n,j)*(GAMMA(mu + 1))/(GAMMA(mu + 1 - j))*(I((t)^(n - j)* f[n , j])*)^(mu)*(x), j = 0..n)

Subscript[\[Delta], n][x] == Sum[Binomial[n,j]*Divide[Gamma[\[Mu]+ 1],Gamma[\[Mu]+ 1 - j]]*(I[(t)^(n - j)* Subscript[f, n , j]]*)^\[Mu]*(x), {j, 0, n}, GenerateConditions->None]
Failure Failure
Failed [300 / 300]
Result: 2.186980427+1.033699533*I
Test Values: {I = 1/2*3^(1/2)+1/2*I, delta = 1/2*3^(1/2)+1/2*I, mu = 1/2*3^(1/2)+1/2*I, t = -1.5, x = 1.5, delta[n] = 1/2*3^(1/2)+1/2*I, f[n,j] = 1/2*3^(1/2)+1/2*I, n = 1}


Result: .6751758732+2.165771578*I
Test Values: {I = 1/2*3^(1/2)+1/2*I, delta = 1/2*3^(1/2)+1/2*I, mu = 1/2*3^(1/2)+1/2*I, t = -1.5, x = 1.5, delta[n] = 1/2*3^(1/2)+1/2*I, f[n,j] = 1/2*3^(1/2)+1/2*I, n = 2}


Result: -.429437374-4.142136088*I
Test Values: {I = 1/2*3^(1/2)+1/2*I, delta = 1/2*3^(1/2)+1/2*I, mu = 1/2*3^(1/2)+1/2*I, t = -1.5, x = 1.5, delta[n] = 1/2*3^(1/2)+1/2*I, f[n,j] = 1/2*3^(1/2)+1/2*I, n = 3}


Result: 1.015338573+1.637942321*I
Test Values: {I = 1/2*3^(1/2)+1/2*I, delta = 1/2*3^(1/2)+1/2*I, mu = 1/2*3^(1/2)+1/2*I, t = -1.5, x = 1.5, delta[n] = 1/2*3^(1/2)+1/2*I, f[n,j] = -1/2+1/2*I*3^(1/2), n = 1}

... skip entries to safe data
 Skipped - Because timed out
2.6.E50 f n ⁢ ( t ) = ( - 1 ) n ⁢ t 1 - n - α 1 + t subscript 𝑓 𝑛 𝑡 superscript 1 𝑛 superscript 𝑡 1 𝑛 𝛼 1 𝑡 {\displaystyle{\displaystyle f_{n}(t)=(-1)^{n}\frac{t^{1-n-\alpha}}{1+t}}}
f_{n}(t) = (-1)^{n}\frac{t^{1-n-\alpha}}{1+t}		 

f[n](t) = (- 1)^(n)*((t)^(1 - n - alpha))/(1 + t)
 
