5.9: Difference between revisions

Latest revision as of 11:12, 28 June 2021

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
5.9.E1	${\displaystyle{\displaystyle\frac{1}{\mu}\Gamma\left(\frac{\nu}{\mu}\right)% \frac{1}{z^{\nu/\mu}}=\int_{0}^{\infty}\exp\left(-zt^{\mu}\right)t^{\nu-1}% \mathrm{d}t}}$ `\frac{1}{\mu}\EulerGamma@{\frac{\nu}{\mu}}\frac{1}{z^{\nu/\mu}} = \int_{0}^{\infty}\exp@{-zt^{\mu}}t^{\nu-1}\diff{t}`		(1)/(mu)GAMMA((nu)/(mu))(1)/((z)^(nu/mu)) = int(exp(- z(t)^(mu))(t)^(nu - 1), t = 0..infinity)	Divide[1,\[Mu]]Gamma[Divide[\[Nu],\[Mu]]]Divide[1,(z)^(\[Nu]/\[Mu])] == Integrate[Exp[- z(t)^\[Mu]](t)^(\[Nu]- 1), {t, 0, Infinity}, GenerateConditions->None]	Failure	Aborted	Skipped - Because timed out	Successful [Tested: 300]
5.9.E2	${\displaystyle{\displaystyle\frac{1}{\Gamma\left(z\right)}=\frac{1}{2\pi i}% \int_{-\infty}^{(0+)}e^{t}t^{-z}\mathrm{d}t}}$ `\frac{1}{\EulerGamma@{z}} = \frac{1}{2\pi i}\int_{-\infty}^{(0+)}e^{t}t^{-z}\diff{t}`	${\displaystyle{\displaystyle\Re z>0}}$	(1)/(GAMMA(z)) = (1)/(2PiI)int(exp(t)(t)^(- z), t = - infinity..(0 +))	Divide[1,Gamma[z]] == Divide[1,2PiI]Integrate[Exp[t](t)^(- z), {t, - Infinity, (0 +)}, GenerateConditions->None]	Error	Failure	-	Error
5.9.E3	${\displaystyle{\displaystyle c^{-z}\Gamma\left(z\right)=\int_{-\infty}^{\infty% }\|t\|^{2z-1}e^{-ct^{2}}\mathrm{d}t}}$ `c^{-z}\EulerGamma@{z} = \int_{-\infty}^{\infty}\|t\|^{2z-1}e^{-ct^{2}}\diff{t}`	${\displaystyle{\displaystyle c>0,\Re z>0}}$	(c)^(- z)* GAMMA(z) = int((abs(t))^(2z - 1) exp(- c*(t)^(2)), t = - infinity..infinity)	(c)^(- z)* Gamma[z] == Integrate[(Abs[t])^(2z - 1) Exp[- c*(t)^(2)], {t, - Infinity, Infinity}, GenerateConditions->None]	Missing Macro Error	Missing Macro Error	-	-
5.9.E4	${\displaystyle{\displaystyle\Gamma\left(z\right)=\int_{1}^{\infty}t^{z-1}e^{-t% }\mathrm{d}t+\sum_{k=0}^{\infty}\frac{(-1)^{k}}{(z+k)k!}}}$ `\EulerGamma@{z} = \int_{1}^{\infty}t^{z-1}e^{-t}\diff{t}+\sum_{k=0}^{\infty}\frac{(-1)^{k}}{(z+k)k!}`	${\displaystyle{\displaystyle\Re z>0}}$	GAMMA(z) = int((t)^(z - 1)* exp(- t), t = 1..infinity)+ sum(((- 1)^(k))/((z + k)*factorial(k)), k = 0..infinity)	Gamma[z] == Integrate[(t)^(z - 1)* Exp[- t], {t, 1, Infinity}, GenerateConditions->None]+ Sum[Divide[(- 1)^(k),(z + k)*(k)!], {k, 0, Infinity}, GenerateConditions->None]	Failure	Successful	Failed [1 / 5] Result: .9999999999-0.*I Test Values: {z = 2, z = 1}	Successful [Tested: 1]
5.9.E5	${\displaystyle{\displaystyle\Gamma\left(z\right)=\int_{0}^{\infty}t^{z-1}\left% (e^{-t}-\sum_{k=0}^{n}\frac{(-1)^{k}t^{k}}{k!}\right)\mathrm{d}t}}$ `\EulerGamma@{z} = \int_{0}^{\infty}t^{z-1}\left(e^{-t}-\sum_{k=0}^{n}\frac{(-1)^{k}t^{k}}{k!}\right)\diff{t}`	${\displaystyle{\displaystyle-n-1<\Re z,\Re z<-n,\Re z>0}}$	GAMMA(z) = int((t)^(z - 1)(exp(- t)- sum(((- 1)^(k) (t)^(k))/(factorial(k)), k = 0..n)), t = 0..infinity)	Gamma[z] == Integrate[(t)^(z - 1)(Exp[- t]- Sum[Divide[(- 1)^(k) (t)^(k),(k)!], {k, 0, n}, GenerateConditions->None]), {t, 0, Infinity}, GenerateConditions->None]	Failure	Aborted	Error	Skip - No test values generated
5.9.E6	${\displaystyle{\displaystyle\Gamma\left(z\right)\cos\left(\tfrac{1}{2}\pi z% \right)=\int_{0}^{\infty}t^{z-1}\cos t\mathrm{d}t}}$ `\EulerGamma@{z}\cos@{\tfrac{1}{2}\pi z} = \int_{0}^{\infty}t^{z-1}\cos@@{t}\diff{t}`	${\displaystyle{\displaystyle 0<\Re z,\Re z<1,\Re z>0}}$	GAMMA(z)cos((1)/(2)Piz) = int((t)^(z - 1) cos(t), t = 0..infinity)	Gamma[z]Cos[Divide[1,2]Piz] == Integrate[(t)^(z - 1) Cos[t], {t, 0, Infinity}, GenerateConditions->None]	Successful	Successful	-	Successful [Tested: 3]
5.9.E7	${\displaystyle{\displaystyle\Gamma\left(z\right)\sin\left(\tfrac{1}{2}\pi z% \right)=\int_{0}^{\infty}t^{z-1}\sin t\mathrm{d}t}}$ `\EulerGamma@{z}\sin@{\tfrac{1}{2}\pi z} = \int_{0}^{\infty}t^{z-1}\sin@@{t}\diff{t}`	${\displaystyle{\displaystyle-1<\Re z,\Re z<1,\Re z>0}}$	GAMMA(z)sin((1)/(2)Piz) = int((t)^(z - 1) sin(t), t = 0..infinity)	Gamma[z]Sin[Divide[1,2]Piz] == Integrate[(t)^(z - 1) Sin[t], {t, 0, Infinity}, GenerateConditions->None]	Successful	Successful	-	Successful [Tested: 3]
5.9.E8	${\displaystyle{\displaystyle\Gamma\left(1+\frac{1}{n}\right)\cos\left(\frac{% \pi}{2n}\right)=\int_{0}^{\infty}\cos\left(t^{n}\right)\mathrm{d}t}}$ `\EulerGamma@{1+\frac{1}{n}}\cos@{\frac{\pi}{2n}} = \int_{0}^{\infty}\cos@{t^{n}}\diff{t}`		GAMMA(1 +(1)/(n))cos((Pi)/(2n)) = int(cos((t)^(n)), t = 0..infinity)	Gamma[1 +Divide[1,n]]Cos[Divide[Pi,2n]] == Integrate[Cos[(t)^(n)], {t, 0, Infinity}, GenerateConditions->None]	Successful	Successful	-	Successful [Tested: 1]
5.9.E9	${\displaystyle{\displaystyle\Gamma\left(1+\frac{1}{n}\right)\sin\left(\frac{% \pi}{2n}\right)=\int_{0}^{\infty}\sin\left(t^{n}\right)\mathrm{d}t}}$ `\EulerGamma@{1+\frac{1}{n}}\sin@{\frac{\pi}{2n}} = \int_{0}^{\infty}\sin@{t^{n}}\diff{t}`		GAMMA(1 +(1)/(n))sin((Pi)/(2n)) = int(sin((t)^(n)), t = 0..infinity)	Gamma[1 +Divide[1,n]]Sin[Divide[Pi,2n]] == Integrate[Sin[(t)^(n)], {t, 0, Infinity}, GenerateConditions->None]	Successful	Successful	-	Successful [Tested: 1]
5.9.E10	${\displaystyle{\displaystyle\operatorname{Ln}\Gamma\left(z\right)=\left(z-% \tfrac{1}{2}\right)\ln z-z+\tfrac{1}{2}\ln\left(2\pi\right)+2\int_{0}^{\infty}% \frac{\operatorname{arctan}\left(t/z\right)}{e^{2\pi t}-1}\mathrm{d}t}}$ `\Ln@@{\EulerGamma@{z}} = \left(z-\tfrac{1}{2}\right)\ln@@{z}-z+\tfrac{1}{2}\ln@{2\pi}+2\int_{0}^{\infty}\frac{\atan@{t/z}}{e^{2\pi t}-1}\diff{t}`	${\displaystyle{\displaystyle\Re z>0}}$	ln(GAMMA(z)) = (z -(1)/(2))ln(z)- z +(1)/(2)ln(2Pi)+ 2int((arctan(t/z))/(exp(2Pit)- 1), t = 0..infinity)	Log[Gamma[z]] == (z -Divide[1,2])Log[z]- z +Divide[1,2]Log[2Pi]+ 2Integrate[Divide[ArcTan[t/z],Exp[2Pit]- 1], {t, 0, Infinity}, GenerateConditions->None]	Failure	Aborted	Successful [Tested: 5]	Skipped - Because timed out
