Results of Gamma Function

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DLMF Formula Maple Mathematica Symbolic
Maple		 Symbolic
Mathematica		 Numeric
Maple		 Numeric
Mathematica
5.2.E1 Γ ⁡ ( z ) = ∫ 0 ∞ e - t ⁢ t z - 1 ⁢ d t Euler-Gamma 𝑧 superscript subscript 0 superscript 𝑒 𝑡 superscript 𝑡 𝑧 1 𝑡 {\displaystyle{\displaystyle\Gamma\left(z\right)=\int_{0}^{\infty}e^{-t}t^{z-1% }\mathrm{d}t}} GAMMA(z)= int(exp(- t)*(t)^(z - 1), t = 0..infinity) Gamma[z]= Integrate[Exp[- t]*(t)^(z - 1), {t, 0, Infinity}] Successful Successful - -
5.2.E2 ψ ⁡ ( z ) = Γ ′ ⁡ ( z ) / Γ ⁡ ( z ) digamma 𝑧 diffop Euler-Gamma 1 𝑧 Euler-Gamma 𝑧 {\displaystyle{\displaystyle\psi\left(z\right)=\Gamma'\left(z\right)/\Gamma% \left(z\right)}} Psi(z)= subs( temp=z, diff( GAMMA(temp), temp$(1) ) )/ GAMMA(z) PolyGamma[z]= (D[Gamma[temp], {temp, 1}]/.temp-> z)/ Gamma[z] Successful Successful - -
5.2#Ex1 ( a ) 0 = 1 Pochhammer 𝑎 0 1 {\displaystyle{\displaystyle{\left(a\right)_{0}}=1}} pochhammer(a, 0)= 1 Pochhammer[a, 0]= 1 Successful Successful - -
5.2.E5 ( a ) n = Γ ⁡ ( a + n ) / Γ ⁡ ( a ) Pochhammer 𝑎 𝑛 Euler-Gamma 𝑎 𝑛 Euler-Gamma 𝑎 {\displaystyle{\displaystyle{\left(a\right)_{n}}=\Gamma\left(a+n\right)/\Gamma% \left(a\right)}} pochhammer(a, n)= GAMMA(a + n)/ GAMMA(a) Pochhammer[a, n]= Gamma[a + n]/ Gamma[a] Successful Successful - -
5.2.E6 ( - a ) n = ( - 1 ) n ⁢ ( a - n + 1 ) n Pochhammer 𝑎 𝑛 superscript 1 𝑛 Pochhammer 𝑎 𝑛 1 𝑛 {\displaystyle{\displaystyle{\left(-a\right)_{n}}=(-1)^{n}{\left(a-n+1\right)_% {n}}}} pochhammer(- a, n)=(- 1)^(n)* pochhammer(a - n + 1, n) Pochhammer[- a, n]=(- 1)^(n)* Pochhammer[a - n + 1, n] Failure Failure Successful Successful
5.2.E7 ( - m ) n = { Pochhammer 𝑚 𝑛 cases {\displaystyle{\displaystyle{\left(-m\right)_{n}}=\begin{cases}\frac{(-1)^{n}m% !}{(m-n)!},&0}}\)\@add@PDF@RDFa@triples\end{document}\end{cases} pochhammer(- m, n)= Pochhammer[- m, n]= Error Failure - -
5.2#Ex3 ( a ) 2 ⁢ n = 2 2 ⁢ n ⁢ ( a 2 ) n ⁢ ( a + 1 2 ) n Pochhammer 𝑎 2 𝑛 superscript 2 2 𝑛 Pochhammer 𝑎 2 𝑛 Pochhammer 𝑎 1 2 𝑛 {\displaystyle{\displaystyle{\left(a\right)_{2n}}=2^{2n}{\left(\frac{a}{2}% \right)_{n}}{\left(\frac{a+1}{2}\right)_{n}}}} pochhammer(a, 2*n)= (2)^(2*n)* pochhammer((a)/(2), n)*pochhammer((a + 1)/(2), n) Pochhammer[a, 2*n]= (2)^(2*n)* Pochhammer[Divide[a,2], n]*Pochhammer[Divide[a + 1,2], n] Successful Successful - -
