Gamma Function - 5.9 Integral Representations

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DLMF Formula Constraints Maple Mathematica Symbolic
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5.9.E1 1 μ ⁢ Γ ⁡ ( ν μ ) ⁢ 1 z ν / μ = ∫ 0 ∞ exp ⁡ ( - z ⁢ t μ ) ⁢ t ν - 1 ⁢ d t 1 𝜇 Euler-Gamma 𝜈 𝜇 1 superscript 𝑧 𝜈 𝜇 superscript subscript 0 𝑧 superscript 𝑡 𝜇 superscript 𝑡 𝜈 1 𝑡 {\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 1 Γ ⁡ ( z ) = 1 2 ⁢ π ⁢ i ⁢ ∫ - ∞ ( 0 + ) e t ⁢ t - z ⁢ d t 1 Euler-Gamma 𝑧 1 2 𝜋 𝑖 superscript subscript limit-from 0 superscript 𝑒 𝑡 superscript 𝑡 𝑧 𝑡 {\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}		 ℜ ⁡ z > 0 𝑧 0 {\displaystyle{\displaystyle\Re z>0}} 
(1)/(GAMMA(z)) = (1)/(2*Pi*I)*int(exp(t)*(t)^(- z), t = - infinity..(0 +))

Divide[1,Gamma[z]] == Divide[1,2*Pi*I]*Integrate[Exp[t]*(t)^(- z), {t, - Infinity, (0 +)}, GenerateConditions->None]
Error Failure - Error
5.9.E3 c - z ⁢ Γ ⁡ ( z ) = ∫ - ∞ ∞ | t | 2 ⁢ z - 1 ⁢ e - c ⁢ t 2 ⁢ d t superscript 𝑐 𝑧 Euler-Gamma 𝑧 superscript subscript superscript 𝑡 2 𝑧 1 superscript 𝑒 𝑐 superscript 𝑡 2 𝑡 {\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}		 c > 0 , ℜ ⁡ z > 0 formulae-sequence 𝑐 0 𝑧 0 {\displaystyle{\displaystyle c>0,\Re z>0}} 
(c)^(- z)* GAMMA(z) = int((abs(t))^(2*z - 1)* exp(- c*(t)^(2)), t = - infinity..infinity)

(c)^(- z)* Gamma[z] == Integrate[(Abs[t])^(2*z - 1)* Exp[- c*(t)^(2)], {t, - Infinity, Infinity}, GenerateConditions->None]
Missing Macro Error Missing Macro Error - -
5.9.E4 Γ ⁡ ( z ) = ∫ 1 ∞ t z - 1 ⁢ e - t ⁢ d t + ∑ k = 0 ∞ ( - 1 ) k ( z + k ) ⁢ k ! Euler-Gamma 𝑧 superscript subscript 1 superscript 𝑡 𝑧 1 superscript 𝑒 𝑡 𝑡 superscript subscript 𝑘 0 superscript 1 𝑘 𝑧 𝑘 𝑘 {\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!}		 ℜ ⁡ z > 0 𝑧 0 {\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 Γ ⁡ ( z ) = ∫ 0 ∞ t z - 1 ⁢ ( e - t - ∑ k = 0 n ( - 1 ) k ⁢ t k k ! ) ⁢ d t Euler-Gamma 𝑧 superscript subscript 0 superscript 𝑡 𝑧 1 superscript 𝑒 𝑡 superscript subscript 𝑘 0 𝑛 superscript 1 𝑘 superscript 𝑡 𝑘 𝑘 𝑡 {\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}		 - n - 1 < ℜ ⁡ z , ℜ ⁡ z < - n , ℜ ⁡ z > 0 formulae-sequence 𝑛 1 𝑧 formulae-sequence 𝑧 𝑛 𝑧 0 {\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 Γ ⁡ ( z ) ⁢ cos ⁡ ( 1 2 ⁢ π ⁢ z ) = ∫ 0 ∞ t z - 1 ⁢ cos ⁡ t ⁢ d t Euler-Gamma 𝑧 1 2 𝜋 𝑧 superscript subscript 0 superscript 𝑡 𝑧 1 𝑡 𝑡 {\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}		 0 < ℜ ⁡ z , ℜ ⁡ z < 1 , ℜ ⁡ z > 0 formulae-sequence 0 𝑧 formulae-sequence 𝑧 1 𝑧 0 {\displaystyle{\displaystyle 0<\Re z,\Re z<1,\Re z>0}} 
GAMMA(z)*cos((1)/(2)*Pi*z) = int((t)^(z - 1)* cos(t), t = 0..infinity)

