5.11: Difference between revisions

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! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
|- style="background: #dfe6e9;"
|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/5.11#Ex1 5.11#Ex1] || [[Item:Q2120|<math>g_{0} = 1</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>g_{0} = 1</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">g[0] = 1</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[g, 0] == 1</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| [https://dlmf.nist.gov/5.11#Ex1 5.11#Ex1] || <math qid="Q2120">g_{0} = 1</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>g_{0} = 1</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">g[0] = 1</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[g, 0] == 1</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
|- style="background: #dfe6e9;"
|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/5.11#Ex2 5.11#Ex2] || [[Item:Q2121|<math>g_{1} = \tfrac{1}{12}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>g_{1} = \tfrac{1}{12}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">g[1] = (1)/(12)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[g, 1] == Divide[1,12]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| [https://dlmf.nist.gov/5.11#Ex2 5.11#Ex2] || <math qid="Q2121">g_{1} = \tfrac{1}{12}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>g_{1} = \tfrac{1}{12}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">g[1] = (1)/(12)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[g, 1] == Divide[1,12]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
|- style="background: #dfe6e9;"
|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/5.11#Ex3 5.11#Ex3] || [[Item:Q2122|<math>g_{2} = \tfrac{1}{288}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>g_{2} = \tfrac{1}{288}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">g[2] = (1)/(288)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[g, 2] == Divide[1,288]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| [https://dlmf.nist.gov/5.11#Ex3 5.11#Ex3] || <math qid="Q2122">g_{2} = \tfrac{1}{288}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>g_{2} = \tfrac{1}{288}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">g[2] = (1)/(288)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[g, 2] == Divide[1,288]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
|- style="background: #dfe6e9;"
|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/5.11#Ex4 5.11#Ex4] || [[Item:Q2123|<math>g_{3} = -\tfrac{139}{51840}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>g_{3} = -\tfrac{139}{51840}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">g[3] = -(139)/(51840)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[g, 3] == -Divide[139,51840]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| [https://dlmf.nist.gov/5.11#Ex4 5.11#Ex4] || <math qid="Q2123">g_{3} = -\tfrac{139}{51840}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>g_{3} = -\tfrac{139}{51840}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">g[3] = -(139)/(51840)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[g, 3] == -Divide[139,51840]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
|- style="background: #dfe6e9;"
|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/5.11#Ex5 5.11#Ex5] || [[Item:Q2124|<math>g_{4} = -\tfrac{571}{24\;88320}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>g_{4} = -\tfrac{571}{24\;88320}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">g[4] = -(571)/(2488320)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[g, 4] == -Divide[571,2488320]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| [https://dlmf.nist.gov/5.11#Ex5 5.11#Ex5] || <math qid="Q2124">g_{4} = -\tfrac{571}{24\;88320}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>g_{4} = -\tfrac{571}{24\;88320}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">g[4] = -(571)/(2488320)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[g, 4] == -Divide[571,2488320]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
|- style="background: #dfe6e9;"