5.9.E11	${\displaystyle{\displaystyle\operatorname{Ln}\Gamma\left(z+1\right)=-\gamma z-% \frac{1}{2\pi i}\int_{-c-\infty i}^{-c+\infty i}\frac{\pi z^{-s}}{s\sin\left(% \pi s\right)}\zeta\left(-s\right)\mathrm{d}s}}$ `\Ln@@{\EulerGamma@{z+1}} = -\EulerConstant z-\frac{1}{2\pi i}\int_{-c-\infty i}^{-c+\infty i}\frac{\pi z^{-s}}{s\sin@{\pi s}}\Riemannzeta@{-s}\diff{s}`	${\displaystyle{\displaystyle\Re(z+1)>0}}$	ln(GAMMA(z + 1)) = - gammaz -(1)/(2PiI)int((Pi(z)^(- s))/(ssin(Pis))Zeta(- s), s = - c - infinityI..- c + infinityI)	Log[Gamma[z + 1]] == - EulerGammaz -Divide[1,2PiI]Integrate[Divide[Pi(z)^(- s),sSin[Pis]]Zeta[- s], {s, - c - InfinityI, - c + InfinityI}, GenerateConditions->None]	Failure	Aborted	Failed [42 / 42] Result: .3627983593+.4645558136I Test Values: {c = -1.5, z = 1/23^(1/2)+1/2I} Result: -.7321808519-.4375773776I Test Values: {c = -1.5, z = -1/2+1/2I3^(1/2)} Result: -.1549651868-.6096201737I Test Values: {c = -1.5, z = 1/2-1/2I3^(1/2)} Result: -.670593886e-1+1.175772123I Test Values: {c = -1.5, z = -1/23^(1/2)-1/2I} ... skip entries to safe data	Skipped - Because timed out
5.9.E12	${\displaystyle{\displaystyle\psi\left(z\right)=\int_{0}^{\infty}\left(\frac{e^% {-t}}{t}-\frac{e^{-zt}}{1-e^{-t}}\right)\mathrm{d}t}}$ `\digamma@{z} = \int_{0}^{\infty}\left(\frac{e^{-t}}{t}-\frac{e^{-zt}}{1-e^{-t}}\right)\diff{t}`		Psi(z) = int((exp(- t))/(t)-(exp(- z*t))/(1 - exp(- t)), t = 0..infinity)	PolyGamma[z] == Integrate[Divide[Exp[- t],t]-Divide[Exp[- z*t],1 - Exp[- t]], {t, 0, Infinity}, GenerateConditions->None]	Failure	Aborted	Failed [2 / 7] Result: Float(infinity)+Float(infinity)I Test Values: {z = -1/2+1/2I3^(1/2)} Result: Float(infinity)+Float(infinity)I Test Values: {z = -1/23^(1/2)-1/2I}	Successful [Tested: 1]
5.9.E13	${\displaystyle{\displaystyle\psi\left(z\right)=\ln z+\int_{0}^{\infty}\left(% \frac{1}{t}-\frac{1}{1-e^{-t}}\right)e^{-tz}\mathrm{d}t}}$ `\digamma@{z} = \ln@@{z}+\int_{0}^{\infty}\left(\frac{1}{t}-\frac{1}{1-e^{-t}}\right)e^{-tz}\diff{t}`		Psi(z) = ln(z)+ int(((1)/(t)-(1)/(1 - exp(- t)))exp(- tz), t = 0..infinity)	PolyGamma[z] == Log[z]+ Integrate[(Divide[1,t]-Divide[1,1 - Exp[- t]])Exp[- tz], {t, 0, Infinity}, GenerateConditions->None]	Failure	Aborted	Failed [2 / 7] Result: Float(infinity)+Float(infinity)I Test Values: {z = -1/2+1/2I3^(1/2)} Result: Float(infinity)+Float(infinity)I Test Values: {z = -1/23^(1/2)-1/2I}	Failed [1 / 1] Result: Indeterminate Test Values: {Rule[z, 1]}
5.9.E14	${\displaystyle{\displaystyle\psi\left(z\right)=\int_{0}^{\infty}\left(e^{-t}-% \frac{1}{(1+t)^{z}}\right)\frac{\mathrm{d}t}{t}}}$ `\digamma@{z} = \int_{0}^{\infty}\left(e^{-t}-\frac{1}{(1+t)^{z}}\right)\frac{\diff{t}}{t}`		Psi(z) = int((exp(- t)-(1)/((1 + t)^(z)))*(1)/(t), t = 0..infinity)	PolyGamma[z] == Integrate[(Exp[- t]-Divide[1,(1 + t)^(z)])*Divide[1,t], {t, 0, Infinity}, GenerateConditions->None]	Failure	Successful	Skipped - Because timed out	Successful [Tested: 7]
5.9.E15	${\displaystyle{\displaystyle\psi\left(z\right)=\ln z-\frac{1}{2z}-2\int_{0}^{% \infty}\frac{t\mathrm{d}t}{(t^{2}+z^{2})(e^{2\pi t}-1)}}}$ `\digamma@{z} = \ln@@{z}-\frac{1}{2z}-2\int_{0}^{\infty}\frac{t\diff{t}}{(t^{2}+z^{2})(e^{2\pi t}-1)}`		Psi(z) = ln(z)-(1)/(2z)- 2int((t)/(((t)^(2)+ (z)^(2))(exp(2Pi*t)- 1)), t = 0..infinity)	PolyGamma[z] == Log[z]-Divide[1,2z]- 2Integrate[Divide[t,((t)^(2)+ (z)^(2))(Exp[2Pi*t]- 1)], {t, 0, Infinity}, GenerateConditions->None]	Failure	Aborted	Failed [2 / 7] Result: .4e-10-.2711020420e-1I Test Values: {z = -1/2+1/2I3^(1/2)} Result: -.2144560970-.1791125126I Test Values: {z = -1/23^(1/2)-1/2I}	Successful [Tested: 1]
5.9.E16	${\displaystyle{\displaystyle\psi\left(z\right)+\gamma=\int_{0}^{\infty}\frac{e% ^{-t}-e^{-zt}}{1-e^{-t}}\mathrm{d}t}}$ `\digamma@{z}+\EulerConstant = \int_{0}^{\infty}\frac{e^{-t}-e^{-zt}}{1-e^{-t}}\diff{t}`		Psi(z)+ gamma = int((exp(- t)- exp(- z*t))/(1 - exp(- t)), t = 0..infinity)	PolyGamma[z]+ EulerGamma == Integrate[Divide[Exp[- t]- Exp[- z*t],1 - Exp[- t]], {t, 0, Infinity}, GenerateConditions->None]	Failure	Failure	Failed [2 / 7] Result: Float(infinity)+Float(infinity)I Test Values: {z = -1/2+1/2I3^(1/2)} Result: Float(infinity)+Float(infinity)I Test Values: {z = -1/23^(1/2)-1/2I}	Successful [Tested: 1]
5.9.E16	${\displaystyle{\displaystyle\int_{0}^{\infty}\frac{e^{-t}-e^{-zt}}{1-e^{-t}}% \mathrm{d}t=\int_{0}^{1}\frac{1-t^{z-1}}{1-t}\mathrm{d}t}}$ `\int_{0}^{\infty}\frac{e^{-t}-e^{-zt}}{1-e^{-t}}\diff{t} = \int_{0}^{1}\frac{1-t^{z-1}}{1-t}\diff{t}`		int((exp(- t)- exp(- z*t))/(1 - exp(- t)), t = 0..infinity) = int((1 - (t)^(z - 1))/(1 - t), t = 0..1)	Integrate[Divide[Exp[- t]- Exp[- z*t],1 - Exp[- t]], {t, 0, Infinity}, GenerateConditions->None] == Integrate[Divide[1 - (t)^(z - 1),1 - t], {t, 0, 1}, GenerateConditions->None]	Failure	Successful	Failed [2 / 7] Result: Float(infinity)+Float(infinity)I Test Values: {z = -1/2+1/2I3^(1/2)} Result: Float(infinity)+Float(infinity)I Test Values: {z = -1/23^(1/2)-1/2I}	Successful [Tested: 7]
5.9.E17	${\displaystyle{\displaystyle\psi\left(z+1\right)=-\gamma+\frac{1}{2\pi i}\int_% {-c-\infty i}^{-c+\infty i}\frac{\pi z^{-s-1}}{\sin\left(\pi s\right)}\zeta% \left(-s\right)\mathrm{d}s}}$ `\digamma@{z+1} = -\EulerConstant+\frac{1}{2\pi i}\int_{-c-\infty i}^{-c+\infty i}\frac{\pi z^{-s-1}}{\sin@{\pi s}}\Riemannzeta@{-s}\diff{s}`		Psi(z + 1) = - gamma +(1)/(2PiI)int((Pi(z)^(- s - 1))/(sin(Pis))Zeta(- s), s = - c - infinityI..- c + infinityI)	PolyGamma[z + 1] == - EulerGamma +Divide[1,2PiI]Integrate[Divide[Pi(z)^(- s - 1),Sin[Pis]]Zeta[- s], {s, - c - InfinityI, - c + InfinityI}, GenerateConditions->None]	Failure	Failure	Failed [42 / 42] Result: .9666504222+.3394950970I Test Values: {c = -1.5, z = 1/23^(1/2)+1/2I} Result: .3622891065+1.557241225I Test Values: {c = -1.5, z = -1/2+1/2I3^(1/2)} Result: .8622891063-.6912158211I Test Values: {c = -1.5, z = 1/2-1/2I3^(1/2)} Result: -.1138310784-2.481210069I Test Values: {c = -1.5, z = -1/23^(1/2)-1/2I} ... skip entries to safe data	Skipped - Because timed out
5.9.E18	${\displaystyle{\displaystyle\gamma=-\int_{0}^{\infty}e^{-t}\ln t\mathrm{d}t}}$ `\EulerConstant = -\int_{0}^{\infty}e^{-t}\ln@@{t}\diff{t}`		gamma = - int(exp(- t)*ln(t), t = 0..infinity)	EulerGamma == - Integrate[Exp[- t]*Log[t], {t, 0, Infinity}, GenerateConditions->None]	Successful	Successful	-	Successful [Tested: 1]
5.9.E18	${\displaystyle{\displaystyle-\int_{0}^{\infty}e^{-t}\ln t\mathrm{d}t=\int_{0}^% {\infty}\left(\frac{1}{1+t}-e^{-t}\right)\frac{\mathrm{d}t}{t}}}$ `-\int_{0}^{\infty}e^{-t}\ln@@{t}\diff{t} = \int_{0}^{\infty}\left(\frac{1}{1+t}-e^{-t}\right)\frac{\diff{t}}{t}`		- int(exp(- t)ln(t), t = 0..infinity) = int(((1)/(1 + t)- exp(- t))(1)/(t), t = 0..infinity)	- Integrate[Exp[- t]Log[t], {t, 0, Infinity}, GenerateConditions->None] == Integrate[(Divide[1,1 + t]- Exp[- t])Divide[1,t], {t, 0, Infinity}, GenerateConditions->None]	Successful	Successful	-	Successful [Tested: 1]