Subscript[f, n][t] == (- 1)^(n)*Divide[(t)^(1 - n - \[Alpha]),1 + t]
 Skipped - no semantic math Skipped - no semantic math - -
2.6.E53 | δ n ⁢ ( x ) | ≤ Γ ⁡ ( μ + 1 ) ⁢ Γ ⁡ ( 1 - α ) Γ ⁡ ( μ + 1 - α ) ⁢ Γ ⁡ ( n + α ) ⁢ ∑ j = 0 n ( n j ) ⁢ Γ ⁡ ( n + α - j ) | Γ ⁡ ( μ + 1 - j ) | ⁢ x μ - α subscript 𝛿 𝑛 𝑥 Euler-Gamma 𝜇 1 Euler-Gamma 1 𝛼 Euler-Gamma 𝜇 1 𝛼 Euler-Gamma 𝑛 𝛼 superscript subscript 𝑗 0 𝑛 binomial 𝑛 𝑗 Euler-Gamma 𝑛 𝛼 𝑗 Euler-Gamma 𝜇 1 𝑗 superscript 𝑥 𝜇 𝛼 {\displaystyle{\displaystyle{\left|\delta_{n}(x)\right|}\leq\frac{\Gamma\left(% \mu+1\right)\Gamma\left(1-\alpha\right)}{\Gamma\left(\mu+1-\alpha\right)\Gamma% \left(n+\alpha\right)}\*\sum_{j=0}^{n}\dbinom{n}{j}\frac{\Gamma\left(n+\alpha-% j\right)}{\left|\Gamma\left(\mu+1-j\right)\right|}x^{\mu-\alpha}}}
{\left|\delta_{n}(x)\right|} \leq \frac{\EulerGamma@{\mu+1}\EulerGamma@{1-\alpha}}{\EulerGamma@{\mu+1-\alpha}\EulerGamma@{n+\alpha}}\*\sum_{j=0}^{n}\dbinom{n}{j}\frac{\EulerGamma@{n+\alpha-j}}{\left|\EulerGamma@{\mu+1-j}\right|}x^{\mu-\alpha}		 ℜ ⁡ ( μ + 1 ) > 0 , ℜ ⁡ ( 1 - α ) > 0 , ℜ ⁡ ( μ + 1 - α ) > 0 , ℜ ⁡ ( n + α ) > 0 , ℜ ⁡ ( n + α - j ) > 0 , ℜ ⁡ ( μ + 1 - j ) > 0 formulae-sequence 𝜇 1 0 formulae-sequence 1 𝛼 0 formulae-sequence 𝜇 1 𝛼 0 formulae-sequence 𝑛 𝛼 0 formulae-sequence 𝑛 𝛼 𝑗 0 𝜇 1 𝑗 0 {\displaystyle{\displaystyle\Re(\mu+1)>0,\Re(1-\alpha)>0,\Re(\mu+1-\alpha)>0,% \Re(n+\alpha)>0,\Re(n+\alpha-j)>0,\Re(\mu+1-j)>0}} 
abs(delta[n](x)) <= (GAMMA(mu + 1)*GAMMA(1 - alpha))/(GAMMA(mu + 1 - alpha)*GAMMA(n + alpha))* sum(binomial(n,j)*(GAMMA(n + alpha - j))/(abs(GAMMA(mu + 1 - j)))*(x)^(mu - alpha), j = 0..n)

Abs[Subscript[\[Delta], n][x]] <= Divide[Gamma[\[Mu]+ 1]*Gamma[1 - \[Alpha]],Gamma[\[Mu]+ 1 - \[Alpha]]*Gamma[n + \[Alpha]]]* Sum[Binomial[n,j]*Divide[Gamma[n + \[Alpha]- j],Abs[Gamma[\[Mu]+ 1 - j]]]*(x)^(\[Mu]- \[Alpha]), {j, 0, n}, GenerateConditions->None]
Failure Failure Successful [Tested: 300] Skipped - Because timed out
2.6.E56 h ⁢ ( t ) = ∑ s = 0 n - 1 b s ⁢ t - s - β + h n ⁢ ( t ) ℎ 𝑡 superscript subscript 𝑠 0 𝑛 1 subscript 𝑏 𝑠 superscript 𝑡 𝑠 𝛽 subscript ℎ 𝑛 𝑡 {\displaystyle{\displaystyle h(t)=\sum_{s=0}^{n-1}b_{s}t^{-s-\beta}+h_{n}(t)}}
h(t) = \sum_{s=0}^{n-1}b_{s}t^{-s-\beta}+h_{n}(t)		 

h(t) = sum(b[s]*(t)^(- s - beta), s = 0..n - 1)+ h[n](t)
 
h[t] == Sum[Subscript[b, s]*(t)^(- s - \[Beta]), {s, 0, n - 1}, GenerateConditions->None]+ Subscript[h, n][t]
 Skipped - no semantic math Skipped - no semantic math - -
2.6.E59 ∫ 0 ∞ t λ ⁢ d t = 0 superscript subscript 0 superscript 𝑡 𝜆 𝑡 0 {\displaystyle{\displaystyle\int_{0}^{\infty}t^{\lambda}\mathrm{d}t=0}}
\int_{0}^{\infty}t^{\lambda}\diff{t} = 0		 

int((t)^(lambda), t = 0..infinity) = 0

Integrate[(t)^\[Lambda], {t, 0, Infinity}, GenerateConditions->None] == 0
Failure Failure
Failed [10 / 10]
Result: Float(undefined)
Test Values: {lambda = 1/2*3^(1/2)+1/2*I, lambda = 1+I}