Gamma[z]*Cos[Divide[1,2]*Pi*z] == Integrate[(t)^(z - 1)* Cos[t], {t, 0, Infinity}, GenerateConditions->None]
Successful Successful - Successful [Tested: 3]
5.9.E7 Γ ⁡ ( z ) ⁢ sin ⁡ ( 1 2 ⁢ π ⁢ z ) = ∫ 0 ∞ t z - 1 ⁢ sin ⁡ t ⁢ d t Euler-Gamma 𝑧 1 2 𝜋 𝑧 superscript subscript 0 superscript 𝑡 𝑧 1 𝑡 𝑡 {\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}		 - 1 < ℜ ⁡ z , ℜ ⁡ z < 1 , ℜ ⁡ z > 0 formulae-sequence 1 𝑧 formulae-sequence 𝑧 1 𝑧 0 {\displaystyle{\displaystyle-1<\Re z,\Re z<1,\Re z>0}} 
GAMMA(z)*sin((1)/(2)*Pi*z) = int((t)^(z - 1)* sin(t), t = 0..infinity)

Gamma[z]*Sin[Divide[1,2]*Pi*z] == Integrate[(t)^(z - 1)* Sin[t], {t, 0, Infinity}, GenerateConditions->None]
Successful Successful - Successful [Tested: 3]
5.9.E8 Γ ⁡ ( 1 + 1 n ) ⁢ cos ⁡ ( π 2 ⁢ n ) = ∫ 0 ∞ cos ⁡ ( t n ) ⁢ d t Euler-Gamma 1 1 𝑛 𝜋 2 𝑛 superscript subscript 0 superscript 𝑡 𝑛 𝑡 {\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)/(2*n)) = int(cos((t)^(n)), t = 0..infinity)

Gamma[1 +Divide[1,n]]*Cos[Divide[Pi,2*n]] == Integrate[Cos[(t)^(n)], {t, 0, Infinity}, GenerateConditions->None]
Successful Successful - Successful [Tested: 1]
5.9.E9 Γ ⁡ ( 1 + 1 n ) ⁢ sin ⁡ ( π 2 ⁢ n ) = ∫ 0 ∞ sin ⁡ ( t n ) ⁢ d t Euler-Gamma 1 1 𝑛 𝜋 2 𝑛 superscript subscript 0 superscript 𝑡 𝑛 𝑡 {\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)/(2*n)) = int(sin((t)^(n)), t = 0..infinity)

Gamma[1 +Divide[1,n]]*Sin[Divide[Pi,2*n]] == Integrate[Sin[(t)^(n)], {t, 0, Infinity}, GenerateConditions->None]
Successful Successful - Successful [Tested: 1]
5.9.E10 Ln ⁡ Γ ⁡ ( z ) = ( z - 1 2 ) ⁢ ln ⁡ z - z + 1 2 ⁢ ln ⁡ ( 2 ⁢ π ) + 2 ⁢ ∫ 0 ∞ arctan ⁡ ( t / z ) e 2 ⁢ π ⁢ t - 1 ⁢ d t multivalued-natural-logarithm Euler-Gamma 𝑧 𝑧 1 2 𝑧 𝑧 1 2 2 𝜋 2 superscript subscript 0 𝑡 𝑧 superscript 𝑒 2 𝜋 𝑡 1 𝑡 {\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}		 ℜ ⁡ z > 0 𝑧 0 {\displaystyle{\displaystyle\Re z>0}} 
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)

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]
Failure Aborted Successful [Tested: 5] Skipped - Because timed out
5.9.E11 Ln ⁡ Γ ⁡ ( z + 1 ) = - γ ⁢ z - 1 2 ⁢ π ⁢ i ⁢ ∫ - c - ∞ ⁢ i - c + ∞ ⁢ i π ⁢ z - s s ⁢ sin ⁡ ( π ⁢ s ) ⁢ ζ ⁡ ( - s ) ⁢ d s multivalued-natural-logarithm Euler-Gamma 𝑧 1 𝑧 1 2 𝜋 𝑖 superscript subscript 𝑐 𝑖 𝑐 𝑖 𝜋 superscript 𝑧 𝑠 𝑠 𝜋 𝑠 Riemann-zeta 𝑠 𝑠 {\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}		 ℜ ⁡ ( z + 1 ) > 0 𝑧 1 0 {\displaystyle{\displaystyle\Re(z+1)>0}} 
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)