5.9.E18	${\displaystyle{\displaystyle\int_{0}^{\infty}\left(\frac{1}{1+t}-e^{-t}\right)% \frac{\mathrm{d}t}{t}=\int_{0}^{1}(1-e^{-t})\frac{\mathrm{d}t}{t}-\int_{1}^{% \infty}e^{-t}\frac{\mathrm{d}t}{t}}}$ `\int_{0}^{\infty}\left(\frac{1}{1+t}-e^{-t}\right)\frac{\diff{t}}{t} = \int_{0}^{1}(1-e^{-t})\frac{\diff{t}}{t}-\int_{1}^{\infty}e^{-t}\frac{\diff{t}}{t}`		int(((1)/(1 + t)- exp(- t))(1)/(t), t = 0..infinity) = int((1 - exp(- t))(1)/(t), t = 0..1)- int(exp(- t)*(1)/(t), t = 1..infinity)	Integrate[(Divide[1,1 + t]- Exp[- t])Divide[1,t], {t, 0, Infinity}, GenerateConditions->None] == Integrate[(1 - Exp[- t])Divide[1,t], {t, 0, 1}, GenerateConditions->None]- Integrate[Exp[- t]*Divide[1,t], {t, 1, Infinity}, GenerateConditions->None]	Successful	Successful	-	Successful [Tested: 1]
5.9.E18	${\displaystyle{\displaystyle\int_{0}^{1}(1-e^{-t})\frac{\mathrm{d}t}{t}-\int_{% 1}^{\infty}e^{-t}\frac{\mathrm{d}t}{t}=\int_{0}^{\infty}\left(\frac{e^{-t}}{1-% e^{-t}}-\frac{e^{-t}}{t}\right)\mathrm{d}t}}$ `\int_{0}^{1}(1-e^{-t})\frac{\diff{t}}{t}-\int_{1}^{\infty}e^{-t}\frac{\diff{t}}{t} = \int_{0}^{\infty}\left(\frac{e^{-t}}{1-e^{-t}}-\frac{e^{-t}}{t}\right)\diff{t}`		int((1 - exp(- t))(1)/(t), t = 0..1)- int(exp(- t)(1)/(t), t = 1..infinity) = int((exp(- t))/(1 - exp(- t))-(exp(- t))/(t), t = 0..infinity)	Integrate[(1 - Exp[- t])Divide[1,t], {t, 0, 1}, GenerateConditions->None]- Integrate[Exp[- t]Divide[1,t], {t, 1, Infinity}, GenerateConditions->None] == Integrate[Divide[Exp[- t],1 - Exp[- t]]-Divide[Exp[- t],t], {t, 0, Infinity}, GenerateConditions->None]	Successful	Successful	-	Successful [Tested: 1]
5.9.E19	${\displaystyle{\displaystyle{\Gamma^{(n)}}\left(z\right)=\int_{0}^{\infty}(\ln t% )^{n}e^{-t}t^{z-1}\mathrm{d}t}}$ `\EulerGamma^{(n)}@{z} = \int_{0}^{\infty}(\ln@@{t})^{n}e^{-t}t^{z-1}\diff{t}`	${\displaystyle{\displaystyle n\geq 0,\Re z>0}}$	diff( GAMMA(z), z$(n) ) = int((ln(t))^(n)* exp(- t)*(t)^(z - 1), t = 0..infinity)	D[Gamma[z], {z, n}] == Integrate[(Log[t])^(n)* Exp[- t]*(t)^(z - 1), {t, 0, Infinity}, GenerateConditions->None]	Successful	Aborted	-	Skipped - Because timed out

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 ! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
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-| [https://dlmf.nist.gov/5.9.E1 5.9.E1] || [[Item:Q2090|<math>\frac{1}{\mu}\EulerGamma@{\frac{\nu}{\mu}}\frac{1}{z^{\nu/\mu}} = \int_{0}^{\infty}\exp@{-zt^{\mu}}t^{\nu-1}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{1}{\mu}\EulerGamma@{\frac{\nu}{\mu}}\frac{1}{z^{\nu/\mu}} = \int_{0}^{\infty}\exp@{-zt^{\mu}}t^{\nu-1}\diff{t}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(1)/(mu)*GAMMA((nu)/(mu))*(1)/((z)^(nu/mu)) = int(exp(- z*(t)^(mu))*(t)^(nu - 1), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[1,\[Mu]]*Gamma[Divide[\[Nu],\[Mu]]]*Divide[1,(z)^(\[Nu]/\[Mu])] == Integrate[Exp[- z*(t)^\[Mu]]*(t)^(\[Nu]- 1), {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Successful [Tested: 300]
+| [https://dlmf.nist.gov/5.9.E1 5.9.E1] || <math qid="Q2090">\frac{1}{\mu}\EulerGamma@{\frac{\nu}{\mu}}\frac{1}{z^{\nu/\mu}} = \int_{0}^{\infty}\exp@{-zt^{\mu}}t^{\nu-1}\diff{t}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{1}{\mu}\EulerGamma@{\frac{\nu}{\mu}}\frac{1}{z^{\nu/\mu}} = \int_{0}^{\infty}\exp@{-zt^{\mu}}t^{\nu-1}\diff{t}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(1)/(mu)*GAMMA((nu)/(mu))*(1)/((z)^(nu/mu)) = int(exp(- z*(t)^(mu))*(t)^(nu - 1), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[1,\[Mu]]*Gamma[Divide[\[Nu],\[Mu]]]*Divide[1,(z)^(\[Nu]/\[Mu])] == Integrate[Exp[- z*(t)^\[Mu]]*(t)^(\[Nu]- 1), {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Successful [Tested: 300]
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-| [https://dlmf.nist.gov/5.9.E2 5.9.E2] || [[Item:Q2091|<math>\frac{1}{\EulerGamma@{z}} = \frac{1}{2\pi i}\int_{-\infty}^{(0+)}e^{t}t^{-z}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{1}{\EulerGamma@{z}} = \frac{1}{2\pi i}\int_{-\infty}^{(0+)}e^{t}t^{-z}\diff{t}</syntaxhighlight> || <math>\realpart@@{z} > 0</math> || <syntaxhighlight lang=mathematica>(1)/(GAMMA(z)) = (1)/(2*Pi*I)*int(exp(t)*(t)^(- z), t = - infinity..(0 +))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[1,Gamma[z]] == Divide[1,2*Pi*I]*Integrate[Exp[t]*(t)^(- z), {t, - Infinity, (0 +)}, GenerateConditions->None]</syntaxhighlight> || Error || Failure || - || Error
+| [https://dlmf.nist.gov/5.9.E2 5.9.E2] || <math qid="Q2091">\frac{1}{\EulerGamma@{z}} = \frac{1}{2\pi i}\int_{-\infty}^{(0+)}e^{t}t^{-z}\diff{t}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{1}{\EulerGamma@{z}} = \frac{1}{2\pi i}\int_{-\infty}^{(0+)}e^{t}t^{-z}\diff{t}</syntaxhighlight> || <math>\realpart@@{z} > 0</math> || <syntaxhighlight lang=mathematica>(1)/(GAMMA(z)) = (1)/(2*Pi*I)*int(exp(t)*(t)^(- z), t = - infinity..(0 +))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[1,Gamma[z]] == Divide[1,2*Pi*I]*Integrate[Exp[t]*(t)^(- z), {t, - Infinity, (0 +)}, GenerateConditions->None]</syntaxhighlight> || Error || Failure || - || Error
 |-
-| [https://dlmf.nist.gov/5.9.E3 5.9.E3] || [[Item:Q2092|<math>c^{-z}\EulerGamma@{z} = \int_{-\infty}^{\infty}|t|^{2z-1}e^{-ct^{2}}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>c^{-z}\EulerGamma@{z} = \int_{-\infty}^{\infty}|t|^{2z-1}e^{-ct^{2}}\diff{t}</syntaxhighlight> || <math>c > 0, \realpart@@{z} > 0</math> || <syntaxhighlight lang=mathematica>(c)^(- z)* GAMMA(z) = int((abs(t))^(2*z - 1)* exp(- c*(t)^(2)), t = - infinity..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(c)^(- z)* Gamma[z] == Integrate[(Abs[t])^(2*z - 1)* Exp[- c*(t)^(2)], {t, - Infinity, Infinity}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Missing Macro Error || - || -
+| [https://dlmf.nist.gov/5.9.E3 5.9.E3] || <math qid="Q2092">c^{-z}\EulerGamma@{z} = \int_{-\infty}^{\infty}|t|^{2z-1}e^{-ct^{2}}\diff{t}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>c^{-z}\EulerGamma@{z} = \int_{-\infty}^{\infty}|t|^{2z-1}e^{-ct^{2}}\diff{t}</syntaxhighlight> || <math>c > 0, \realpart@@{z} > 0</math> || <syntaxhighlight lang=mathematica>(c)^(- z)* GAMMA(z) = int((abs(t))^(2*z - 1)* exp(- c*(t)^(2)), t = - infinity..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(c)^(- z)* Gamma[z] == Integrate[(Abs[t])^(2*z - 1)* Exp[- c*(t)^(2)], {t, - Infinity, Infinity}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Missing Macro Error || - || -