|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/5.11#Ex6 5.11#Ex6] || [[Item:Q2125|<math>g_{5} = \tfrac{1\;63879}{2090\;18880}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>g_{5} = \tfrac{1\;63879}{2090\;18880}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">g[5] = (163879)/(209018880)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[g, 5] == Divide[163879,209018880]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| [https://dlmf.nist.gov/5.11#Ex6 5.11#Ex6] || <math qid="Q2125">g_{5} = \tfrac{1\;63879}{2090\;18880}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>g_{5} = \tfrac{1\;63879}{2090\;18880}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">g[5] = (163879)/(209018880)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[g, 5] == Divide[163879,209018880]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
|- style="background: #dfe6e9;"
|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/5.11#Ex7 5.11#Ex7] || [[Item:Q2126|<math>g_{6} = \tfrac{52\;46819}{7\;52467\;96800}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>g_{6} = \tfrac{52\;46819}{7\;52467\;96800}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">g[6] = (5246819)/(75246796800)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[g, 6] == Divide[5246819,75246796800]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| [https://dlmf.nist.gov/5.11#Ex7 5.11#Ex7] || <math qid="Q2126">g_{6} = \tfrac{52\;46819}{7\;52467\;96800}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>g_{6} = \tfrac{52\;46819}{7\;52467\;96800}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">g[6] = (5246819)/(75246796800)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[g, 6] == Divide[5246819,75246796800]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
|-  
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| [https://dlmf.nist.gov/5.11.E5 5.11.E5] || [[Item:Q2127|<math>g_{k} = \sqrt{2}\Pochhammersym{\tfrac{1}{2}}{k}a_{2k}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>g_{k} = \sqrt{2}\Pochhammersym{\tfrac{1}{2}}{k}a_{2k}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>g[k] = sqrt(2)*pochhammer((1)/(2), k)*a[2*k]</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[g, k] == Sqrt[2]*Pochhammer[Divide[1,2], k]*Subscript[a, 2*k]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .2536529683+.1464466095*I
| [https://dlmf.nist.gov/5.11.E5 5.11.E5] || <math qid="Q2127">g_{k} = \sqrt{2}\Pochhammersym{\tfrac{1}{2}}{k}a_{2k}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>g_{k} = \sqrt{2}\Pochhammersym{\tfrac{1}{2}}{k}a_{2k}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>g[k] = sqrt(2)*pochhammer((1)/(2), k)*a[2*k]</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[g, k] == Sqrt[2]*Pochhammer[Divide[1,2], k]*Subscript[a, 2*k]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .2536529683+.1464466095*I
Test Values: {a[2*k] = 1/2*3^(1/2)+1/2*I, g[k] = 1/2*3^(1/2)+1/2*I, k = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.5253324962e-1-.303300858e-1*I
Test Values: {a[2*k] = 1/2*3^(1/2)+1/2*I, g[k] = 1/2*3^(1/2)+1/2*I, k = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.5253324962e-1-.303300858e-1*I
Test Values: {a[2*k] = 1/2*3^(1/2)+1/2*I, g[k] = 1/2*3^(1/2)+1/2*I, k = 2}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -1.430371229-.8258252140*I
Test Values: {a[2*k] = 1/2*3^(1/2)+1/2*I, g[k] = 1/2*3^(1/2)+1/2*I, k = 2}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -1.430371229-.8258252140*I
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Test Values: {Rule[k, 2], Rule[Subscript[a, Times[2, k]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[g, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
Test Values: {Rule[k, 2], Rule[Subscript[a, Times[2, k]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[g, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-  
|-  
