Zeta and Related Functions - 25.4 Reflection Formulas
| DLMF | Formula | Constraints | Maple | Mathematica | Symbolic Maple |
Symbolic Mathematica |
Numeric Maple |
Numeric Mathematica |
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| 25.4.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Riemannzeta@{1-s} = 2(2\pi)^{-s}\cos@{\tfrac{1}{2}\pi s}\EulerGamma@{s}\Riemannzeta@{s}}
\Riemannzeta@{1-s} = 2(2\pi)^{-s}\cos@{\tfrac{1}{2}\pi s}\EulerGamma@{s}\Riemannzeta@{s} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{s} > 0} | Zeta(1 - s) = 2*(2*Pi)^(- s)* cos((1)/(2)*Pi*s)*GAMMA(s)*Zeta(s)
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Zeta[1 - s] == 2*(2*Pi)^(- s)* Cos[Divide[1,2]*Pi*s]*Gamma[s]*Zeta[s]
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Failure | Successful | Successful [Tested: 3] | Successful [Tested: 3] |
| 25.4.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Riemannzeta@{s} = 2(2\pi)^{s-1}\sin@{\tfrac{1}{2}\pi s}\EulerGamma@{1-s}\Riemannzeta@{1-s}}
\Riemannzeta@{s} = 2(2\pi)^{s-1}\sin@{\tfrac{1}{2}\pi s}\EulerGamma@{1-s}\Riemannzeta@{1-s} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{(1-s)} > 0} | Zeta(s) = 2*(2*Pi)^(s - 1)* sin((1)/(2)*Pi*s)*GAMMA(1 - s)*Zeta(1 - s)
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Zeta[s] == 2*(2*Pi)^(s - 1)* Sin[Divide[1,2]*Pi*s]*Gamma[1 - s]*Zeta[1 - s]
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Failure | Successful | Successful [Tested: 4] | Successful [Tested: 4] |
| 25.4.E3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Riemannxi@{s} = \Riemannxi@{1-s}}
\Riemannxi@{s} = \Riemannxi@{1-s} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | (s)*(s-1)*GAMMA((s)/2)*Pi^(-(s)/2)*Zeta(s)/2 = (1 - s)*(1 - s-1)*GAMMA((1 - s)/2)*Pi^(-(1 - s)/2)*Zeta(1 - s)/2
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RiemannXi[s] == RiemannXi[1 - s]
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Failure | Failure | Failed [1 / 6] Result: Float(undefined)+Float(undefined)*I
Test Values: {s = -2}
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Failed [1 / 6]
Result: Indeterminate
Test Values: {Rule[s, -2]}
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| 25.4.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Riemannxi@{s} = \tfrac{1}{2}s(s-1)\EulerGamma@{\tfrac{1}{2}s}\pi^{-s/2}\Riemannzeta@{s}}
\Riemannxi@{s} = \tfrac{1}{2}s(s-1)\EulerGamma@{\tfrac{1}{2}s}\pi^{-s/2}\Riemannzeta@{s} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{(\tfrac{1}{2}s)} > 0} | (s)*(s-1)*GAMMA((s)/2)*Pi^(-(s)/2)*Zeta(s)/2 = (1)/(2)*s*(s - 1)*GAMMA((1)/(2)*s)*(Pi)^(- s/2)* Zeta(s)
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RiemannXi[s] == Divide[1,2]*s*(s - 1)*Gamma[Divide[1,2]*s]*(Pi)^(- s/2)* Zeta[s]
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Successful | Successful | - | Successful [Tested: 3] |
| 25.4.E5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle (-1)^{k}\Riemannzeta^{(k)}@{1-s} = \frac{2}{(2\pi)^{s}}\sum_{m=0}^{k}\sum_{r=0}^{m}\binom{k}{m}\binom{m}{r}\left(\realpart@{c^{k-m}}\cos@{\tfrac{1}{2}\pi s}+\imagpart@{c^{k-m}}\sin@{\tfrac{1}{2}\pi s}\right)\EulerGamma^{(r)}@{s}\Riemannzeta^{(m-r)}@{s}}
(-1)^{k}\Riemannzeta^{(k)}@{1-s} = \frac{2}{(2\pi)^{s}}\sum_{m=0}^{k}\sum_{r=0}^{m}\binom{k}{m}\binom{m}{r}\left(\realpart@{c^{k-m}}\cos@{\tfrac{1}{2}\pi s}+\imagpart@{c^{k-m}}\sin@{\tfrac{1}{2}\pi s}\right)\EulerGamma^{(r)}@{s}\Riemannzeta^{(m-r)}@{s} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{s} > 0} | (- 1)^(k)* subs( temp=1 - s, diff( Zeta(temp), temp$(k) ) ) = (2)/((2*Pi)^(s))*sum(sum(binomial(k,m)*binomial(m,r)*(Re((c)^(k - m))*cos((1)/(2)*Pi*s)+ Im((c)^(k - m))*sin((1)/(2)*Pi*s))*diff( GAMMA(s), s$(r) )*diff( Zeta(s), s$(m - r) ), r = 0..m), m = 0..k)
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(- 1)^(k)* (D[Zeta[temp], {temp, k}]/.temp-> 1 - s) == Divide[2,(2*Pi)^(s)]*Sum[Sum[Binomial[k,m]*Binomial[m,r]*(Re[(c)^(k - m)]*Cos[Divide[1,2]*Pi*s]+ Im[(c)^(k - m)]*Sin[Divide[1,2]*Pi*s])*D[Gamma[s], {s, r}]*D[Zeta[s], {s, m - r}], {r, 0, m}, GenerateConditions->None], {m, 0, k}, GenerateConditions->None]
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Aborted | Failure | Skipped - Because timed out | Skipped - Because timed out |