Zeta and Related Functions - 25.4 Reflection Formulas

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
25.4.E1 ζ ⁡ ( 1 - s ) = 2 ⁢ ( 2 ⁢ π ) - s ⁢ cos ⁡ ( 1 2 ⁢ π ⁢ s ) ⁢ Γ ⁡ ( s ) ⁢ ζ ⁡ ( s ) Riemann-zeta 1 𝑠 2 superscript 2 𝜋 𝑠 1 2 𝜋 𝑠 Euler-Gamma 𝑠 Riemann-zeta 𝑠 {\displaystyle{\displaystyle\zeta\left(1-s\right)=2(2\pi)^{-s}\cos\left(\tfrac% {1}{2}\pi s\right)\Gamma\left(s\right)\zeta\left(s\right)}}
\Riemannzeta@{1-s} = 2(2\pi)^{-s}\cos@{\tfrac{1}{2}\pi s}\EulerGamma@{s}\Riemannzeta@{s}		 ℜ ⁡ s > 0 𝑠 0 {\displaystyle{\displaystyle\Re s>0}} 
Zeta(1 - s) = 2*(2*Pi)^(- s)* cos((1)/(2)*Pi*s)*GAMMA(s)*Zeta(s)

Zeta[1 - s] == 2*(2*Pi)^(- s)* Cos[Divide[1,2]*Pi*s]*Gamma[s]*Zeta[s]
Failure Successful Successful [Tested: 3] Successful [Tested: 3]
25.4.E2 ζ ⁡ ( s ) = 2 ⁢ ( 2 ⁢ π ) s - 1 ⁢ sin ⁡ ( 1 2 ⁢ π ⁢ s ) ⁢ Γ ⁡ ( 1 - s ) ⁢ ζ ⁡ ( 1 - s ) Riemann-zeta 𝑠 2 superscript 2 𝜋 𝑠 1 1 2 𝜋 𝑠 Euler-Gamma 1 𝑠 Riemann-zeta 1 𝑠 {\displaystyle{\displaystyle\zeta\left(s\right)=2(2\pi)^{s-1}\sin\left(\tfrac{% 1}{2}\pi s\right)\Gamma\left(1-s\right)\zeta\left(1-s\right)}}
\Riemannzeta@{s} = 2(2\pi)^{s-1}\sin@{\tfrac{1}{2}\pi s}\EulerGamma@{1-s}\Riemannzeta@{1-s}		 ℜ ⁡ ( 1 - s ) > 0 1 𝑠 0 {\displaystyle{\displaystyle\Re(1-s)>0}} 
Zeta(s) = 2*(2*Pi)^(s - 1)* sin((1)/(2)*Pi*s)*GAMMA(1 - s)*Zeta(1 - s)

Zeta[s] == 2*(2*Pi)^(s - 1)* Sin[Divide[1,2]*Pi*s]*Gamma[1 - s]*Zeta[1 - s]
Failure Successful Successful [Tested: 4] Successful [Tested: 4]
25.4.E3 ξ ⁡ ( s ) = ξ ⁡ ( 1 - s ) Riemann-xi 𝑠 Riemann-xi 1 𝑠 {\displaystyle{\displaystyle\xi\left(s\right)=\xi\left(1-s\right)}}
\Riemannxi@{s} = \Riemannxi@{1-s}		 

(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

RiemannXi[s] == RiemannXi[1 - s]
Failure Failure
Failed [1 / 6]
Result: Float(undefined)+Float(undefined)*I
Test Values: {s = -2}

Failed [1 / 6]
Result: Indeterminate
Test Values: {Rule[s, -2]}

25.4.E4 ξ ⁡ ( s ) = 1 2 ⁢ s ⁢ ( s - 1 ) ⁢ Γ ⁡ ( 1 2 ⁢ s ) ⁢ π - s / 2 ⁢ ζ ⁡ ( s ) Riemann-xi 𝑠 1 2 𝑠 𝑠 1 Euler-Gamma 1 2 𝑠 superscript 𝜋 𝑠 2 Riemann-zeta 𝑠 {\displaystyle{\displaystyle\xi\left(s\right)=\tfrac{1}{2}s(s-1)\Gamma\left(% \tfrac{1}{2}s\right)\pi^{-s/2}\zeta\left(s\right)}}
\Riemannxi@{s} = \tfrac{1}{2}s(s-1)\EulerGamma@{\tfrac{1}{2}s}\pi^{-s/2}\Riemannzeta@{s}		 ℜ ⁡ ( 1 2 ⁢ s ) > 0 1 2 𝑠 0 {\displaystyle{\displaystyle\Re(\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)

RiemannXi[s] == Divide[1,2]*s*(s - 1)*Gamma[Divide[1,2]*s]*(Pi)^(- s/2)* Zeta[s]
Successful Successful - Successful [Tested: 3]
25.4.E5 ( - 1 ) k ⁢ ζ ( k ) ⁡ ( 1 - s ) = 2 ( 2 ⁢ π ) s ⁢ ∑ m = 0 k ∑ r = 0 m ( k m ) ⁢ ( m r ) ⁢ ( ℜ ⁡ ( c k - m ) ⁢ cos ⁡ ( 1 2 ⁢ π ⁢ s ) + ℑ ⁡ ( c k - m ) ⁢ sin ⁡ ( 1 2 ⁢ π ⁢ s ) ) ⁢ Γ ( r ) ⁡ ( s ) ⁢ ζ ( m - r ) ⁡ ( s ) superscript 1 𝑘 Riemann-zeta 𝑘 1 𝑠 2 superscript 2 𝜋 𝑠 superscript subscript 𝑚 0 𝑘 superscript subscript 𝑟 0 𝑚 binomial 𝑘 𝑚 binomial 𝑚 𝑟 superscript 𝑐 𝑘 𝑚 1 2 𝜋 𝑠 superscript 𝑐 𝑘 𝑚 1 2 𝜋 𝑠 Euler-Gamma 𝑟 𝑠 Riemann-zeta 𝑚 𝑟 𝑠 {\displaystyle{\displaystyle(-1)^{k}{\zeta^{(k)}}\left(1-s\right)=\frac{2}{(2% \pi)^{s}}\sum_{m=0}^{k}\sum_{r=0}^{m}\genfrac{(}{)}{0.0pt}{}{k}{m}\genfrac{(}{% )}{0.0pt}{}{m}{r}\left(\Re\left(c^{k-m}\right)\cos\left(\tfrac{1}{2}\pi s% \right)+\Im\left(c^{k-m}\right)\sin\left(\tfrac{1}{2}\pi s\right)\right){% \Gamma^{(r)}}\left(s\right){\zeta^{(m-r)}}\left(s\right)}}
(-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}		 ℜ ⁡ s > 0 𝑠 0 {\displaystyle{\displaystyle\Re 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)

(- 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]
Aborted Failure Skipped - Because timed out Skipped - Because timed out
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