 |-
-| [https://dlmf.nist.gov/5.9.E4 5.9.E4] || [[Item:Q2093|<math>\EulerGamma@{z} = \int_{1}^{\infty}t^{z-1}e^{-t}\diff{t}+\sum_{k=0}^{\infty}\frac{(-1)^{k}}{(z+k)k!}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerGamma@{z} = \int_{1}^{\infty}t^{z-1}e^{-t}\diff{t}+\sum_{k=0}^{\infty}\frac{(-1)^{k}}{(z+k)k!}</syntaxhighlight> || <math>\realpart@@{z} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA(z) = int((t)^(z - 1)* exp(- t), t = 1..infinity)+ sum(((- 1)^(k))/((z + k)*factorial(k)), k = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[z] == Integrate[(t)^(z - 1)* Exp[- t], {t, 1, Infinity}, GenerateConditions->None]+ Sum[Divide[(- 1)^(k),(z + k)*(k)!], {k, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Successful || <div class="toccolours mw-collapsible mw-collapsed">Failed [1 / 5]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .9999999999-0.*I
+| [https://dlmf.nist.gov/5.9.E4 5.9.E4] || <math qid="Q2093">\EulerGamma@{z} = \int_{1}^{\infty}t^{z-1}e^{-t}\diff{t}+\sum_{k=0}^{\infty}\frac{(-1)^{k}}{(z+k)k!}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerGamma@{z} = \int_{1}^{\infty}t^{z-1}e^{-t}\diff{t}+\sum_{k=0}^{\infty}\frac{(-1)^{k}}{(z+k)k!}</syntaxhighlight> || <math>\realpart@@{z} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA(z) = int((t)^(z - 1)* exp(- t), t = 1..infinity)+ sum(((- 1)^(k))/((z + k)*factorial(k)), k = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[z] == Integrate[(t)^(z - 1)* Exp[- t], {t, 1, Infinity}, GenerateConditions->None]+ Sum[Divide[(- 1)^(k),(z + k)*(k)!], {k, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Successful || <div class="toccolours mw-collapsible mw-collapsed">Failed [1 / 5]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .9999999999-0.*I
 Test Values: {z = 2, z = 1}</syntaxhighlight><br></div></div> || Successful [Tested: 1]
 |-
-| [https://dlmf.nist.gov/5.9.E5 5.9.E5] || [[Item:Q2094|<math>\EulerGamma@{z} = \int_{0}^{\infty}t^{z-1}\left(e^{-t}-\sum_{k=0}^{n}\frac{(-1)^{k}t^{k}}{k!}\right)\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerGamma@{z} = \int_{0}^{\infty}t^{z-1}\left(e^{-t}-\sum_{k=0}^{n}\frac{(-1)^{k}t^{k}}{k!}\right)\diff{t}</syntaxhighlight> || <math>-n-1 < \realpart@@{z}, \realpart@@{z} < -n, \realpart@@{z} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA(z) = int((t)^(z - 1)*(exp(- t)- sum(((- 1)^(k)* (t)^(k))/(factorial(k)), k = 0..n)), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[z] == Integrate[(t)^(z - 1)*(Exp[- t]- Sum[Divide[(- 1)^(k)* (t)^(k),(k)!], {k, 0, n}, GenerateConditions->None]), {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Error || Skip - No test values generated
+| [https://dlmf.nist.gov/5.9.E5 5.9.E5] || <math qid="Q2094">\EulerGamma@{z} = \int_{0}^{\infty}t^{z-1}\left(e^{-t}-\sum_{k=0}^{n}\frac{(-1)^{k}t^{k}}{k!}\right)\diff{t}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerGamma@{z} = \int_{0}^{\infty}t^{z-1}\left(e^{-t}-\sum_{k=0}^{n}\frac{(-1)^{k}t^{k}}{k!}\right)\diff{t}</syntaxhighlight> || <math>-n-1 < \realpart@@{z}, \realpart@@{z} < -n, \realpart@@{z} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA(z) = int((t)^(z - 1)*(exp(- t)- sum(((- 1)^(k)* (t)^(k))/(factorial(k)), k = 0..n)), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[z] == Integrate[(t)^(z - 1)*(Exp[- t]- Sum[Divide[(- 1)^(k)* (t)^(k),(k)!], {k, 0, n}, GenerateConditions->None]), {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Error || Skip - No test values generated
 |-
-| [https://dlmf.nist.gov/5.9.E6 5.9.E6] || [[Item:Q2095|<math>\EulerGamma@{z}\cos@{\tfrac{1}{2}\pi z} = \int_{0}^{\infty}t^{z-1}\cos@@{t}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerGamma@{z}\cos@{\tfrac{1}{2}\pi z} = \int_{0}^{\infty}t^{z-1}\cos@@{t}\diff{t}</syntaxhighlight> || <math>0 < \realpart@@{z}, \realpart@@{z} < 1, \realpart@@{z} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA(z)*cos((1)/(2)*Pi*z) = int((t)^(z - 1)* cos(t), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[z]*Cos[Divide[1,2]*Pi*z] == Integrate[(t)^(z - 1)* Cos[t], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 3]
+| [https://dlmf.nist.gov/5.9.E6 5.9.E6] || <math qid="Q2095">\EulerGamma@{z}\cos@{\tfrac{1}{2}\pi z} = \int_{0}^{\infty}t^{z-1}\cos@@{t}\diff{t}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerGamma@{z}\cos@{\tfrac{1}{2}\pi z} = \int_{0}^{\infty}t^{z-1}\cos@@{t}\diff{t}</syntaxhighlight> || <math>0 < \realpart@@{z}, \realpart@@{z} < 1, \realpart@@{z} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA(z)*cos((1)/(2)*Pi*z) = int((t)^(z - 1)* cos(t), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[z]*Cos[Divide[1,2]*Pi*z] == Integrate[(t)^(z - 1)* Cos[t], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 3]
 |-
-| [https://dlmf.nist.gov/5.9.E7 5.9.E7] || [[Item:Q2096|<math>\EulerGamma@{z}\sin@{\tfrac{1}{2}\pi z} = \int_{0}^{\infty}t^{z-1}\sin@@{t}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerGamma@{z}\sin@{\tfrac{1}{2}\pi z} = \int_{0}^{\infty}t^{z-1}\sin@@{t}\diff{t}</syntaxhighlight> || <math>-1 < \realpart@@{z}, \realpart@@{z} < 1, \realpart@@{z} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA(z)*sin((1)/(2)*Pi*z) = int((t)^(z - 1)* sin(t), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[z]*Sin[Divide[1,2]*Pi*z] == Integrate[(t)^(z - 1)* Sin[t], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 3]
+| [https://dlmf.nist.gov/5.9.E7 5.9.E7] || <math qid="Q2096">\EulerGamma@{z}\sin@{\tfrac{1}{2}\pi z} = \int_{0}^{\infty}t^{z-1}\sin@@{t}\diff{t}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerGamma@{z}\sin@{\tfrac{1}{2}\pi z} = \int_{0}^{\infty}t^{z-1}\sin@@{t}\diff{t}</syntaxhighlight> || <math>-1 < \realpart@@{z}, \realpart@@{z} < 1, \realpart@@{z} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA(z)*sin((1)/(2)*Pi*z) = int((t)^(z - 1)* sin(t), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[z]*Sin[Divide[1,2]*Pi*z] == Integrate[(t)^(z - 1)* Sin[t], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 3]
 |-