Result: Float(undefined)
Test Values: {lambda = -1/2+1/2*I*3^(1/2), lambda = 1+I}


Result: Float(undefined)
Test Values: {lambda = 1/2-1/2*I*3^(1/2), lambda = 1+I}


Result: Float(undefined)
Test Values: {lambda = -1/2*3^(1/2)-1/2*I, lambda = 1+I}

... skip entries to safe data
Failed [1 / 1]
Result: Complex[2.105124266860741235376093541450691432144791*^+55894, -3.724980817286574983657738842232337454559011*^+55894]
Test Values: {Rule[λ, Complex[1, 1]]}

2.6#Ex2 δ n ⁢ ( x ) = ∫ 0 ∞ f n ⁢ ( t ) ⁢ h n ⁢ ( x ⁢ t ) ⁢ d t subscript 𝛿 𝑛 𝑥 superscript subscript 0 subscript 𝑓 𝑛 𝑡 subscript ℎ 𝑛 𝑥 𝑡 𝑡 {\displaystyle{\displaystyle\delta_{n}(x)=\int_{0}^{\infty}f_{n}(t)h_{n}(xt)% \mathrm{d}t}}
\delta_{n}(x) = \int_{0}^{\infty}f_{n}(t)h_{n}(xt)\diff{t}		 

delta[n](x) = int(f[n](t)* h[n](x*t), t = 0..infinity)

Subscript[\[Delta], n][x] == Integrate[Subscript[f, n][t]* Subscript[h, n][x*t], {t, 0, Infinity}, GenerateConditions->None]
Failure Failure
Failed [300 / 300]
Result: Float(infinity)+Float(infinity)*I
Test Values: {delta = 1/2*3^(1/2)+1/2*I, x = 1.5, delta[n] = 1/2*3^(1/2)+1/2*I, f[n] = 1/2*3^(1/2)+1/2*I, h[n] = 1/2*3^(1/2)+1/2*I, n = 1}


Result: Float(infinity)+Float(infinity)*I
Test Values: {delta = 1/2*3^(1/2)+1/2*I, x = 1.5, delta[n] = 1/2*3^(1/2)+1/2*I, f[n] = 1/2*3^(1/2)+1/2*I, h[n] = 1/2*3^(1/2)+1/2*I, n = 2}


Result: Float(infinity)+Float(infinity)*I
Test Values: {delta = 1/2*3^(1/2)+1/2*I, x = 1.5, delta[n] = 1/2*3^(1/2)+1/2*I, f[n] = 1/2*3^(1/2)+1/2*I, h[n] = 1/2*3^(1/2)+1/2*I, n = 3}


Result: Float(infinity)+Float(infinity)*I
Test Values: {delta = 1/2*3^(1/2)+1/2*I, x = 1.5, delta[n] = 1/2*3^(1/2)+1/2*I, f[n] = 1/2*3^(1/2)+1/2*I, h[n] = -1/2+1/2*I*3^(1/2), n = 1}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-1.59977280929447116972275470162594*^+83839, -2.77088778626521950864048398971341*^+83839]
Test Values: {Rule[n, 1], Rule[x, 1.5], Rule[δ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[h, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[δ, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-1.59977280929447116972275470162594*^+83839, -2.77088778626521950864048398971341*^+83839]
Test Values: {Rule[n, 2], Rule[x, 1.5], Rule[δ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[h, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[δ, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
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