25.16: 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
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| [https://dlmf.nist.gov/25.16.E10 25.16.E10] || [[Item:Q7766|<math>\frac{1}{2}\Riemannzeta@{1-2a} = -\frac{\BernoullinumberB{2a}}{4a}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{1}{2}\Riemannzeta@{1-2a} = -\frac{\BernoullinumberB{2a}}{4a}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(1)/(2)*Zeta(1 - 2*a) = -(bernoulli(2*a))/(4*a)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[1,2]*Zeta[1 - 2*a] == -Divide[BernoulliB[2*a],4*a]</syntaxhighlight> || Failure || Failure || Successful [Tested: 1] || Successful [Tested: 1]
| [https://dlmf.nist.gov/25.16.E10 25.16.E10] || <math qid="Q7766">\frac{1}{2}\Riemannzeta@{1-2a} = -\frac{\BernoullinumberB{2a}}{4a}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{1}{2}\Riemannzeta@{1-2a} = -\frac{\BernoullinumberB{2a}}{4a}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(1)/(2)*Zeta(1 - 2*a) = -(bernoulli(2*a))/(4*a)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[1,2]*Zeta[1 - 2*a] == -Divide[BernoulliB[2*a],4*a]</syntaxhighlight> || Failure || Failure || Successful [Tested: 1] || Successful [Tested: 1]
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| [https://dlmf.nist.gov/25.16.E13 25.16.E13] || [[Item:Q7769|<math>\sum_{n=1}^{\infty}\left(\frac{h(n)}{n}\right)^{2} = \frac{17}{4}\Riemannzeta@{4}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{n=1}^{\infty}\left(\frac{h(n)}{n}\right)^{2} = \frac{17}{4}\Riemannzeta@{4}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sum(((h(n))/(n))^(2), n = 1..infinity) = (17)/(4)*Zeta(4)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[(Divide[h[n],n])^(2), {n, 1, Infinity}, GenerateConditions->None] == Divide[17,4]*Zeta[4]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [10 / 10]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Float(infinity)+Float(infinity)*I
| [https://dlmf.nist.gov/25.16.E13 25.16.E13] || <math qid="Q7769">\sum_{n=1}^{\infty}\left(\frac{h(n)}{n}\right)^{2} = \frac{17}{4}\Riemannzeta@{4}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{n=1}^{\infty}\left(\frac{h(n)}{n}\right)^{2} = \frac{17}{4}\Riemannzeta@{4}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sum(((h(n))/(n))^(2), n = 1..infinity) = (17)/(4)*Zeta(4)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[(Divide[h[n],n])^(2), {n, 1, Infinity}, GenerateConditions->None] == Divide[17,4]*Zeta[4]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [10 / 10]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Float(infinity)+Float(infinity)*I
Test Values: {h = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Float(infinity)+Float(infinity)*I
Test Values: {h = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Float(infinity)+Float(infinity)*I
Test Values: {h = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [10 / 10]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[-4.599873743272337, NSum[Power[E, Times[Complex[0, Rational[1, 3]], Pi]]
Test Values: {h = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [10 / 10]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[-4.599873743272337, NSum[Power[E, Times[Complex[0, Rational[1, 3]], Pi]]
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Test Values: {n, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], {Rule[h, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
Test Values: {n, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], {Rule[h, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/25.16.E14 25.16.E14] || [[Item:Q7770|<math>\sum_{r=1}^{\infty}\sum_{k=1}^{r}\frac{1}{rk(r+k)} = \frac{5}{4}\Riemannzeta@{3}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{r=1}^{\infty}\sum_{k=1}^{r}\frac{1}{rk(r+k)} = \frac{5}{4}\Riemannzeta@{3}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sum(sum((1)/(r*k*(r + k)), k = 1..r), r = 1..infinity) = (5)/(4)*Zeta(3)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[Sum[Divide[1,r*k*(r + k)], {k, 1, r}, GenerateConditions->None], {r, 1, Infinity}, GenerateConditions->None] == Divide[5,4]*Zeta[3]</syntaxhighlight> || Failure || Aborted || Error || Successful [Tested: 1]
| [https://dlmf.nist.gov/25.16.E14 25.16.E14] || <math qid="Q7770">\sum_{r=1}^{\infty}\sum_{k=1}^{r}\frac{1}{rk(r+k)} = \frac{5}{4}\Riemannzeta@{3}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{r=1}^{\infty}\sum_{k=1}^{r}\frac{1}{rk(r+k)} = \frac{5}{4}\Riemannzeta@{3}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sum(sum((1)/(r*k*(r + k)), k = 1..r), r = 1..infinity) = (5)/(4)*Zeta(3)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[Sum[Divide[1,r*k*(r + k)], {k, 1, r}, GenerateConditions->None], {r, 1, Infinity}, GenerateConditions->None] == Divide[5,4]*Zeta[3]</syntaxhighlight> || Failure || Aborted || Error || Successful [Tested: 1]
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Latest revision as of 12:05, 28 June 2021


DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
25.16.E10 1 2 ⁢ ζ ⁡ ( 1 - 2 ⁢ a ) = - B 2 ⁢ a 4 ⁢ a 1 2 Riemann-zeta 1 2 𝑎 Bernoulli-number-B 2 𝑎 4 𝑎 {\displaystyle{\displaystyle\frac{1}{2}\zeta\left(1-2a\right)=-\frac{B_{2a}}{4% a}}}
\frac{1}{2}\Riemannzeta@{1-2a} = -\frac{\BernoullinumberB{2a}}{4a}		 

(1)/(2)*Zeta(1 - 2*a) = -(bernoulli(2*a))/(4*a)

Divide[1,2]*Zeta[1 - 2*a] == -Divide[BernoulliB[2*a],4*a]
Failure Failure Successful [Tested: 1] Successful [Tested: 1]
25.16.E13 ∑ n = 1 ∞ ( h ⁢ ( n ) n ) 2 = 17 4 ⁢ ζ ⁡ ( 4 ) superscript subscript 𝑛 1 superscript ℎ 𝑛 𝑛 2 17 4 Riemann-zeta 4 {\displaystyle{\displaystyle\sum_{n=1}^{\infty}\left(\frac{h(n)}{n}\right)^{2}% =\frac{17}{4}\zeta\left(4\right)}}
\sum_{n=1}^{\infty}\left(\frac{h(n)}{n}\right)^{2} = \frac{17}{4}\Riemannzeta@{4}		 

sum(((h(n))/(n))^(2), n = 1..infinity) = (17)/(4)*Zeta(4)

Sum[(Divide[h[n],n])^(2), {n, 1, Infinity}, GenerateConditions->None] == Divide[17,4]*Zeta[4]
Failure Failure
Failed [10 / 10]
Result: Float(infinity)+Float(infinity)*I
Test Values: {h = 1/2*3^(1/2)+1/2*I}


Result: Float(infinity)+Float(infinity)*I
Test Values: {h = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [10 / 10]
Result: Plus[-4.599873743272337, NSum[Power[E, Times[Complex[0, Rational[1, 3]], Pi]]
Test Values: {n, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], {Rule[h, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Plus[-4.599873743272337, NSum[Power[E, Times[Complex[0, Rational[-2, 3]], Pi]]
Test Values: {n, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], {Rule[h, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
25.16.E14 ∑ r = 1 ∞ ∑ k = 1 r 1 r ⁢ k ⁢ ( r + k ) = 5 4 ⁢ ζ ⁡ ( 3 ) superscript subscript 𝑟 1 superscript subscript 𝑘 1 𝑟 1 𝑟 𝑘 𝑟 𝑘 5 4 Riemann-zeta 3 {\displaystyle{\displaystyle\sum_{r=1}^{\infty}\sum_{k=1}^{r}\frac{1}{rk(r+k)}% =\frac{5}{4}\zeta\left(3\right)}}
\sum_{r=1}^{\infty}\sum_{k=1}^{r}\frac{1}{rk(r+k)} = \frac{5}{4}\Riemannzeta@{3}		 

sum(sum((1)/(r*k*(r + k)), k = 1..r), r = 1..infinity) = (5)/(4)*Zeta(3)

Sum[Sum[Divide[1,r*k*(r + k)], {k, 1, r}, GenerateConditions->None], {r, 1, Infinity}, GenerateConditions->None] == Divide[5,4]*Zeta[3]
Failure Aborted Error Successful [Tested: 1]
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