-| [https://dlmf.nist.gov/5.9.E8 5.9.E8] || [[Item:Q2097|<math>\EulerGamma@{1+\frac{1}{n}}\cos@{\frac{\pi}{2n}} = \int_{0}^{\infty}\cos@{t^{n}}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerGamma@{1+\frac{1}{n}}\cos@{\frac{\pi}{2n}} = \int_{0}^{\infty}\cos@{t^{n}}\diff{t}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>GAMMA(1 +(1)/(n))*cos((Pi)/(2*n)) = int(cos((t)^(n)), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[1 +Divide[1,n]]*Cos[Divide[Pi,2*n]] == Integrate[Cos[(t)^(n)], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 1]
+| [https://dlmf.nist.gov/5.9.E8 5.9.E8] || <math qid="Q2097">\EulerGamma@{1+\frac{1}{n}}\cos@{\frac{\pi}{2n}} = \int_{0}^{\infty}\cos@{t^{n}}\diff{t}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerGamma@{1+\frac{1}{n}}\cos@{\frac{\pi}{2n}} = \int_{0}^{\infty}\cos@{t^{n}}\diff{t}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>GAMMA(1 +(1)/(n))*cos((Pi)/(2*n)) = int(cos((t)^(n)), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[1 +Divide[1,n]]*Cos[Divide[Pi,2*n]] == Integrate[Cos[(t)^(n)], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 1]
 |-
-| [https://dlmf.nist.gov/5.9.E9 5.9.E9] || [[Item:Q2098|<math>\EulerGamma@{1+\frac{1}{n}}\sin@{\frac{\pi}{2n}} = \int_{0}^{\infty}\sin@{t^{n}}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerGamma@{1+\frac{1}{n}}\sin@{\frac{\pi}{2n}} = \int_{0}^{\infty}\sin@{t^{n}}\diff{t}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>GAMMA(1 +(1)/(n))*sin((Pi)/(2*n)) = int(sin((t)^(n)), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[1 +Divide[1,n]]*Sin[Divide[Pi,2*n]] == Integrate[Sin[(t)^(n)], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 1]
+| [https://dlmf.nist.gov/5.9.E9 5.9.E9] || <math qid="Q2098">\EulerGamma@{1+\frac{1}{n}}\sin@{\frac{\pi}{2n}} = \int_{0}^{\infty}\sin@{t^{n}}\diff{t}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerGamma@{1+\frac{1}{n}}\sin@{\frac{\pi}{2n}} = \int_{0}^{\infty}\sin@{t^{n}}\diff{t}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>GAMMA(1 +(1)/(n))*sin((Pi)/(2*n)) = int(sin((t)^(n)), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[1 +Divide[1,n]]*Sin[Divide[Pi,2*n]] == Integrate[Sin[(t)^(n)], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 1]
 |-
-| [https://dlmf.nist.gov/5.9.E10 5.9.E10] || [[Item:Q2099|<math>\Ln@@{\EulerGamma@{z}} = \left(z-\tfrac{1}{2}\right)\ln@@{z}-z+\tfrac{1}{2}\ln@{2\pi}+2\int_{0}^{\infty}\frac{\atan@{t/z}}{e^{2\pi t}-1}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Ln@@{\EulerGamma@{z}} = \left(z-\tfrac{1}{2}\right)\ln@@{z}-z+\tfrac{1}{2}\ln@{2\pi}+2\int_{0}^{\infty}\frac{\atan@{t/z}}{e^{2\pi t}-1}\diff{t}</syntaxhighlight> || <math>\realpart@@{z} > 0</math> || <syntaxhighlight lang=mathematica>ln(GAMMA(z)) = (z -(1)/(2))*ln(z)- z +(1)/(2)*ln(2*Pi)+ 2*int((arctan(t/z))/(exp(2*Pi*t)- 1), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Log[Gamma[z]] == (z -Divide[1,2])*Log[z]- z +Divide[1,2]*Log[2*Pi]+ 2*Integrate[Divide[ArcTan[t/z],Exp[2*Pi*t]- 1], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Successful [Tested: 5] || Skipped - Because timed out
+| [https://dlmf.nist.gov/5.9.E10 5.9.E10] || <math qid="Q2099">\Ln@@{\EulerGamma@{z}} = \left(z-\tfrac{1}{2}\right)\ln@@{z}-z+\tfrac{1}{2}\ln@{2\pi}+2\int_{0}^{\infty}\frac{\atan@{t/z}}{e^{2\pi t}-1}\diff{t}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Ln@@{\EulerGamma@{z}} = \left(z-\tfrac{1}{2}\right)\ln@@{z}-z+\tfrac{1}{2}\ln@{2\pi}+2\int_{0}^{\infty}\frac{\atan@{t/z}}{e^{2\pi t}-1}\diff{t}</syntaxhighlight> || <math>\realpart@@{z} > 0</math> || <syntaxhighlight lang=mathematica>ln(GAMMA(z)) = (z -(1)/(2))*ln(z)- z +(1)/(2)*ln(2*Pi)+ 2*int((arctan(t/z))/(exp(2*Pi*t)- 1), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Log[Gamma[z]] == (z -Divide[1,2])*Log[z]- z +Divide[1,2]*Log[2*Pi]+ 2*Integrate[Divide[ArcTan[t/z],Exp[2*Pi*t]- 1], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Successful [Tested: 5] || Skipped - Because timed out
 |-
-| [https://dlmf.nist.gov/5.9.E11 5.9.E11] || [[Item:Q2100|<math>\Ln@@{\EulerGamma@{z+1}} = -\EulerConstant z-\frac{1}{2\pi i}\int_{-c-\infty i}^{-c+\infty i}\frac{\pi z^{-s}}{s\sin@{\pi s}}\Riemannzeta@{-s}\diff{s}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Ln@@{\EulerGamma@{z+1}} = -\EulerConstant z-\frac{1}{2\pi i}\int_{-c-\infty i}^{-c+\infty i}\frac{\pi z^{-s}}{s\sin@{\pi s}}\Riemannzeta@{-s}\diff{s}</syntaxhighlight> || <math>\realpart@@{(z+1)} > 0</math> || <syntaxhighlight lang=mathematica>ln(GAMMA(z + 1)) = - gamma*z -(1)/(2*Pi*I)*int((Pi*(z)^(- s))/(s*sin(Pi*s))*Zeta(- s), s = - c - infinity*I..- c + infinity*I)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Log[Gamma[z + 1]] == - EulerGamma*z -Divide[1,2*Pi*I]*Integrate[Divide[Pi*(z)^(- s),s*Sin[Pi*s]]*Zeta[- s], {s, - c - Infinity*I, - c + Infinity*I}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || <div class="toccolours mw-collapsible mw-collapsed">Failed [42 / 42]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .3627983593+.4645558136*I
+| [https://dlmf.nist.gov/5.9.E11 5.9.E11] || <math qid="Q2100">\Ln@@{\EulerGamma@{z+1}} = -\EulerConstant z-\frac{1}{2\pi i}\int_{-c-\infty i}^{-c+\infty i}\frac{\pi z^{-s}}{s\sin@{\pi s}}\Riemannzeta@{-s}\diff{s}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Ln@@{\EulerGamma@{z+1}} = -\EulerConstant z-\frac{1}{2\pi i}\int_{-c-\infty i}^{-c+\infty i}\frac{\pi z^{-s}}{s\sin@{\pi s}}\Riemannzeta@{-s}\diff{s}</syntaxhighlight> || <math>\realpart@@{(z+1)} > 0</math> || <syntaxhighlight lang=mathematica>ln(GAMMA(z + 1)) = - gamma*z -(1)/(2*Pi*I)*int((Pi*(z)^(- s))/(s*sin(Pi*s))*Zeta(- s), s = - c - infinity*I..- c + infinity*I)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Log[Gamma[z + 1]] == - EulerGamma*z -Divide[1,2*Pi*I]*Integrate[Divide[Pi*(z)^(- s),s*Sin[Pi*s]]*Zeta[- s], {s, - c - Infinity*I, - c + Infinity*I}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || <div class="toccolours mw-collapsible mw-collapsed">Failed [42 / 42]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .3627983593+.4645558136*I
 Test Values: {c = -1.5, z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.7321808519-.4375773776*I
 Test Values: {c = -1.5, z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.1549651868-.6096201737*I
@@ Line 41: / Line 41: @@
 Test Values: {c = -1.5, z = -1/2*3^(1/2)-1/2*I}</syntaxhighlight><br>... skip entries to safe data</div></div> || Skipped - Because timed out
 |-
-| [https://dlmf.nist.gov/5.9.E12 5.9.E12] || [[Item:Q2101|<math>\digamma@{z} = \int_{0}^{\infty}\left(\frac{e^{-t}}{t}-\frac{e^{-zt}}{1-e^{-t}}\right)\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\digamma@{z} = \int_{0}^{\infty}\left(\frac{e^{-t}}{t}-\frac{e^{-zt}}{1-e^{-t}}\right)\diff{t}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Psi(z) = int((exp(- t))/(t)-(exp(- z*t))/(1 - exp(- t)), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>PolyGamma[z] == Integrate[Divide[Exp[- t],t]-Divide[Exp[- z*t],1 - Exp[- t]], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Float(infinity)+Float(infinity)*I
+| [https://dlmf.nist.gov/5.9.E12 5.9.E12] || <math qid="Q2101">\digamma@{z} = \int_{0}^{\infty}\left(\frac{e^{-t}}{t}-\frac{e^{-zt}}{1-e^{-t}}\right)\diff{t}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\digamma@{z} = \int_{0}^{\infty}\left(\frac{e^{-t}}{t}-\frac{e^{-zt}}{1-e^{-t}}\right)\diff{t}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Psi(z) = int((exp(- t))/(t)-(exp(- z*t))/(1 - exp(- t)), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>PolyGamma[z] == Integrate[Divide[Exp[- t],t]-Divide[Exp[- z*t],1 - Exp[- t]], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Float(infinity)+Float(infinity)*I
 Test Values: {z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Float(infinity)+Float(infinity)*I
 Test Values: {z = -1/2*3^(1/2)-1/2*I}</syntaxhighlight><br></div></div> || Successful [Tested: 1]
 |-
-| [https://dlmf.nist.gov/5.9.E13 5.9.E13] || [[Item:Q2102|<math>\digamma@{z} = \ln@@{z}+\int_{0}^{\infty}\left(\frac{1}{t}-\frac{1}{1-e^{-t}}\right)e^{-tz}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\digamma@{z} = \ln@@{z}+\int_{0}^{\infty}\left(\frac{1}{t}-\frac{1}{1-e^{-t}}\right)e^{-tz}\diff{t}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Psi(z) = ln(z)+ int(((1)/(t)-(1)/(1 - exp(- t)))*exp(- t*z), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>PolyGamma[z] == Log[z]+ Integrate[(Divide[1,t]-Divide[1,1 - Exp[- t]])*Exp[- t*z], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Float(infinity)+Float(infinity)*I
+| [https://dlmf.nist.gov/5.9.E13 5.9.E13] || <math qid="Q2102">\digamma@{z} = \ln@@{z}+\int_{0}^{\infty}\left(\frac{1}{t}-\frac{1}{1-e^{-t}}\right)e^{-tz}\diff{t}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\digamma@{z} = \ln@@{z}+\int_{0}^{\infty}\left(\frac{1}{t}-\frac{1}{1-e^{-t}}\right)e^{-tz}\diff{t}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Psi(z) = ln(z)+ int(((1)/(t)-(1)/(1 - exp(- t)))*exp(- t*z), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>PolyGamma[z] == Log[z]+ Integrate[(Divide[1,t]-Divide[1,1 - Exp[- t]])*Exp[- t*z], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Float(infinity)+Float(infinity)*I
 Test Values: {z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Float(infinity)+Float(infinity)*I
 Test Values: {z = -1/2*3^(1/2)-1/2*I}</syntaxhighlight><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [1 / 1]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Indeterminate
 Test Values: {Rule[z, 1]}</syntaxhighlight><br></div></div>
 |-
-| [https://dlmf.nist.gov/5.9.E14 5.9.E14] || [[Item:Q2103|<math>\digamma@{z} = \int_{0}^{\infty}\left(e^{-t}-\frac{1}{(1+t)^{z}}\right)\frac{\diff{t}}{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\digamma@{z} = \int_{0}^{\infty}\left(e^{-t}-\frac{1}{(1+t)^{z}}\right)\frac{\diff{t}}{t}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Psi(z) = int((exp(- t)-(1)/((1 + t)^(z)))*(1)/(t), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>PolyGamma[z] == Integrate[(Exp[- t]-Divide[1,(1 + t)^(z)])*Divide[1,t], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Successful || Skipped - Because timed out || Successful [Tested: 7]
+| [https://dlmf.nist.gov/5.9.E14 5.9.E14] || <math qid="Q2103">\digamma@{z} = \int_{0}^{\infty}\left(e^{-t}-\frac{1}{(1+t)^{z}}\right)\frac{\diff{t}}{t}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\digamma@{z} = \int_{0}^{\infty}\left(e^{-t}-\frac{1}{(1+t)^{z}}\right)\frac{\diff{t}}{t}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Psi(z) = int((exp(- t)-(1)/((1 + t)^(z)))*(1)/(t), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>PolyGamma[z] == Integrate[(Exp[- t]-Divide[1,(1 + t)^(z)])*Divide[1,t], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Successful || Skipped - Because timed out || Successful [Tested: 7]
 |-
-| [https://dlmf.nist.gov/5.9.E15 5.9.E15] || [[Item:Q2104|<math>\digamma@{z} = \ln@@{z}-\frac{1}{2z}-2\int_{0}^{\infty}\frac{t\diff{t}}{(t^{2}+z^{2})(e^{2\pi t}-1)}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\digamma@{z} = \ln@@{z}-\frac{1}{2z}-2\int_{0}^{\infty}\frac{t\diff{t}}{(t^{2}+z^{2})(e^{2\pi t}-1)}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Psi(z) = ln(z)-(1)/(2*z)- 2*int((t)/(((t)^(2)+ (z)^(2))*(exp(2*Pi*t)- 1)), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>PolyGamma[z] == Log[z]-Divide[1,2*z]- 2*Integrate[Divide[t,((t)^(2)+ (z)^(2))*(Exp[2*Pi*t]- 1)], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .4e-10-.2711020420e-1*I
+| [https://dlmf.nist.gov/5.9.E15 5.9.E15] || <math qid="Q2104">\digamma@{z} = \ln@@{z}-\frac{1}{2z}-2\int_{0}^{\infty}\frac{t\diff{t}}{(t^{2}+z^{2})(e^{2\pi t}-1)}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\digamma@{z} = \ln@@{z}-\frac{1}{2z}-2\int_{0}^{\infty}\frac{t\diff{t}}{(t^{2}+z^{2})(e^{2\pi t}-1)}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Psi(z) = ln(z)-(1)/(2*z)- 2*int((t)/(((t)^(2)+ (z)^(2))*(exp(2*Pi*t)- 1)), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>PolyGamma[z] == Log[z]-Divide[1,2*z]- 2*Integrate[Divide[t,((t)^(2)+ (z)^(2))*(Exp[2*Pi*t]- 1)], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .4e-10-.2711020420e-1*I
 Test Values: {z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.2144560970-.1791125126*I
 Test Values: {z = -1/2*3^(1/2)-1/2*I}</syntaxhighlight><br></div></div> || Successful [Tested: 1]
 |-
-| [https://dlmf.nist.gov/5.9.E16 5.9.E16] || [[Item:Q2105|<math>\digamma@{z}+\EulerConstant = \int_{0}^{\infty}\frac{e^{-t}-e^{-zt}}{1-e^{-t}}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\digamma@{z}+\EulerConstant = \int_{0}^{\infty}\frac{e^{-t}-e^{-zt}}{1-e^{-t}}\diff{t}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Psi(z)+ gamma = int((exp(- t)- exp(- z*t))/(1 - exp(- t)), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>PolyGamma[z]+ EulerGamma == Integrate[Divide[Exp[- t]- Exp[- z*t],1 - Exp[- t]], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Float(infinity)+Float(infinity)*I