5.2#Ex4 ( a ) 2 ⁢ n + 1 = 2 2 ⁢ n + 1 ⁢ ( a 2 ) n + 1 ⁢ ( a + 1 2 ) n Pochhammer 𝑎 2 𝑛 1 superscript 2 2 𝑛 1 Pochhammer 𝑎 2 𝑛 1 Pochhammer 𝑎 1 2 𝑛 {\displaystyle{\displaystyle{\left(a\right)_{2n+1}}=2^{2n+1}{\left(\frac{a}{2}% \right)_{n+1}}{\left(\frac{a+1}{2}\right)_{n}}}} pochhammer(a, 2*n + 1)= (2)^(2*n + 1)* pochhammer((a)/(2), n + 1)*pochhammer((a + 1)/(2), n) Pochhammer[a, 2*n + 1]= (2)^(2*n + 1)* Pochhammer[Divide[a,2], n + 1]*Pochhammer[Divide[a + 1,2], n] Successful Successful - -
5.4#Ex1 Γ ⁡ ( 1 ) = 1 Euler-Gamma 1 1 {\displaystyle{\displaystyle\Gamma\left(1\right)=1}} GAMMA(1)= 1 Gamma[1]= 1 Successful Successful - -
5.4#Ex2 n ! = Γ ⁡ ( n + 1 ) 𝑛 Euler-Gamma 𝑛 1 {\displaystyle{\displaystyle n!=\Gamma\left(n+1\right)}} factorial(n)= GAMMA(n + 1) (n)!= Gamma[n + 1] Successful Successful - -
5.4.E3 | Γ ⁡ ( i ⁢ y ) | = ( π y ⁢ sinh ⁡ ( π ⁢ y ) ) 1 / 2 Euler-Gamma 𝑖 𝑦 superscript 𝜋 𝑦 𝜋 𝑦 1 2 {\displaystyle{\displaystyle|\Gamma\left(iy\right)|=\left(\frac{\pi}{y\sinh% \left(\pi y\right)}\right)^{1/2}}} abs(GAMMA(I*y))=((Pi)/(y*sinh(Pi*y)))^(1/ 2) Abs[Gamma[I*y]]=(Divide[Pi,y*Sinh[Pi*y]])^(1/ 2) Failure Failure Successful Successful
5.4.E4 Γ ⁡ ( 1 2 + i ⁢ y ) ⁢ Γ ⁡ ( 1 2 - i ⁢ y ) = | Γ ⁡ ( 1 2 + i ⁢ y ) | 2 Euler-Gamma 1 2 imaginary-unit 𝑦 Euler-Gamma 1 2 imaginary-unit 𝑦 superscript Euler-Gamma 1 2 imaginary-unit 𝑦 2 {\displaystyle{\displaystyle\Gamma\left(\tfrac{1}{2}+\mathrm{i}y\right)\Gamma% \left(\tfrac{1}{2}-\mathrm{i}y\right)=\left|\Gamma\left(\tfrac{1}{2}+\mathrm{i% }y\right)\right|^{2}}} GAMMA((1)/(2)+ I*y)*GAMMA((1)/(2)- I*y)=(abs(GAMMA((1)/(2)+ I*y)))^(2) Gamma[Divide[1,2]+ I*y]*Gamma[Divide[1,2]- I*y]=(Abs[Gamma[Divide[1,2]+ I*y]])^(2) Failure Failure Successful Successful
5.4.E4 | Γ ⁡ ( 1 2 + i ⁢ y ) | 2 = π cosh ⁡ ( π ⁢ y ) superscript Euler-Gamma 1 2 imaginary-unit 𝑦 2 𝜋 𝜋 𝑦 {\displaystyle{\displaystyle\left|\Gamma\left(\tfrac{1}{2}+\mathrm{i}y\right)% \right|^{2}=\frac{\pi}{\cosh\left(\pi y\right)}}} (abs(GAMMA((1)/(2)+ I*y)))^(2)=(Pi)/(cosh(Pi*y)) (Abs[Gamma[Divide[1,2]+ I*y]])^(2)=Divide[Pi,Cosh[Pi*y]] Failure Failure Successful Successful