| [https://dlmf.nist.gov/5.11.E10 5.11.E10] || [[Item:Q2132|<math>\EulerGamma@{z} = e^{-z}z^{z}\left(\frac{2\pi}{z}\right)^{1/2}\left(\sum_{k=0}^{K-1}\frac{g_{k}}{z^{k}}+R_{K}(z)\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerGamma@{z} = e^{-z}z^{z}\left(\frac{2\pi}{z}\right)^{1/2}\left(\sum_{k=0}^{K-1}\frac{g_{k}}{z^{k}}+R_{K}(z)\right)</syntaxhighlight> || <math>\realpart@@{z} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA(z) = exp(- z)*(z)^(z)*((2*Pi)/(z))^(1/2)*(sum((g[k])/((z)^(k)), k = 0..K - 1)+ R[K](z))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[z] == Exp[- z]*(z)^(z)*(Divide[2*Pi,z])^(1/2)*(Sum[Divide[Subscript[g, k],(z)^(k)], {k, 0, K - 1}, GenerateConditions->None]+ Subscript[R, K][z])</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -1.892613380-.1706947928*I
| [https://dlmf.nist.gov/5.11.E10 5.11.E10] || <math qid="Q2132">\EulerGamma@{z} = e^{-z}z^{z}\left(\frac{2\pi}{z}\right)^{1/2}\left(\sum_{k=0}^{K-1}\frac{g_{k}}{z^{k}}+R_{K}(z)\right)</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerGamma@{z} = e^{-z}z^{z}\left(\frac{2\pi}{z}\right)^{1/2}\left(\sum_{k=0}^{K-1}\frac{g_{k}}{z^{k}}+R_{K}(z)\right)</syntaxhighlight> || <math>\realpart@@{z} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA(z) = exp(- z)*(z)^(z)*((2*Pi)/(z))^(1/2)*(sum((g[k])/((z)^(k)), k = 0..K - 1)+ R[K](z))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[z] == Exp[- z]*(z)^(z)*(Divide[2*Pi,z])^(1/2)*(Sum[Divide[Subscript[g, k],(z)^(k)], {k, 0, K - 1}, GenerateConditions->None]+ Subscript[R, K][z])</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -1.892613380-.1706947928*I
Test Values: {K = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, R[K] = 1/2*3^(1/2)+1/2*I, g[k] = 1/2*3^(1/2)+1/2*I, K = 3}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.4529896033-2.955992714*I
Test Values: {K = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, R[K] = 1/2*3^(1/2)+1/2*I, g[k] = 1/2*3^(1/2)+1/2*I, K = 3}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.4529896033-2.955992714*I
Test Values: {K = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, R[K] = 1/2*3^(1/2)+1/2*I, g[k] = -1/2+1/2*I*3^(1/2), K = 3}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .8926845412+1.268928985*I
Test Values: {K = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, R[K] = 1/2*3^(1/2)+1/2*I, g[k] = -1/2+1/2*I*3^(1/2), K = 3}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .8926845412+1.268928985*I
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Test Values: {Rule[K, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[g, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[R, K], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
Test Values: {Rule[K, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[g, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[R, K], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- style="background: #dfe6e9;"
|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/5.11#Ex8 5.11#Ex8] || [[Item:Q2137|<math>G_{0}(a,b) = 1</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>G_{0}(a,b) = 1</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">G[0](a , b) = 1</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[G, 0][a , b] == 1</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| [https://dlmf.nist.gov/5.11#Ex8 5.11#Ex8] || <math qid="Q2137">G_{0}(a,b) = 1</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>G_{0}(a,b) = 1</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">G[0](a , b) = 1</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[G, 0][a , b] == 1</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
|- style="background: #dfe6e9;"
|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/5.11#Ex9 5.11#Ex9] || [[Item:Q2138|<math>G_{1}(a,b) = \tfrac{1}{2}(a-b)(a+b-1)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>G_{1}(a,b) = \tfrac{1}{2}(a-b)(a+b-1)</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">G[1](a , b) = (1)/(2)*(a - b)*(a + b - 1)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[G, 1][a , b] == Divide[1,2]*(a - b)*(a + b - 1)</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| [https://dlmf.nist.gov/5.11#Ex9 5.11#Ex9] || <math qid="Q2138">G_{1}(a,b) = \tfrac{1}{2}(a-b)(a+b-1)</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>G_{1}(a,b) = \tfrac{1}{2}(a-b)(a+b-1)</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">G[1](a , b) = (1)/(2)*(a - b)*(a + b - 