+| [https://dlmf.nist.gov/5.9.E16 5.9.E16] || <math qid="Q2105">\digamma@{z}+\EulerConstant = \int_{0}^{\infty}\frac{e^{-t}-e^{-zt}}{1-e^{-t}}\diff{t}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\digamma@{z}+\EulerConstant = \int_{0}^{\infty}\frac{e^{-t}-e^{-zt}}{1-e^{-t}}\diff{t}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Psi(z)+ gamma = int((exp(- t)- exp(- z*t))/(1 - exp(- t)), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>PolyGamma[z]+ EulerGamma == Integrate[Divide[Exp[- t]- Exp[- z*t],1 - Exp[- t]], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Float(infinity)+Float(infinity)*I
 Test Values: {z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Float(infinity)+Float(infinity)*I
 Test Values: {z = -1/2*3^(1/2)-1/2*I}</syntaxhighlight><br></div></div> || Successful [Tested: 1]
 |-
-| [https://dlmf.nist.gov/5.9.E16 5.9.E16] || [[Item:Q2105|<math>\int_{0}^{\infty}\frac{e^{-t}-e^{-zt}}{1-e^{-t}}\diff{t} = \int_{0}^{1}\frac{1-t^{z-1}}{1-t}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\infty}\frac{e^{-t}-e^{-zt}}{1-e^{-t}}\diff{t} = \int_{0}^{1}\frac{1-t^{z-1}}{1-t}\diff{t}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>int((exp(- t)- exp(- z*t))/(1 - exp(- t)), t = 0..infinity) = int((1 - (t)^(z - 1))/(1 - t), t = 0..1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Divide[Exp[- t]- Exp[- z*t],1 - Exp[- t]], {t, 0, Infinity}, GenerateConditions->None] == Integrate[Divide[1 - (t)^(z - 1),1 - t], {t, 0, 1}, GenerateConditions->None]</syntaxhighlight> || Failure || Successful || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Float(infinity)+Float(infinity)*I
+| [https://dlmf.nist.gov/5.9.E16 5.9.E16] || <math qid="Q2105">\int_{0}^{\infty}\frac{e^{-t}-e^{-zt}}{1-e^{-t}}\diff{t} = \int_{0}^{1}\frac{1-t^{z-1}}{1-t}\diff{t}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\infty}\frac{e^{-t}-e^{-zt}}{1-e^{-t}}\diff{t} = \int_{0}^{1}\frac{1-t^{z-1}}{1-t}\diff{t}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>int((exp(- t)- exp(- z*t))/(1 - exp(- t)), t = 0..infinity) = int((1 - (t)^(z - 1))/(1 - t), t = 0..1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Divide[Exp[- t]- Exp[- z*t],1 - Exp[- t]], {t, 0, Infinity}, GenerateConditions->None] == Integrate[Divide[1 - (t)^(z - 1),1 - t], {t, 0, 1}, GenerateConditions->None]</syntaxhighlight> || Failure || Successful || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Float(infinity)+Float(infinity)*I
 Test Values: {z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Float(infinity)+Float(infinity)*I
 Test Values: {z = -1/2*3^(1/2)-1/2*I}</syntaxhighlight><br></div></div> || Successful [Tested: 7]
 |-
-| [https://dlmf.nist.gov/5.9.E17 5.9.E17] || [[Item:Q2106|<math>\digamma@{z+1} = -\EulerConstant+\frac{1}{2\pi i}\int_{-c-\infty i}^{-c+\infty i}\frac{\pi z^{-s-1}}{\sin@{\pi s}}\Riemannzeta@{-s}\diff{s}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\digamma@{z+1} = -\EulerConstant+\frac{1}{2\pi i}\int_{-c-\infty i}^{-c+\infty i}\frac{\pi z^{-s-1}}{\sin@{\pi s}}\Riemannzeta@{-s}\diff{s}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Psi(z + 1) = - gamma +(1)/(2*Pi*I)*int((Pi*(z)^(- s - 1))/(sin(Pi*s))*Zeta(- s), s = - c - infinity*I..- c + infinity*I)</syntaxhighlight> || <syntaxhighlight lang=mathematica>PolyGamma[z + 1] == - EulerGamma +Divide[1,2*Pi*I]*Integrate[Divide[Pi*(z)^(- s - 1),Sin[Pi*s]]*Zeta[- s], {s, - c - Infinity*I, - c + Infinity*I}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [42 / 42]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .9666504222+.3394950970*I
+| [https://dlmf.nist.gov/5.9.E17 5.9.E17] || <math qid="Q2106">\digamma@{z+1} = -\EulerConstant+\frac{1}{2\pi i}\int_{-c-\infty i}^{-c+\infty i}\frac{\pi z^{-s-1}}{\sin@{\pi s}}\Riemannzeta@{-s}\diff{s}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\digamma@{z+1} = -\EulerConstant+\frac{1}{2\pi i}\int_{-c-\infty i}^{-c+\infty i}\frac{\pi z^{-s-1}}{\sin@{\pi s}}\Riemannzeta@{-s}\diff{s}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Psi(z + 1) = - gamma +(1)/(2*Pi*I)*int((Pi*(z)^(- s - 1))/(sin(Pi*s))*Zeta(- s), s = - c - infinity*I..- c + infinity*I)</syntaxhighlight> || <syntaxhighlight lang=mathematica>PolyGamma[z + 1] == - EulerGamma +Divide[1,2*Pi*I]*Integrate[Divide[Pi*(z)^(- s - 1),Sin[Pi*s]]*Zeta[- s], {s, - c - Infinity*I, - c + Infinity*I}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [42 / 42]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .9666504222+.3394950970*I
 Test Values: {c = -1.5, z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .3622891065+1.557241225*I
 Test Values: {c = -1.5, z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .8622891063-.6912158211*I
@@ Line 70: / Line 70: @@
 Test Values: {c = -1.5, z = -1/2*3^(1/2)-1/2*I}</syntaxhighlight><br>... skip entries to safe data</div></div> || Skipped - Because timed out
 |-
-| [https://dlmf.nist.gov/5.9.E18 5.9.E18] || [[Item:Q2107|<math>\EulerConstant = -\int_{0}^{\infty}e^{-t}\ln@@{t}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerConstant = -\int_{0}^{\infty}e^{-t}\ln@@{t}\diff{t}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>gamma = - int(exp(- t)*ln(t), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>EulerGamma == - Integrate[Exp[- t]*Log[t], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 1]
+| [https://dlmf.nist.gov/5.9.E18 5.9.E18] || <math qid="Q2107">\EulerConstant = -\int_{0}^{\infty}e^{-t}\ln@@{t}\diff{t}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerConstant = -\int_{0}^{\infty}e^{-t}\ln@@{t}\diff{t}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>gamma = - int(exp(- t)*ln(t), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>EulerGamma == - Integrate[Exp[- t]*Log[t], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 1]
 |-