5.4.E5 Γ ⁡ ( 1 4 + i ⁢ y ) ⁢ Γ ⁡ ( 3 4 - i ⁢ y ) = π ⁢ 2 cosh ⁡ ( π ⁢ y ) + i ⁢ sinh ⁡ ( π ⁢ y ) Euler-Gamma 1 4 imaginary-unit 𝑦 Euler-Gamma 3 4 imaginary-unit 𝑦 𝜋 2 𝜋 𝑦 imaginary-unit 𝜋 𝑦 {\displaystyle{\displaystyle\Gamma\left(\tfrac{1}{4}+\mathrm{i}y\right)\Gamma% \left(\tfrac{3}{4}-\mathrm{i}y\right)=\frac{\pi\sqrt{2}}{\cosh\left(\pi y% \right)+\mathrm{i}\sinh\left(\pi y\right)}}} GAMMA((1)/(4)+ I*y)*GAMMA((3)/(4)- I*y)=(Pi*sqrt(2))/(cosh(Pi*y)+ I*sinh(Pi*y)) Gamma[Divide[1,4]+ I*y]*Gamma[Divide[3,4]- I*y]=Divide[Pi*Sqrt[2],Cosh[Pi*y]+ I*Sinh[Pi*y]] Failure Successful Successful -
5.4.E11 Γ ′ ⁡ ( 1 ) = - γ diffop Euler-Gamma 1 1 {\displaystyle{\displaystyle\Gamma'\left(1\right)=-\gamma}} subs( temp=1, diff( GAMMA(temp), temp$(1) ) )= - gamma (D[Gamma[temp], {temp, 1}]/.temp-> 1)= - EulerGamma Successful Successful - -
5.4#Ex3 ψ ⁡ ( 1 ) = - γ digamma 1 {\displaystyle{\displaystyle\psi\left(1\right)=-\gamma}} Psi(1)= - gamma PolyGamma[1]= - EulerGamma Successful Successful - -
5.4#Ex4 ψ ′ ⁡ ( 1 ) = 1 6 ⁢ π 2 diffop digamma 1 1 1 6 superscript 𝜋 2 {\displaystyle{\displaystyle\psi'\left(1\right)=\tfrac{1}{6}\pi^{2}}} subs( temp=1, diff( Psi(temp), temp$(1) ) )=(1)/(6)*(Pi)^(2) (D[PolyGamma[temp], {temp, 1}]/.temp-> 1)=Divide[1,6]*(Pi)^(2) Successful Successful - -
5.4#Ex5 ψ ⁡ ( 1 2 ) = - γ - 2 ⁢ ln ⁡ 2 digamma 1 2 2 2 {\displaystyle{\displaystyle\psi\left(\tfrac{1}{2}\right)=-\gamma-2\ln 2}} Psi((1)/(2))= - gamma - 2*ln(2) PolyGamma[Divide[1,2]]= - EulerGamma - 2*Log[2] Successful Successful - -
5.4#Ex6 ψ ′ ⁡ ( 1 2 ) = 1 2 ⁢ π 2 diffop digamma 1 1 2 1 2 superscript 𝜋 2 {\displaystyle{\displaystyle\psi'\left(\tfrac{1}{2}\right)=\tfrac{1}{2}\pi^{2}}} subs( temp=(1)/(2), diff( Psi(temp), temp$(1) ) )=(1)/(2)*(Pi)^(2) (D[PolyGamma[temp], {temp, 1}]/.temp-> Divide[1,2])=Divide[1,2]*(Pi)^(2) Successful Successful - -
5.4.E14 ψ ⁡ ( n + 1 ) = ∑ k = 1 n 1 k - γ digamma 𝑛 1 superscript subscript 𝑘 1 𝑛 1 𝑘 {\displaystyle{\displaystyle\psi\left(n+1\right)=\sum_{k=1}^{n}\frac{1}{k}-% \gamma}} Psi(n + 1)= sum((1)/(k), k = 1..n)- gamma PolyGamma[n + 1]= Sum[Divide[1,k], {k, 1, n}]- EulerGamma Successful Successful - -
