5.17

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
5.17#Ex1	${\displaystyle{\displaystyle G\left(z+1\right)=\Gamma\left(z\right)G\left(z\right)}}$ \BarnesG@{z+1} = \EulerGamma@{z}\BarnesG@{z}	${\displaystyle{\displaystyle\Re z>0}}$	Error	BarnesG[z + 1] == Gamma[z]*BarnesG[z]	Missing Macro Error	Failure	-	Successful [Tested: 5]
5.17#Ex2	${\displaystyle{\displaystyle G\left(1\right)=1}}$ \BarnesG@{1} = 1		Error	BarnesG[1] == 1	Missing Macro Error	Successful	-	Successful [Tested: 1]
5.17.E3	${\displaystyle{\displaystyle G\left(z+1\right)=(2\pi)^{z/2}\exp\left(-\tfrac{1}{2}z(z+1)-\tfrac{1}{2}\gamma z^{2}\right)\*\prod_{k=1}^{\infty}\left(\left(1+\frac{z}{k}\right)^{k}\exp\left(-z+\frac{z^{2}}{2k}\right)\right)}}$ \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	${\displaystyle{\displaystyle\operatorname{Ln}G\left(z+1\right)=\tfrac{1}{2}z\ln\left(2\pi\right)-\tfrac{1}{2}z(z+1)+z\operatorname{Ln}\Gamma\left(z+1\right)-\int_{0}^{z}\operatorname{Ln}\Gamma\left(t+1\right)\mathrm{d}t}}$ \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}	${\displaystyle{\displaystyle\Re(z+1)>0,\Re(t+1)>0}}$	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	${\displaystyle{\displaystyle A=e^{C}}}$ A = e^{C}		A = exp(C)	A == Exp[C]	Skipped - no semantic math	Skipped - no semantic math	-	-
5.17.E7	${\displaystyle{\displaystyle C=\lim_{n\to\infty}\left(\sum_{k=1}^{n}k\ln k-\left(\tfrac{1}{2}n^{2}+\tfrac{1}{2}n+\tfrac{1}{12}\right)\ln n+\tfrac{1}{4}n^{2}\right)}}$ 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	${\displaystyle{\displaystyle\lim_{n\to\infty}\left(\sum_{k=1}^{n}k\ln k-\left(\tfrac{1}{2}n^{2}+\tfrac{1}{2}n+\tfrac{1}{12}\right)\ln n+\tfrac{1}{4}n^{2}\right)=\frac{\gamma+\ln\left(2\pi\right)}{12}-\frac{\zeta'\left(2\right)}{2\pi^{2}}}}$ \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	${\displaystyle{\displaystyle\frac{\gamma+\ln\left(2\pi\right)}{12}-\frac{\zeta'\left(2\right)}{2\pi^{2}}=\frac{1}{12}-\zeta'\left(-1\right)}}$ \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]