1)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[G, 1][a , b] == Divide[1,2]*(a - b)*(a + b - 1)</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/5.11#Ex10 5.11#Ex10] || [[Item:Q2139|<math>G_{2}(a,b) = \frac{1}{12}\binom{a-b}{2}(3(a+b-1)^{2}-(a-b+1))</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>G_{2}(a,b) = \frac{1}{12}\binom{a-b}{2}(3(a+b-1)^{2}-(a-b+1))</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>G[2](a , b) = (1)/(12)*binomial(a - b,2)*(3*(a + b - 1)^(2)-(a - b + 1))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[G, 2][a , b] == Divide[1,12]*Binomial[a - b,2]*(3*(a + b - 1)^(2)-(a - b + 1))</syntaxhighlight> || Failure || Failure || Error || Error
| [https://dlmf.nist.gov/5.11#Ex10 5.11#Ex10] || <math qid="Q2139">G_{2}(a,b) = \frac{1}{12}\binom{a-b}{2}(3(a+b-1)^{2}-(a-b+1))</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>G_{2}(a,b) = \frac{1}{12}\binom{a-b}{2}(3(a+b-1)^{2}-(a-b+1))</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>G[2](a , b) = (1)/(12)*binomial(a - b,2)*(3*(a + b - 1)^(2)-(a - b + 1))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[G, 2][a , b] == Divide[1,12]*Binomial[a - b,2]*(3*(a + b - 1)^(2)-(a - b + 1))</syntaxhighlight> || Failure || Failure || Error || Error
|- style="background: #dfe6e9;"
|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/5.11#Ex11 5.11#Ex11] || [[Item:Q2140|<math>H_{0}(a,b) = 1</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>H_{0}(a,b) = 1</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">H[0](a , b) = 1</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[H, 0][a , b] == 1</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| [https://dlmf.nist.gov/5.11#Ex11 5.11#Ex11] || <math qid="Q2140">H_{0}(a,b) = 1</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>H_{0}(a,b) = 1</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">H[0](a , b) = 1</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[H, 0][a , b] == 1</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
|-  
|-  
| [https://dlmf.nist.gov/5.11#Ex12 5.11#Ex12] || [[Item:Q2141|<math>H_{1}(a,b) = -\frac{1}{12}\binom{a-b}{2}(a-b+1)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>H_{1}(a,b) = -\frac{1}{12}\binom{a-b}{2}(a-b+1)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>H[1](a , b) = -(1)/(12)*binomial(a - b,2)*(a - b + 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[H, 1][a , b] == -Divide[1,12]*Binomial[a - b,2]*(a - b + 1)</syntaxhighlight> || Failure || Failure || Error || Error
| [https://dlmf.nist.gov/5.11#Ex12 5.11#Ex12] || <math qid="Q2141">H_{1}(a,b) = -\frac{1}{12}\binom{a-b}{2}(a-b+1)</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>H_{1}(a,b) = -\frac{1}{12}\binom{a-b}{2}(a-b+1)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>H[1](a , b) = -(1)/(12)*binomial(a - b,2)*(a - b + 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[H, 1][a , b] == -Divide[1,12]*Binomial[a - b,2]*(a - b + 1)</syntaxhighlight> || Failure || Failure || Error || Error
|-  
|-  
| [https://dlmf.nist.gov/5.11#Ex13 5.11#Ex13] || [[Item:Q2142|<math>H_{2}(a,b) = \frac{1}{240}\binom{a-b}{4}(2(a-b+1)+5(a-b+1)^{2})</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>H_{2}(a,b) = \frac{1}{240}\binom{a-b}{4}(2(a-b+1)+5(a-b+1)^{2})</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>H[2](a , b) = (1)/(240)*binomial(a - b,4)*(2*(a - b + 1)+ 5*(a - b + 1)^(2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[H, 2][a , b] == Divide[1,240]*Binomial[a - b,4]*(2*(a - b + 1)+ 5*(a - b + 1)^(2))</syntaxhighlight> || Failure || Failure || Error || Error
| [https://dlmf.nist.gov/5.11#Ex13 5.11#Ex13] || <math qid="Q2142">H_{2}(a,b) = \frac{1}{240}\binom{a-b}{4}(2(a-b+1)+5(a-b+1)^{2})</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>H_{2}(a,b) = \frac{1}{240}\binom{a-b}{4}(2(a-b+1)+5(a-b+1)^{2})</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>H[2](a , b) = (1)/(240)*binomial(a - b,4)*(2*(a - b + 1)+ 5*(a - b + 1)^(2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[H, 2][a , b] == Divide[1,240]*Binomial[a - b,4]*(2*(a - b + 1)+ 5*(a - b + 1)^(2))</syntaxhighlight> || Failure || Failure || Error || Error
|}
|}
</div>
</div>