5.4.E16 ℑ ⁡ ψ ⁡ ( i ⁢ y ) = 1 2 ⁢ y + π 2 ⁢ coth ⁡ ( π ⁢ y ) digamma 𝑖 𝑦 1 2 𝑦 𝜋 2 hyperbolic-cotangent 𝜋 𝑦 {\displaystyle{\displaystyle\Im\psi\left(iy\right)=\frac{1}{2y}+\frac{\pi}{2}% \coth\left(\pi y\right)}} Im(Psi(I*y))=(1)/(2*y)+(Pi)/(2)*coth(Pi*y) Im[PolyGamma[I*y]]=Divide[1,2*y]+Divide[Pi,2]*Coth[Pi*y] Failure Failure Successful Successful
5.4.E17 ℑ ⁡ ψ ⁡ ( 1 2 + i ⁢ y ) = π 2 ⁢ tanh ⁡ ( π ⁢ y ) digamma 1 2 𝑖 𝑦 𝜋 2 𝜋 𝑦 {\displaystyle{\displaystyle\Im\psi\left(\tfrac{1}{2}+iy\right)=\frac{\pi}{2}% \tanh\left(\pi y\right)}} Im(Psi((1)/(2)+ I*y))=(Pi)/(2)*tanh(Pi*y) Im[PolyGamma[Divide[1,2]+ I*y]]=Divide[Pi,2]*Tanh[Pi*y] Failure Failure Successful Successful
5.4.E18 ℑ ⁡ ψ ⁡ ( 1 + i ⁢ y ) = - 1 2 ⁢ y + π 2 ⁢ coth ⁡ ( π ⁢ y ) digamma 1 𝑖 𝑦 1 2 𝑦 𝜋 2 hyperbolic-cotangent 𝜋 𝑦 {\displaystyle{\displaystyle\Im\psi\left(1+iy\right)=-\frac{1}{2y}+\frac{\pi}{% 2}\coth\left(\pi y\right)}} Im(Psi(1 + I*y))= -(1)/(2*y)+(Pi)/(2)*coth(Pi*y) Im[PolyGamma[1 + I*y]]= -Divide[1,2*y]+Divide[Pi,2]*Coth[Pi*y] Failure Failure Successful Successful
5.4.E19 ψ ⁡ ( p q ) = - γ - ln ⁡ q - π 2 ⁢ cot ⁡ ( π ⁢ p q ) + 1 2 ⁢ ∑ k = 1 q - 1 cos ⁡ ( 2 ⁢ π ⁢ k ⁢ p q ) ⁢ ln ⁡ ( 2 - 2 ⁢ cos ⁡ ( 2 ⁢ π ⁢ k q ) ) digamma 𝑝 𝑞 𝑞 𝜋 2 𝜋 𝑝 𝑞 1 2 superscript subscript 𝑘 1 𝑞 1 2 𝜋 𝑘 𝑝 𝑞 2 2 2 𝜋 𝑘 𝑞 {\displaystyle{\displaystyle\psi\left(\frac{p}{q}\right)=-\gamma-\ln q-\frac{% \pi}{2}\cot\left(\frac{\pi p}{q}\right)+\frac{1}{2}\sum_{k=1}^{q-1}\cos\left(% \frac{2\pi kp}{q}\right)\ln\left(2-2\cos\left(\frac{2\pi k}{q}\right)\right)}} Psi((p)/(q))= - gamma - ln(q)-(Pi)/(2)*cot((Pi*p)/(q))+(1)/(2)*sum(cos((2*Pi*k*p)/(q))*ln(2 - 2*cos((2*Pi*k)/(q))), k = 1..q - 1) PolyGamma[Divide[p,q]]= - EulerGamma - Log[q]-Divide[Pi,2]*Cot[Divide[Pi*p,q]]+Divide[1,2]*Sum[Cos[Divide[2*Pi*k*p,q]]*Log[2 - 2*Cos[Divide[2*Pi*k,q]]], {k, 1, q - 1}] Failure Failure Skip
Fail
DirectedInfinity[] <- {Rule[k, 1], Rule[p, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[q, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
DirectedInfinity[] <- {Rule[k, 2], Rule[p, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[q, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