-| [https://dlmf.nist.gov/5.9.E18 5.9.E18] || [[Item:Q2107|<math>-\int_{0}^{\infty}e^{-t}\ln@@{t}\diff{t} = \int_{0}^{\infty}\left(\frac{1}{1+t}-e^{-t}\right)\frac{\diff{t}}{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>-\int_{0}^{\infty}e^{-t}\ln@@{t}\diff{t} = \int_{0}^{\infty}\left(\frac{1}{1+t}-e^{-t}\right)\frac{\diff{t}}{t}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>- int(exp(- t)*ln(t), t = 0..infinity) = int(((1)/(1 + t)- exp(- t))*(1)/(t), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>- Integrate[Exp[- t]*Log[t], {t, 0, Infinity}, GenerateConditions->None] == Integrate[(Divide[1,1 + t]- Exp[- t])*Divide[1,t], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 1]
+| [https://dlmf.nist.gov/5.9.E18 5.9.E18] || <math qid="Q2107">-\int_{0}^{\infty}e^{-t}\ln@@{t}\diff{t} = \int_{0}^{\infty}\left(\frac{1}{1+t}-e^{-t}\right)\frac{\diff{t}}{t}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>-\int_{0}^{\infty}e^{-t}\ln@@{t}\diff{t} = \int_{0}^{\infty}\left(\frac{1}{1+t}-e^{-t}\right)\frac{\diff{t}}{t}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>- int(exp(- t)*ln(t), t = 0..infinity) = int(((1)/(1 + t)- exp(- t))*(1)/(t), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>- Integrate[Exp[- t]*Log[t], {t, 0, Infinity}, GenerateConditions->None] == Integrate[(Divide[1,1 + t]- Exp[- t])*Divide[1,t], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 1]
 |-
-| [https://dlmf.nist.gov/5.9.E18 5.9.E18] || [[Item:Q2107|<math>\int_{0}^{\infty}\left(\frac{1}{1+t}-e^{-t}\right)\frac{\diff{t}}{t} = \int_{0}^{1}(1-e^{-t})\frac{\diff{t}}{t}-\int_{1}^{\infty}e^{-t}\frac{\diff{t}}{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\infty}\left(\frac{1}{1+t}-e^{-t}\right)\frac{\diff{t}}{t} = \int_{0}^{1}(1-e^{-t})\frac{\diff{t}}{t}-\int_{1}^{\infty}e^{-t}\frac{\diff{t}}{t}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>int(((1)/(1 + t)- exp(- t))*(1)/(t), t = 0..infinity) = int((1 - exp(- t))*(1)/(t), t = 0..1)- int(exp(- t)*(1)/(t), t = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[(Divide[1,1 + t]- Exp[- t])*Divide[1,t], {t, 0, Infinity}, GenerateConditions->None] == Integrate[(1 - Exp[- t])*Divide[1,t], {t, 0, 1}, GenerateConditions->None]- Integrate[Exp[- t]*Divide[1,t], {t, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 1]
+| [https://dlmf.nist.gov/5.9.E18 5.9.E18] || <math qid="Q2107">\int_{0}^{\infty}\left(\frac{1}{1+t}-e^{-t}\right)\frac{\diff{t}}{t} = \int_{0}^{1}(1-e^{-t})\frac{\diff{t}}{t}-\int_{1}^{\infty}e^{-t}\frac{\diff{t}}{t}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\infty}\left(\frac{1}{1+t}-e^{-t}\right)\frac{\diff{t}}{t} = \int_{0}^{1}(1-e^{-t})\frac{\diff{t}}{t}-\int_{1}^{\infty}e^{-t}\frac{\diff{t}}{t}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>int(((1)/(1 + t)- exp(- t))*(1)/(t), t = 0..infinity) = int((1 - exp(- t))*(1)/(t), t = 0..1)- int(exp(- t)*(1)/(t), t = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[(Divide[1,1 + t]- Exp[- t])*Divide[1,t], {t, 0, Infinity}, GenerateConditions->None] == Integrate[(1 - Exp[- t])*Divide[1,t], {t, 0, 1}, GenerateConditions->None]- Integrate[Exp[- t]*Divide[1,t], {t, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 1]
 |-
-| [https://dlmf.nist.gov/5.9.E18 5.9.E18] || [[Item:Q2107|<math>\int_{0}^{1}(1-e^{-t})\frac{\diff{t}}{t}-\int_{1}^{\infty}e^{-t}\frac{\diff{t}}{t} = \int_{0}^{\infty}\left(\frac{e^{-t}}{1-e^{-t}}-\frac{e^{-t}}{t}\right)\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{1}(1-e^{-t})\frac{\diff{t}}{t}-\int_{1}^{\infty}e^{-t}\frac{\diff{t}}{t} = \int_{0}^{\infty}\left(\frac{e^{-t}}{1-e^{-t}}-\frac{e^{-t}}{t}\right)\diff{t}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>int((1 - exp(- t))*(1)/(t), t = 0..1)- int(exp(- t)*(1)/(t), t = 1..infinity) = int((exp(- t))/(1 - exp(- t))-(exp(- t))/(t), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[(1 - Exp[- t])*Divide[1,t], {t, 0, 1}, GenerateConditions->None]- Integrate[Exp[- t]*Divide[1,t], {t, 1, Infinity}, GenerateConditions->None] == Integrate[Divide[Exp[- t],1 - Exp[- t]]-Divide[Exp[- t],t], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 1]
+| [https://dlmf.nist.gov/5.9.E18 5.9.E18] || <math qid="Q2107">\int_{0}^{1}(1-e^{-t})\frac{\diff{t}}{t}-\int_{1}^{\infty}e^{-t}\frac{\diff{t}}{t} = \int_{0}^{\infty}\left(\frac{e^{-t}}{1-e^{-t}}-\frac{e^{-t}}{t}\right)\diff{t}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{1}(1-e^{-t})\frac{\diff{t}}{t}-\int_{1}^{\infty}e^{-t}\frac{\diff{t}}{t} = \int_{0}^{\infty}\left(\frac{e^{-t}}{1-e^{-t}}-\frac{e^{-t}}{t}\right)\diff{t}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>int((1 - exp(- t))*(1)/(t), t = 0..1)- int(exp(- t)*(1)/(t), t = 1..infinity) = int((exp(- t))/(1 - exp(- t))-(exp(- t))/(t), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[(1 - Exp[- t])*Divide[1,t], {t, 0, 1}, GenerateConditions->None]- Integrate[Exp[- t]*Divide[1,t], {t, 1, Infinity}, GenerateConditions->None] == Integrate[Divide[Exp[- t],1 - Exp[- t]]-Divide[Exp[- t],t], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 1]
 |-
-| [https://dlmf.nist.gov/5.9.E19 5.9.E19] || [[Item:Q2108|<math>\EulerGamma^{(n)}@{z} = \int_{0}^{\infty}(\ln@@{t})^{n}e^{-t}t^{z-1}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerGamma^{(n)}@{z} = \int_{0}^{\infty}(\ln@@{t})^{n}e^{-t}t^{z-1}\diff{t}</syntaxhighlight> || <math>n \geq 0, \realpart@@{z} > 0</math> || <syntaxhighlight lang=mathematica>diff( GAMMA(z), z$(n) ) = int((ln(t))^(n)* exp(- t)*(t)^(z - 1), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[Gamma[z], {z, n}] == Integrate[(Log[t])^(n)* Exp[- t]*(t)^(z - 1), {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Successful || Aborted || - || Skipped - Because timed out
+| [https://dlmf.nist.gov/5.9.E19 5.9.E19] || <math qid="Q2108">\EulerGamma^{(n)}@{z} = \int_{0}^{\infty}(\ln@@{t})^{n}e^{-t}t^{z-1}\diff{t}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerGamma^{(n)}@{z} = \int_{0}^{\infty}(\ln@@{t})^{n}e^{-t}t^{z-1}\diff{t}</syntaxhighlight> || <math>n \geq 0, \realpart@@{z} > 0</math> || <syntaxhighlight lang=mathematica>diff( GAMMA(z), z$(n) ) = int((ln(t))^(n)* exp(- t)*(t)^(z - 1), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[Gamma[z], {z, n}] == Integrate[(Log[t])^(n)* Exp[- t]*(t)^(z - 1), {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Successful || Aborted || - || Skipped - Because timed out
 |}
 </div>

5.9: Difference between revisions

Latest revision as of 11:12, 28 June 2021

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