Latest revision as of 11:13, 28 June 2021


DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
5.11#Ex1 g 0 = 1 subscript 𝑔 0 1 {\displaystyle{\displaystyle g_{0}=1}}
g_{0} = 1		 

g[0] = 1
 
Subscript[g, 0] == 1
 Skipped - no semantic math Skipped - no semantic math - -
5.11#Ex2 g 1 = 1 12 subscript 𝑔 1 1 12 {\displaystyle{\displaystyle g_{1}=\tfrac{1}{12}}}
g_{1} = \tfrac{1}{12}		 

g[1] = (1)/(12)
 
Subscript[g, 1] == Divide[1,12]
 Skipped - no semantic math Skipped - no semantic math - -
5.11#Ex3 g 2 = 1 288 subscript 𝑔 2 1 288 {\displaystyle{\displaystyle g_{2}=\tfrac{1}{288}}}
g_{2} = \tfrac{1}{288}		 

g[2] = (1)/(288)
 
Subscript[g, 2] == Divide[1,288]
 Skipped - no semantic math Skipped - no semantic math - -
5.11#Ex4 g 3 = - 139 51840 subscript 𝑔 3 139 51840 {\displaystyle{\displaystyle g_{3}=-\tfrac{139}{51840}}}
g_{3} = -\tfrac{139}{51840}		 

g[3] = -(139)/(51840)
 
Subscript[g, 3] == -Divide[139,51840]
 Skipped - no semantic math Skipped - no semantic math - -
5.11#Ex5 g 4 = - 571 24 88320 subscript 𝑔 4 571 24 88320 {\displaystyle{\displaystyle g_{4}=-\tfrac{571}{24\;88320}}}
g_{4} = -\tfrac{571}{24\;88320}		 

g[4] = -(571)/(2488320)
 
Subscript[g, 4] == -Divide[571,2488320]
 Skipped - no semantic math Skipped - no semantic math - -
5.11#Ex6 g 5 = 1 63879 2090 18880 subscript 𝑔 5 1 63879 2090 18880 {\displaystyle{\displaystyle g_{5}=\tfrac{1\;63879}{2090\;18880}}}
g_{5} = \tfrac{1\;63879}{2090\;18880}		 

g[5] = (163879)/(209018880)
 
Subscript[g, 5] == Divide[163879,209018880]
 Skipped - no semantic math Skipped - no semantic math - -
5.11#Ex7 g 6 = 52 46819 7 52467 96800 subscript 𝑔 6 52 46819 7 52467 96800 {\displaystyle{\displaystyle g_{6}=\tfrac{52\;46819}{7\;52467\;96800}}}
g_{6} = \tfrac{52\;46819}{7\;52467\;96800}		 

g[6] = (5246819)/(75246796800)
 
Subscript[g, 6] == Divide[5246819,75246796800]
 Skipped - no semantic math Skipped - no semantic math - -
5.11.E5 g k = 2 ⁢ ( 1 2 ) k ⁢ a 2 ⁢ k subscript 𝑔 𝑘 2 Pochhammer 1 2 𝑘 subscript 𝑎 2 𝑘 {\displaystyle{\displaystyle g_{k}=\sqrt{2}{\left(\tfrac{1}{2}\right)_{k}}a_{2% k}}}
g_{k} = \sqrt{2}\Pochhammersym{\tfrac{1}{2}}{k}a_{2k}		 

g[k] = sqrt(2)*pochhammer((1)/(2), k)*a[2*k]

Subscript[g, k] == Sqrt[2]*Pochhammer[Divide[1,2], k]*Subscript[a, 2*k]
Failure Failure
Failed [300 / 300]
Result: .2536529683+.1464466095*I
Test Values: {a[2*k] = 1/2*3^(1/2)+1/2*I, g[k] = 1/2*3^(1/2)+1/2*I, k = 1}


Result: -.5253324962e-1-.303300858e-1*I
Test Values: {a[2*k] = 1/2*3^(1/2)+1/2*I, g[k] = 1/2*3^(1/2)+1/2*I, k = 2}


Result: -1.430371229-.8258252140*I
Test Values: {a[2*k] = 1/2*3^(1/2)+1/2*I, g[k] = 1/2*3^(1/2)+1/2*I, k = 3}


Result: -1.112372436+.5124720135*I
Test Values: {a[2*k] = 1/2*3^(1/2)+1/2*I, g[k] = -1/2+1/2*I*3^(1/2), k = 1}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[0.25365296808864424, 0.14644660940672627]
Test Values: {Rule[k, 1], Rule[Subscript[a, Times[2, k]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[g, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.05253324975925311, -0.03033008588991054]
Test Values: {Rule[k, 2], Rule[Subscript[a, Times[2, k]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[g, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
5.11.E10 Γ ⁡ ( z ) = e - z ⁢ z z ⁢ ( 2 ⁢ π z ) 1 / 2 ⁢ ( ∑ k = 0 K - 1 g k z k + R K ⁢ ( z ) ) Euler-Gamma 𝑧 superscript 𝑒 𝑧 superscript 𝑧 𝑧 superscript 2 𝜋 𝑧 1 2 superscript subscript 𝑘 0 𝐾 1 subscript 𝑔 𝑘 superscript 𝑧 𝑘 subscript 𝑅 𝐾 𝑧 {\displaystyle{\displaystyle\Gamma\left(z\right)=e^{-z}z^{z}\left(\frac{2\pi}{% z}\right)^{1/2}\left(\sum_{k=0}^{K-1}\frac{g_{k}}{z^{k}}+R_{K}(z)\right)}}
\EulerGamma@{z} = e^{-z}z^{z}\left(\frac{2\pi}{z}\right)^{1/2}\left(\sum_{k=0}^{K-1}\frac{g_{k}}{z^{k}}+R_{K}(z)\right)		 ℜ ⁡ z > 0 𝑧 0 {\displaystyle{\displaystyle\Re z>0}} 
GAMMA(z) = exp(- z)*(z)^(z)*((2*Pi)/(z))^(1/2)*(sum((g[k])/((z)^(k)), k = 0..K - 1)+ R[K](z))

Gamma[z] == Exp[- z]*(z)^(z)*(Divide[2*Pi,z])^(1/2)*(Sum[Divide[Subscript[g, k],(z)^(k)], {k, 0, K - 1}, GenerateConditions->None]+ Subscript[R, K][z])
Failure Failure
Failed [300 / 300]
Result: -1.892613380-.1706947928*I
Test Values: {K = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, R[K] = 1/2*3^(1/2)+1/2*I, g[k] = 1/2*3^(1/2)+1/2*I, K = 3}


Result: -.4529896033-2.955992714*I
Test Values: {K = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, R[K] = 1/2*3^(1/2)+1/2*I, g[k] = -1/2+1/2*I*3^(1/2), K = 3}


Result: .8926845412+1.268928985*I
Test Values: {K = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, R[K] = 1/2*3^(1/2)+1/2*I, g[k] = 1/2-1/2*I*3^(1/2), K = 3}


Result: 2.332308320-1.516368937*I
Test Values: {K = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, R[K] = 1/2*3^(1/2)+1/2*I, g[k] = -1/2*3^(1/2)-1/2*I, K = 3}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-1.8926133813331316, -0.17069479199840365]
Test Values: {Rule[K, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[g, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[R, K], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.7462398809799414, -0.22409723911500246]
Test Values: {Rule[K, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[g, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[R, K], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
5.11#Ex8 G 0 ⁢ ( a , b ) = 1 subscript 𝐺 0 𝑎 𝑏 1 {\displaystyle{\displaystyle G_{0}(a,b)=1}}
G_{0}(a,b) = 1		 