DirectedInfinity[] <- {Rule[k, 3], Rule[p, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[q, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
DirectedInfinity[] <- {Rule[k, 1], Rule[p, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[q, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
5.5.E1 Γ ⁡ ( z + 1 ) = z ⁢ Γ ⁡ ( z ) Euler-Gamma 𝑧 1 𝑧 Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(z+1\right)=z\Gamma\left(z\right)}} GAMMA(z + 1)= z*GAMMA(z) Gamma[z + 1]= z*Gamma[z] Successful Successful - -
5.5.E2 ψ ⁡ ( z + 1 ) = ψ ⁡ ( z ) + 1 z digamma 𝑧 1 digamma 𝑧 1 𝑧 {\displaystyle{\displaystyle\psi\left(z+1\right)=\psi\left(z\right)+\frac{1}{z% }}} Psi(z + 1)= Psi(z)+(1)/(z) PolyGamma[z + 1]= PolyGamma[z]+Divide[1,z] Successful Successful - -
5.5.E3 Γ ⁡ ( z ) ⁢ Γ ⁡ ( 1 - z ) = π / sin ⁡ ( π ⁢ z ) Euler-Gamma 𝑧 Euler-Gamma 1 𝑧 𝜋 𝜋 𝑧 {\displaystyle{\displaystyle\Gamma\left(z\right)\Gamma\left(1-z\right)=\pi/% \sin\left(\pi z\right)}} GAMMA(z)*GAMMA(1 - z)= Pi/ sin(Pi*z) Gamma[z]*Gamma[1 - z]= Pi/ Sin[Pi*z] Successful Successful - -
5.5.E4 ψ ⁡ ( z ) - ψ ⁡ ( 1 - z ) = - π / tan ⁡ ( π ⁢ z ) digamma 𝑧 digamma 1 𝑧 𝜋 𝜋 𝑧 {\displaystyle{\displaystyle\psi\left(z\right)-\psi\left(1-z\right)=-\pi/\tan% \left(\pi z\right)}} Psi(z)- Psi(1 - z)= - Pi/ tan(Pi*z) PolyGamma[z]- PolyGamma[1 - z]= - Pi/ Tan[Pi*z] Successful Successful - -
5.5.E5 Γ ⁡ ( 2 ⁢ z ) = π - 1 / 2 ⁢ 2 2 ⁢ z - 1 ⁢ Γ ⁡ ( z ) ⁢ Γ ⁡ ( z + 1 2 ) Euler-Gamma 2 𝑧 superscript 𝜋 1 2 superscript 2 2 𝑧 1 Euler-Gamma 𝑧 Euler-Gamma 𝑧 1 2 {\displaystyle{\displaystyle\Gamma\left(2z\right)=\pi^{-1/2}2^{2z-1}\Gamma% \left(z\right)\Gamma\left(z+\tfrac{1}{2}\right)}} GAMMA(2*z)= (Pi)^(- 1/ 2)* (2)^(2*z - 1)* GAMMA(z)*GAMMA(z +(1)/(2)) Gamma[2*z]= (Pi)^(- 1/ 2)* (2)^(2*z - 1)* Gamma[z]*Gamma[z +Divide[1,2]] Successful Successful - -
5.5.E6 Γ ⁡ ( n ⁢ z ) = ( 2 ⁢ π ) ( 1 - n ) / 2 ⁢ n n ⁢ z - ( 1 / 2 ) ⁢ ∏ k = 0 n - 1 Γ ⁡ ( z + k n ) Euler-Gamma 𝑛 𝑧 superscript 2 𝜋 1 𝑛 2 superscript 𝑛 𝑛 𝑧 1 2 superscript subscript product 𝑘 0 𝑛 1 Euler-Gamma 𝑧 𝑘 𝑛 {\displaystyle{\displaystyle\Gamma\left(nz\right)=(2\pi)^{(1-n)/2}n^{nz-(1/2)}% \prod_{k=0}^{n-1}\Gamma\left(z+\frac{k}{n}\right)}} GAMMA(n*z)=(2*Pi)^((1 - n)/ 2)* (n)^(n*z -(1/ 2))* product(GAMMA(z +(k)/(n)), k = 0..n - 1) Gamma[n*z]=(2*Pi)^((1 - n)/ 2)* (n)^(n*z -(1/ 2))* Product[Gamma[z +Divide[k,n]], {k, 0, n - 1}] Failure Successful Skip -
