Gamma Function - 5.17 Barnes’ -Function (Double Gamma Function)

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
5.17#Ex1 G ⁡ ( z + 1 ) = Γ ⁡ ( z ) ⁢ G ⁡ ( z ) Barnes-Gamma 𝑧 1 Euler-Gamma 𝑧 Barnes-Gamma 𝑧 {\displaystyle{\displaystyle G\left(z+1\right)=\Gamma\left(z\right)G\left(z% \right)}}
\BarnesG@{z+1} = \EulerGamma@{z}\BarnesG@{z}		 ℜ ⁡ z > 0 𝑧 0 {\displaystyle{\displaystyle\Re z>0}} 
Error

BarnesG[z + 1] == Gamma[z]*BarnesG[z]
Missing Macro Error Failure - Successful [Tested: 5]
5.17#Ex2 G ⁡ ( 1 ) = 1 Barnes-Gamma 1 1 {\displaystyle{\displaystyle G\left(1\right)=1}}
\BarnesG@{1} = 1		 

Error

BarnesG[1] == 1
Missing Macro Error Successful - Successful [Tested: 1]
5.17.E3 G ⁡ ( z + 1 ) = ( 2 ⁢ π ) z / 2 ⁢ exp ⁡ ( - 1 2 ⁢ z ⁢ ( z + 1 ) - 1 2 ⁢ γ ⁢ z 2 ) ⁢ ∏ k = 1 ∞ ( ( 1 + z k ) k ⁢ exp ⁡ ( - z + z 2 2 ⁢ k ) ) Barnes-Gamma 𝑧 1 superscript 2 𝜋 𝑧 2 1 2 𝑧 𝑧 1 1 2 superscript 𝑧 2 superscript subscript product 𝑘 1 superscript 1 𝑧 𝑘 𝑘 𝑧 superscript 𝑧 2 2 𝑘 {\displaystyle{\displaystyle G\left(z+1\right)=(2\pi)^{z/2}\exp\left(-\tfrac{1% }{2}z(z+1)-\tfrac{1}{2}\gamma z^{2}\right)\*\prod_{k=1}^{\infty}\left(\left(1+% \frac{z}{k}\right)^{k}\exp\left(-z+\frac{z^{2}}{2k}\right)\right)}}
\BarnesG@{z+1} = (2\pi)^{z/2}\exp@{-\tfrac{1}{2}z(z+1)-\tfrac{1}{2}\EulerConstant z^{2}}\*\prod_{k=1}^{\infty}\left(\left(1+\frac{z}{k}\right)^{k}\exp@{-z+\frac{z^{2}}{2k}}\right)		 

Error

BarnesG[z + 1] == (2*Pi)^(z/2)* Exp[-Divide[1,2]*z*(z + 1)-Divide[1,2]*EulerGamma*(z)^(2)]* Product[(1 +Divide[z,k])^(k)* Exp[- z +Divide[(z)^(2),2*k]], {k, 1, Infinity}, GenerateConditions->None]
Missing Macro Error Successful - Successful [Tested: 7]
5.17.E4 Ln ⁡ G ⁡ ( z + 1 ) = 1 2 ⁢ z ⁢ ln ⁡ ( 2 ⁢ π ) - 1 2 ⁢ z ⁢ ( z + 1 ) + z ⁢ Ln ⁡ Γ ⁡ ( z + 1 ) - ∫ 0 z Ln ⁡ Γ ⁡ ( t + 1 ) ⁢ d t multivalued-natural-logarithm Barnes-Gamma 𝑧 1 1 2 𝑧 2 𝜋 1 2 𝑧 𝑧 1 𝑧 multivalued-natural-logarithm Euler-Gamma 𝑧 1 superscript subscript 0 𝑧 multivalued-natural-logarithm Euler-Gamma 𝑡 1 𝑡 {\displaystyle{\displaystyle\operatorname{Ln}G\left(z+1\right)=\tfrac{1}{2}z% \ln\left(2\pi\right)-\tfrac{1}{2}z(z+1)+z\operatorname{Ln}\Gamma\left(z+1% \right)-\int_{0}^{z}\operatorname{Ln}\Gamma\left(t+1\right)\mathrm{d}t}}
\Ln@@{\BarnesG@{z+1}} = \tfrac{1}{2}z\ln@{2\pi}-\tfrac{1}{2}z(z+1)+z\Ln@@{\EulerGamma@{z+1}}-\int_{0}^{z}\Ln@@{\EulerGamma@{t+1}}\diff{t}		 ℜ ⁡ ( z + 1 ) > 0 , ℜ ⁡ ( t + 1 ) > 0 formulae-sequence 𝑧 1 0 𝑡 1 0 {\displaystyle{\displaystyle\Re(z+1)>0,\Re(t+1)>0}} 
Error

Log[BarnesG[z + 1]] == Divide[1,2]*z*Log[2*Pi]-Divide[1,2]*z*(z + 1)+ z*Log[Gamma[z + 1]]- Integrate[Log[Gamma[t + 1]], {t, 0, z}, GenerateConditions->None]
Missing Macro Error Failure - Successful [Tested: 7]
5.17.E6 A = e C 𝐴 superscript 𝑒 𝐶 {\displaystyle{\displaystyle A=e^{C}}}
A = e^{C}		 

A = exp(C)
 
A == Exp[C]
 Skipped - no semantic math Skipped - no semantic math - -
5.17.E7 C = lim n → ∞ ⁡ ( ∑ k = 1 n k ⁢ ln ⁡ k - ( 1 2 ⁢ n 2 + 1 2 ⁢ n + 1 12 ) ⁢ ln ⁡ n + 1 4 ⁢ n 2 ) 𝐶 subscript → 𝑛 superscript subscript 𝑘 1 𝑛 𝑘 𝑘 1 2 superscript 𝑛 2 1 2 𝑛 1 12 𝑛 1 4 superscript 𝑛 2 {\displaystyle{\displaystyle C=\lim_{n\to\infty}\left(\sum_{k=1}^{n}k\ln k-% \left(\tfrac{1}{2}n^{2}+\tfrac{1}{2}n+\tfrac{1}{12}\right)\ln n+\tfrac{1}{4}n^% {2}\right)}}
C = \lim_{n\to\infty}\left(\sum_{k=1}^{n}k\ln@@{k}-\left(\tfrac{1}{2}n^{2}+\tfrac{1}{2}n+\tfrac{1}{12}\right)\ln@@{n}+\tfrac{1}{4}n^{2}\right)		 

C = limit(sum(k*ln(k), k = 1..n)-((1)/(2)*(n)^(2)+(1)/(2)*n +(1)/(12))*ln(n)+(1)/(4)*(n)^(2), n = infinity)

C == Limit[Sum[k*Log[k], {k, 1, n}, GenerateConditions->None]-(Divide[1,2]*(n)^(2)+Divide[1,2]*n +Divide[1,12])*Log[n]+Divide[1,4]*(n)^(2), n -> Infinity, GenerateConditions->None]
Failure Failure
Failed [10 / 10]
Result: .6172709270+.5000000000*I
Test Values: {C = 1/2*3^(1/2)+1/2*I}