G[0](a , b) = 1
 
Subscript[G, 0][a , b] == 1
 Skipped - no semantic math Skipped - no semantic math - -
5.11#Ex9 G 1 ⁢ ( a , b ) = 1 2 ⁢ ( a - b ) ⁢ ( a + b - 1 ) subscript 𝐺 1 𝑎 𝑏 1 2 𝑎 𝑏 𝑎 𝑏 1 {\displaystyle{\displaystyle G_{1}(a,b)=\tfrac{1}{2}(a-b)(a+b-1)}}
G_{1}(a,b) = \tfrac{1}{2}(a-b)(a+b-1)		 

G[1](a , b) = (1)/(2)*(a - b)*(a + b - 1)
 
Subscript[G, 1][a , b] == Divide[1,2]*(a - b)*(a + b - 1)
 Skipped - no semantic math Skipped - no semantic math - -
5.11#Ex10 G 2 ⁢ ( a , b ) = 1 12 ⁢ ( a - b 2 ) ⁢ ( 3 ⁢ ( a + b - 1 ) 2 - ( a - b + 1 ) ) subscript 𝐺 2 𝑎 𝑏 1 12 binomial 𝑎 𝑏 2 3 superscript 𝑎 𝑏 1 2 𝑎 𝑏 1 {\displaystyle{\displaystyle G_{2}(a,b)=\frac{1}{12}\genfrac{(}{)}{0.0pt}{}{a-% b}{2}(3(a+b-1)^{2}-(a-b+1))}}
G_{2}(a,b) = \frac{1}{12}\binom{a-b}{2}(3(a+b-1)^{2}-(a-b+1))		 

G[2](a , b) = (1)/(12)*binomial(a - b,2)*(3*(a + b - 1)^(2)-(a - b + 1))

Subscript[G, 2][a , b] == Divide[1,12]*Binomial[a - b,2]*(3*(a + b - 1)^(2)-(a - b + 1))
Failure Failure Error Error
5.11#Ex11 H 0 ⁢ ( a , b ) = 1 subscript 𝐻 0 𝑎 𝑏 1 {\displaystyle{\displaystyle H_{0}(a,b)=1}}
H_{0}(a,b) = 1		 

H[0](a , b) = 1
 
Subscript[H, 0][a , b] == 1
 Skipped - no semantic math Skipped - no semantic math - -
5.11#Ex12 H 1 ⁢ ( a , b ) = - 1 12 ⁢ ( a - b 2 ) ⁢ ( a - b + 1 ) subscript 𝐻 1 𝑎 𝑏 1 12 binomial 𝑎 𝑏 2 𝑎 𝑏 1 {\displaystyle{\displaystyle H_{1}(a,b)=-\frac{1}{12}\genfrac{(}{)}{0.0pt}{}{a% -b}{2}(a-b+1)}}
H_{1}(a,b) = -\frac{1}{12}\binom{a-b}{2}(a-b+1)		 

H[1](a , b) = -(1)/(12)*binomial(a - b,2)*(a - b + 1)

Subscript[H, 1][a , b] == -Divide[1,12]*Binomial[a - b,2]*(a - b + 1)
Failure Failure Error Error
5.11#Ex13 H 2 ⁢ ( a , b ) = 1 240 ⁢ ( a - b 4 ) ⁢ ( 2 ⁢ ( a - b + 1 ) + 5 ⁢ ( a - b + 1 ) 2 ) subscript 𝐻 2 𝑎 𝑏 1 240 binomial 𝑎 𝑏 4 2 𝑎 𝑏 1 5 superscript 𝑎 𝑏 1 2 {\displaystyle{\displaystyle H_{2}(a,b)=\frac{1}{240}\genfrac{(}{)}{0.0pt}{}{a% -b}{4}(2(a-b+1)+5(a-b+1)^{2})}}
H_{2}(a,b) = \frac{1}{240}\binom{a-b}{4}(2(a-b+1)+5(a-b+1)^{2})		 

H[2](a , b) = (1)/(240)*binomial(a - b,4)*(2*(a - b + 1)+ 5*(a - b + 1)^(2))

Subscript[H, 2][a , b] == Divide[1,240]*Binomial[a - b,4]*(2*(a - b + 1)+ 5*(a - b + 1)^(2))
Failure Failure Error Error
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