5.5.E7 ∏ k = 1 n - 1 Γ ⁡ ( k n ) = ( 2 ⁢ π ) ( n - 1 ) / 2 ⁢ n - 1 / 2 superscript subscript product 𝑘 1 𝑛 1 Euler-Gamma 𝑘 𝑛 superscript 2 𝜋 𝑛 1 2 superscript 𝑛 1 2 {\displaystyle{\displaystyle\prod_{k=1}^{n-1}\Gamma\left(\frac{k}{n}\right)=(2% \pi)^{(n-1)/2}n^{-1/2}}} product(GAMMA((k)/(n)), k = 1..n - 1)=(2*Pi)^((n - 1)/ 2)* (n)^(- 1/ 2) Product[Gamma[Divide[k,n]], {k, 1, n - 1}]=(2*Pi)^((n - 1)/ 2)* (n)^(- 1/ 2) Failure Failure Skip
Fail
Complex[1.4142135623730951, 1.4142135623730951] <- {Rule[Indeterminate, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[1.4142135623730951, -1.4142135623730951] <- {Rule[Indeterminate, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.4142135623730951, -1.4142135623730951] <- {Rule[Indeterminate, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.4142135623730951, 1.4142135623730951] <- {Rule[Indeterminate, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
5.5.E8 ψ ⁡ ( 2 ⁢ z ) = 1 2 ⁢ ( ψ ⁡ ( z ) + ψ ⁡ ( z + 1 2 ) ) + ln ⁡ 2 digamma 2 𝑧 1 2 digamma 𝑧 digamma 𝑧 1 2 2 {\displaystyle{\displaystyle\psi\left(2z\right)=\tfrac{1}{2}\left(\psi\left(z% \right)+\psi\left(z+\tfrac{1}{2}\right)\right)+\ln 2}} Psi(2*z)=(1)/(2)*(Psi(z)+ Psi(z +(1)/(2)))+ ln(2) PolyGamma[2*z]=Divide[1,2]*(PolyGamma[z]+ PolyGamma[z +Divide[1,2]])+ Log[2] Successful Successful - -
5.5.E9 ψ ⁡ ( n ⁢ z ) = 1 n ⁢ ∑ k = 0 n - 1 ψ ⁡ ( z + k n ) + ln ⁡ n digamma 𝑛 𝑧 1 𝑛 superscript subscript 𝑘 0 𝑛 1 digamma 𝑧 𝑘 𝑛 𝑛 {\displaystyle{\displaystyle\psi\left(nz\right)=\frac{1}{n}\sum_{k=0}^{n-1}% \psi\left(z+\frac{k}{n}\right)+\ln n}} Psi(n*z)=(1)/(n)*sum(Psi(z +(k)/(n)), k = 0..n - 1)+ ln(n) PolyGamma[n*z]=Divide[1,n]*Sum[PolyGamma[z +Divide[k,n]], {k, 0, n - 1}]+ Log[n] Failure Successful Skip -
5.6.E1 1 < ( 2 ⁢ π ) - 1 / 2 ⁢ x ( 1 / 2 ) - x ⁢ e x ⁢ Γ ⁡ ( x ) 1 superscript 2 𝜋 1 2 superscript 𝑥 1 2 𝑥 superscript 𝑒 𝑥 Euler-Gamma 𝑥 {\displaystyle{\displaystyle 1<(2\pi)^{-1/2}x^{(1/2)-x}e^{x}\Gamma\left(x% \right)}} 1 <(2*Pi)^(- 1/ 2)* (x)^((1/ 2)- x)* exp(x)*GAMMA(x) 1 <(2*Pi)^(- 1/ 2)* (x)^((1/ 2)- x)* Exp[x]*Gamma[x] Failure Failure Successful Successful