Result: -.7487544770+.8660254040*I
Test Values: {C = -1/2+1/2*I*3^(1/2)}


Result: .2512455230-.8660254040*I
Test Values: {C = 1/2-1/2*I*3^(1/2)}


Result: -1.114779881-.5000000000*I
Test Values: {C = -1/2*3^(1/2)-1/2*I}

... skip entries to safe data
Failed [10 / 10]
Result: Complex[0.6172709267506544, 0.49999999999999994]
Test Values: {Rule[C, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.7487544770337841, 0.8660254037844387]
Test Values: {Rule[C, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
5.17.E7 lim n → ∞ ⁡ ( ∑ k = 1 n k ⁢ ln ⁡ k - ( 1 2 ⁢ n 2 + 1 2 ⁢ n + 1 12 ) ⁢ ln ⁡ n + 1 4 ⁢ n 2 ) = γ + ln ⁡ ( 2 ⁢ π ) 12 - ζ ′ ⁡ ( 2 ) 2 ⁢ π 2 subscript → 𝑛 superscript subscript 𝑘 1 𝑛 𝑘 𝑘 1 2 superscript 𝑛 2 1 2 𝑛 1 12 𝑛 1 4 superscript 𝑛 2 2 𝜋 12 diffop Riemann-zeta 1 2 2 superscript 𝜋 2 {\displaystyle{\displaystyle\lim_{n\to\infty}\left(\sum_{k=1}^{n}k\ln k-\left(% \tfrac{1}{2}n^{2}+\tfrac{1}{2}n+\tfrac{1}{12}\right)\ln n+\tfrac{1}{4}n^{2}% \right)=\frac{\gamma+\ln\left(2\pi\right)}{12}-\frac{\zeta'\left(2\right)}{2% \pi^{2}}}}
\lim_{n\to\infty}\left(\sum_{k=1}^{n}k\ln@@{k}-\left(\tfrac{1}{2}n^{2}+\tfrac{1}{2}n+\tfrac{1}{12}\right)\ln@@{n}+\tfrac{1}{4}n^{2}\right) = \frac{\EulerConstant+\ln@{2\pi}}{12}-\frac{\Riemannzeta'@{2}}{2\pi^{2}}		 

limit(sum(k*ln(k), k = 1..n)-((1)/(2)*(n)^(2)+(1)/(2)*n +(1)/(12))*ln(n)+(1)/(4)*(n)^(2), n = infinity) = (gamma + ln(2*Pi))/(12)-(subs( temp=2, diff( Zeta(temp), temp$(1) ) ))/(2*(Pi)^(2))

Limit[Sum[k*Log[k], {k, 1, n}, GenerateConditions->None]-(Divide[1,2]*(n)^(2)+Divide[1,2]*n +Divide[1,12])*Log[n]+Divide[1,4]*(n)^(2), n -> Infinity, GenerateConditions->None] == Divide[EulerGamma + Log[2*Pi],12]-Divide[D[Zeta[temp], {temp, 1}]/.temp-> 2,2*(Pi)^(2)]
Failure Successful Successful [Tested: 0] Successful [Tested: 1]
5.17.E7 γ + ln ⁡ ( 2 ⁢ π ) 12 - ζ ′ ⁡ ( 2 ) 2 ⁢ π 2 = 1 12 - ζ ′ ⁡ ( - 1 ) 2 𝜋 12 diffop Riemann-zeta 1 2 2 superscript 𝜋 2 1 12 diffop Riemann-zeta 1 1 {\displaystyle{\displaystyle\frac{\gamma+\ln\left(2\pi\right)}{12}-\frac{\zeta% '\left(2\right)}{2\pi^{2}}=\frac{1}{12}-\zeta'\left(-1\right)}}
\frac{\EulerConstant+\ln@{2\pi}}{12}-\frac{\Riemannzeta'@{2}}{2\pi^{2}} = \frac{1}{12}-\Riemannzeta'@{-1}		 

(gamma + ln(2*Pi))/(12)-(subs( temp=2, diff( Zeta(temp), temp$(1) ) ))/(2*(Pi)^(2)) = (1)/(12)- subs( temp=- 1, diff( Zeta(temp), temp$(1) ) )

Divide[EulerGamma + Log[2*Pi],12]-Divide[D[Zeta[temp], {temp, 1}]/.temp-> 2,2*(Pi)^(2)] == Divide[1,12]- (D[Zeta[temp], {temp, 1}]/.temp-> - 1)
Failure Successful Successful [Tested: 0] Successful [Tested: 1]
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