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]
Failure Aborted
Failed [42 / 42]
Result: .3627983593+.4645558136*I
Test Values: {c = -1.5, z = 1/2*3^(1/2)+1/2*I}


Result: -.7321808519-.4375773776*I
Test Values: {c = -1.5, z = -1/2+1/2*I*3^(1/2)}


Result: -.1549651868-.6096201737*I
Test Values: {c = -1.5, z = 1/2-1/2*I*3^(1/2)}


Result: -.670593886e-1+1.175772123*I
Test Values: {c = -1.5, z = -1/2*3^(1/2)-1/2*I}

... skip entries to safe data
 Skipped - Because timed out
5.9.E12 ψ ⁡ ( z ) = ∫ 0 ∞ ( e - t t - e - z ⁢ t 1 - e - t ) ⁢ d t digamma 𝑧 superscript subscript 0 superscript 𝑒 𝑡 𝑡 superscript 𝑒 𝑧 𝑡 1 superscript 𝑒 𝑡 𝑡 {\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+1/2*I*3^(1/2)}


Result: Float(infinity)+Float(infinity)*I
Test Values: {z = -1/2*3^(1/2)-1/2*I}

Successful [Tested: 1]
5.9.E13 ψ ⁡ ( z ) = ln ⁡ z + ∫ 0 ∞ ( 1 t - 1 1 - e - t ) ⁢ e - t ⁢ z ⁢ d t digamma 𝑧 𝑧 superscript subscript 0 1 𝑡 1 1 superscript 𝑒 𝑡 superscript 𝑒 𝑡 𝑧 𝑡 {\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(- t*z), t = 0..infinity)

PolyGamma[z] == Log[z]+ Integrate[(Divide[1,t]-Divide[1,1 - Exp[- t]])*Exp[- t*z], {t, 0, Infinity}, GenerateConditions->None]
Failure Aborted
Failed [2 / 7]
Result: Float(infinity)+Float(infinity)*I
Test Values: {z = -1/2+1/2*I*3^(1/2)}


Result: Float(infinity)+Float(infinity)*I
Test Values: {z = -1/2*3^(1/2)-1/2*I}

Failed [1 / 1]
Result: Indeterminate
Test Values: {Rule[z, 1]}

5.9.E14 ψ ⁡ ( z ) = ∫ 0 ∞ ( e - t - 1 ( 1 + t ) z ) ⁢ d t t digamma 𝑧 superscript subscript 0 superscript 𝑒 𝑡 1 superscript 1 𝑡 𝑧 𝑡 𝑡 {\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 ψ ⁡ ( z ) = ln ⁡ z - 1 2 ⁢ z - 2 ⁢ ∫ 0 ∞ t ⁢ d t ( t 2 + z 2 ) ⁢ ( e 2 ⁢ π ⁢ t - 1 ) digamma 𝑧 𝑧 1 2 𝑧 2 superscript subscript 0 𝑡 𝑡 superscript 𝑡 2 superscript 𝑧 2 superscript 𝑒 2 𝜋 𝑡 1 {\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)/(2*z)- 2*int((t)/(((t)^(2)+ (z)^(2))*(exp(2*Pi*t)- 1)), t = 0..infinity)

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]
Failure Aborted
Failed [2 / 7]
Result: .4e-10-.2711020420e-1*I
Test Values: {z = -1/2+1/2*I*3^(1/2)}


Result: -.2144560970-.1791125126*I
Test Values: {z = -1/2*3^(1/2)-1/2*I}

Successful [Tested: 1]
5.9.E16 ψ ⁡ ( z ) + γ = ∫ 0 ∞ e - t - e - z ⁢ t 1 - e - t ⁢ d t digamma 𝑧 superscript subscript 0 superscript 𝑒 𝑡 superscript 𝑒 𝑧 𝑡 1 superscript 𝑒 𝑡 𝑡 {\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+1/2*I*3^(1/2)}