5.6.E1 ( 2 ⁢ π ) - 1 / 2 ⁢ x ( 1 / 2 ) - x ⁢ e x ⁢ Γ ⁡ ( x ) < e 1 / ( 12 ⁢ x ) superscript 2 𝜋 1 2 superscript 𝑥 1 2 𝑥 superscript 𝑒 𝑥 Euler-Gamma 𝑥 superscript 𝑒 1 12 𝑥 {\displaystyle{\displaystyle(2\pi)^{-1/2}x^{(1/2)-x}e^{x}\Gamma\left(x\right)<% e^{1/(12x)}}} (2*Pi)^(- 1/ 2)* (x)^((1/ 2)- x)* exp(x)*GAMMA(x)< exp(1/(12*x)) (2*Pi)^(- 1/ 2)* (x)^((1/ 2)- x)* Exp[x]*Gamma[x]< Exp[1/(12*x)] Failure Failure Successful Successful
5.6.E2 1 Γ ⁡ ( x ) + 1 Γ ⁡ ( 1 / x ) ≤ 2 1 Euler-Gamma 𝑥 1 Euler-Gamma 1 𝑥 2 {\displaystyle{\displaystyle\frac{1}{\Gamma\left(x\right)}+\frac{1}{\Gamma% \left(1/x\right)}<=2}} (1)/(GAMMA(x))+(1)/(GAMMA(1/ x))< = 2 Divide[1,Gamma[x]]+Divide[1,Gamma[1/ x]]< = 2 Failure Failure Successful Successful
5.6.E3 1 ( Γ ⁡ ( x ) ) 2 + 1 ( Γ ⁡ ( 1 / x ) ) 2 ≤ 2 1 superscript Euler-Gamma 𝑥 2 1 superscript Euler-Gamma 1 𝑥 2 2 {\displaystyle{\displaystyle\frac{1}{(\Gamma\left(x\right))^{2}}+\frac{1}{(% \Gamma\left(1/x\right))^{2}}<=2}} (1)/((GAMMA(x))^(2))+(1)/((GAMMA(1/ x))^(2))< = 2 Divide[1,(Gamma[x])^(2)]+Divide[1,(Gamma[1/ x])^(2)]< = 2 Failure Failure Successful Successful
5.6.E4 x 1 - s < Γ ⁡ ( x + 1 ) Γ ⁡ ( x + s ) superscript 𝑥 1 𝑠 Euler-Gamma 𝑥 1 Euler-Gamma 𝑥 𝑠 {\displaystyle{\displaystyle x^{1-s}<\frac{\Gamma\left(x+1\right)}{\Gamma\left% (x+s\right)}}} (x)^(1 - s)<(GAMMA(x + 1))/(GAMMA(x + s)) (x)^(1 - s)<Divide[Gamma[x + 1],Gamma[x + s]] Failure Failure Successful Successful
5.6.E4 Γ ⁡ ( x + 1 ) Γ ⁡ ( x + s ) < ( x + 1 ) 1 - s Euler-Gamma 𝑥 1 Euler-Gamma 𝑥 𝑠 superscript 𝑥 1 1 𝑠 {\displaystyle{\displaystyle\frac{\Gamma\left(x+1\right)}{\Gamma\left(x+s% \right)}<(x+1)^{1-s}}} (GAMMA(x + 1))/(GAMMA(x + s))<(x + 1)^(1 - s) Divide[Gamma[x + 1],Gamma[x + s]]<(x + 1)^(1 - s) Failure Failure Successful Successful
5.6.E5 exp ⁡ ( ( 1 - s ) ⁢ ψ ⁡ ( x + s 1 / 2 ) ) ≤ Γ ⁡ ( x + 1 ) Γ ⁡ ( x + s ) 1 𝑠 digamma 𝑥 superscript 𝑠 1 2 Euler-Gamma 𝑥 1 Euler-Gamma 𝑥 𝑠 {\displaystyle{\displaystyle\exp\left((1-s)\psi\left(x+s^{1/2}\right)\right)<=% \frac{\Gamma\left(x+1\right)}{\Gamma\left(x+s\right)}}} exp((1 - s)* Psi(x + (s)^(1/ 2)))< =(GAMMA(x + 1))/(GAMMA(x + s)) Exp[(1 - s)* PolyGamma[x + (s)^(1/ 2)]]< =Divide[Gamma[x + 1],Gamma[x + s]] