Bernoulli and Euler Polynomials - 25.2 Definition and Expansions
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| DLMF | Formula | Constraints | Maple | Mathematica | Symbolic Maple |
Symbolic Mathematica |
Numeric Maple |
Numeric Mathematica |
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| 25.2.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Riemannzeta@{s} = \sum_{n=1}^{\infty}\frac{1}{n^{s}}}
\Riemannzeta@{s} = \sum_{n=1}^{\infty}\frac{1}{n^{s}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | Zeta(s) = sum((1)/((n)^(s)), n = 1..infinity)
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Zeta[s] == Sum[Divide[1,(n)^(s)], {n, 1, Infinity}, GenerateConditions->None]
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Failure | Successful | Failed [4 / 6] Result: Float(-infinity)
Test Values: {s = -3/2}
Result: Float(-infinity)
Test Values: {s = -1/2}
... skip entries to safe data |
Successful [Tested: 6] |
| 25.2.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Riemannzeta@{s} = \frac{1}{1-2^{-s}}\sum_{n=0}^{\infty}\frac{1}{(2n+1)^{s}}}
\Riemannzeta@{s} = \frac{1}{1-2^{-s}}\sum_{n=0}^{\infty}\frac{1}{(2n+1)^{s}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{s} > 1} | Zeta(s) = (1)/(1 - (2)^(- s))*sum((1)/((2*n + 1)^(s)), n = 0..infinity)
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Zeta[s] == Divide[1,1 - (2)^(- s)]*Sum[Divide[1,(2*n + 1)^(s)], {n, 0, Infinity}, GenerateConditions->None]
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Successful | Successful | - | Successful [Tested: 2] |
| 25.2.E3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Riemannzeta@{s} = \frac{1}{1-2^{1-s}}\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n^{s}}}
\Riemannzeta@{s} = \frac{1}{1-2^{1-s}}\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n^{s}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{s} > 0} | Zeta(s) = (1)/(1 - (2)^(1 - s))*sum(((- 1)^(n - 1))/((n)^(s)), n = 1..infinity)
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Zeta[s] == Divide[1,1 - (2)^(1 - s)]*Sum[Divide[(- 1)^(n - 1),(n)^(s)], {n, 1, Infinity}, GenerateConditions->None]
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Failure | Successful | Successful [Tested: 3] | Successful [Tested: 3] |
| 25.2.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Riemannzeta@{s} = \frac{1}{s-1}+\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!}\StieltjesConstants{n}(s-1)^{n}}
\Riemannzeta@{s} = \frac{1}{s-1}+\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!}\StieltjesConstants{n}(s-1)^{n} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | Zeta(s) = (1)/(s - 1)+ sum(((- 1)^(n))/(factorial(n))*gamma(n)*(s - 1)^(n), n = 0..infinity)
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Error
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Failure | Missing Macro Error | Successful [Tested: 6] | - |
| 25.2.E5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \StieltjesConstants{n} = \lim_{m\to\infty}\left(\sum_{k=1}^{m}\frac{(\ln@@{k})^{n}}{k}-\frac{(\ln@@{m})^{n+1}}{n+1}\right)}
\StieltjesConstants{n} = \lim_{m\to\infty}\left(\sum_{k=1}^{m}\frac{(\ln@@{k})^{n}}{k}-\frac{(\ln@@{m})^{n+1}}{n+1}\right) |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | gamma(n) = limit(sum(((ln(k))^(n))/(k), k = 1..m)-((ln(m))^(n + 1))/(n + 1), m = infinity)
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Error
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Successful | Missing Macro Error | - | - |
| 25.2.E6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Riemannzeta'@{s} = -\sum_{n=2}^{\infty}(\ln@@{n})n^{-s}}
\Riemannzeta'@{s} = -\sum_{n=2}^{\infty}(\ln@@{n})n^{-s} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{s} > 1} | diff( Zeta(s), s$(1) ) = - sum((ln(n))*(n)^(- s), n = 2..infinity)
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D[Zeta[s], {s, 1}] == - Sum[(Log[n])*(n)^(- s), {n, 2, Infinity}, GenerateConditions->None]
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Successful | Successful | - | Successful [Tested: 2] |
| 25.2.E7 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Riemannzeta^{(k)}@{s} = (-1)^{k}\sum_{n=2}^{\infty}(\ln@@{n})^{k}n^{-s}}
\Riemannzeta^{(k)}@{s} = (-1)^{k}\sum_{n=2}^{\infty}(\ln@@{n})^{k}n^{-s} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{s} > 1} | diff( Zeta(s), s$(k) ) = (- 1)^(k)* sum((ln(n))^(k)* (n)^(- s), n = 2..infinity)
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D[Zeta[s], {s, k}] == (- 1)^(k)* Sum[(Log[n])^(k)* (n)^(- s), {n, 2, Infinity}, GenerateConditions->None]
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Failure | Failure | Successful [Tested: 2] | Successful [Tested: 2] |
| 25.2.E8 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Riemannzeta@{s} = \sum_{k=1}^{N}\frac{1}{k^{s}}+\frac{N^{1-s}}{s-1}-s\int_{N}^{\infty}\frac{x-\floor{x}}{x^{s+1}}\diff{x}}
\Riemannzeta@{s} = \sum_{k=1}^{N}\frac{1}{k^{s}}+\frac{N^{1-s}}{s-1}-s\int_{N}^{\infty}\frac{x-\floor{x}}{x^{s+1}}\diff{x} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{s} > 0} | Zeta(s) = sum((1)/((k)^(s)), k = 1..N)+((N)^(1 - s))/(s - 1)- s*int((x - floor(x))/((x)^(s + 1)), x = N..infinity)
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Zeta[s] == Sum[Divide[1,(k)^(s)], {k, 1, N}, GenerateConditions->None]+Divide[(N)^(1 - s),s - 1]- s*Integrate[Divide[x - Floor[x],(x)^(s + 1)], {x, N, Infinity}, GenerateConditions->None]
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Failure | Aborted | Failed [3 / 3] Result: .2180864797
Test Values: {s = 3/2, N = 3}
Result: Float(infinity)
Test Values: {s = 1/2, N = 3}
... skip entries to safe data |
Skipped - Because timed out |
| 25.2.E11 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Riemannzeta@{s} = \prod_{p}(1-p^{-s})^{-1}}
\Riemannzeta@{s} = \prod_{p}(1-p^{-s})^{-1} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{s} > 1} | Zeta(s) = product((1 - (p)^(- s))^(- 1), p = - infinity..infinity)
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Zeta[s] == Product[(1 - (p)^(- s))^(- 1), {p, - Infinity, Infinity}, GenerateConditions->None]
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Failure | Failure | Error | Failed [2 / 2]
Result: Plus[2.612375348685488, Times[-1.0, NProduct[Power[Plus[1, Times[-1, Power[p, -1.5]]], -1]
Test Values: {p, DirectedInfinity[-1], DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[s, 1.5]}
Result: Plus[1.6449340668482262, Times[-1.0, NProduct[Power[Plus[1, Times[-1, Power[p, -2]]], -1]
Test Values: {p, DirectedInfinity[-1], DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[s, 2]}
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| 25.2.E12 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Riemannzeta@{s} = \frac{(2\pi)^{s}e^{-s-(\EulerConstant s/2)}}{2(s-1)\EulerGamma@{\tfrac{1}{2}s+1}}\prod_{\rho}\left(1-\frac{s}{\rho}\right)e^{s/\rho}}
\Riemannzeta@{s} = \frac{(2\pi)^{s}e^{-s-(\EulerConstant s/2)}}{2(s-1)\EulerGamma@{\tfrac{1}{2}s+1}}\prod_{\rho}\left(1-\frac{s}{\rho}\right)e^{s/\rho} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{(\tfrac{1}{2}s+1)} > 0} | Zeta(s) = ((2*Pi)^(s)* exp(- s -(gamma*s/2)))/(2*(s - 1)*GAMMA((1)/(2)*s + 1))*product((1 -(s)/(rho))*exp(s/rho), rho = - infinity..infinity)
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Zeta[s] == Divide[(2*Pi)^(s)* Exp[- s -(EulerGamma*s/2)],2*(s - 1)*Gamma[Divide[1,2]*s + 1]]*Product[(1 -Divide[s,\[Rho]])*Exp[s/\[Rho]], {\[Rho], - Infinity, Infinity}, GenerateConditions->None]
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Failure | Failure | Error | Failed [5 / 5]
Result: Plus[Complex[-0.02548520188983307, 3.121037092000815*^-18], Times[0.02420092827533985, NProduct[Times[Power[E, Times[-1.5, Power[ρ, -1]]], Plus[1, Times[1.5, Power[ρ, -1]]]]
Test Values: {ρ, DirectedInfinity[-1], DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[s, -1.5]}
Result: Plus[2.612375348685488, Times[-2.4801151038890965, NProduct[Times[Power[E, Times[1.5, Power[ρ, -1]]], Plus[1, Times[-1.5, Power[ρ, -1]]]]
Test Values: {ρ, DirectedInfinity[-1], DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[s, 1.5]}
... skip entries to safe data |