Result: Float(infinity)+Float(infinity)*I
Test Values: {z = -1/2*3^(1/2)-1/2*I}

Successful [Tested: 1]
5.9.E16 ∫ 0 ∞ e - t - e - z ⁢ t 1 - e - t ⁢ d t = ∫ 0 1 1 - t z - 1 1 - t ⁢ d t superscript subscript 0 superscript 𝑒 𝑡 superscript 𝑒 𝑧 𝑡 1 superscript 𝑒 𝑡 𝑡 superscript subscript 0 1 1 superscript 𝑡 𝑧 1 1 𝑡 𝑡 {\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+1/2*I*3^(1/2)}


Result: Float(infinity)+Float(infinity)*I
Test Values: {z = -1/2*3^(1/2)-1/2*I}

Successful [Tested: 7]
5.9.E17 ψ ⁡ ( z + 1 ) = - γ + 1 2 ⁢ π ⁢ i ⁢ ∫ - c - ∞ ⁢ i - c + ∞ ⁢ i π ⁢ z - s - 1 sin ⁡ ( π ⁢ s ) ⁢ ζ ⁡ ( - s ) ⁢ d s digamma 𝑧 1 1 2 𝜋 𝑖 superscript subscript 𝑐 𝑖 𝑐 𝑖 𝜋 superscript 𝑧 𝑠 1 𝜋 𝑠 Riemann-zeta 𝑠 𝑠 {\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)/(2*Pi*I)*int((Pi*(z)^(- s - 1))/(sin(Pi*s))*Zeta(- s), s = - c - infinity*I..- c + infinity*I)

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]
Failure Failure
Failed [42 / 42]
Result: .9666504222+.3394950970*I
Test Values: {c = -1.5, z = 1/2*3^(1/2)+1/2*I}


Result: .3622891065+1.557241225*I
Test Values: {c = -1.5, z = -1/2+1/2*I*3^(1/2)}


Result: .8622891063-.6912158211*I
Test Values: {c = -1.5, z = 1/2-1/2*I*3^(1/2)}


Result: -.1138310784-2.481210069*I
Test Values: {c = -1.5, z = -1/2*3^(1/2)-1/2*I}

... skip entries to safe data
 Skipped - Because timed out
5.9.E18 γ = - ∫ 0 ∞ e - t ⁢ ln ⁡ t ⁢ d t superscript subscript 0 superscript 𝑒 𝑡 𝑡 𝑡 {\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 - ∫ 0 ∞ e - t ⁢ ln ⁡ t ⁢ d t = ∫ 0 ∞ ( 1 1 + t - e - t ) ⁢ d t t superscript subscript 0 superscript 𝑒 𝑡 𝑡 𝑡 superscript subscript 0 1 1 𝑡 superscript 𝑒 𝑡 𝑡 𝑡 {\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 ∫ 0 ∞ ( 1 1 + t - e - t ) ⁢ d t t = ∫ 0 1 ( 1 - e - t ) ⁢ d t t - ∫ 1 ∞ e - t ⁢ d t t superscript subscript 0 1 1 𝑡 superscript 𝑒 𝑡 𝑡 𝑡 superscript subscript 0 1 1 superscript 𝑒 𝑡 𝑡 𝑡 superscript subscript 1 superscript 𝑒 𝑡 𝑡 𝑡 {\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 ∫ 0 1 ( 1 - e - t ) ⁢ d t t - ∫ 1 ∞ e - t ⁢ d t t = ∫ 0 ∞ ( e - t 1 - e - t - e - t t ) ⁢ d t superscript subscript 0 1 1 superscript 𝑒 𝑡 𝑡 𝑡 superscript subscript 1 superscript 𝑒 𝑡 𝑡 𝑡 superscript subscript 0 superscript 𝑒 𝑡 1 superscript 𝑒 𝑡 superscript 𝑒 𝑡 𝑡 𝑡 {\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 Γ ( n ) ⁡ ( z ) = ∫ 0 ∞ ( ln ⁡ t ) n ⁢ e - t ⁢ t z - 1 ⁢ d t Euler-Gamma 𝑛 𝑧 superscript subscript 0 superscript 𝑡 𝑛 superscript 𝑒 𝑡 superscript 𝑡 𝑧 1 𝑡 {\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}		 n ≥ 0 , ℜ ⁡ z > 0 formulae-sequence 𝑛 0 𝑧 0 {\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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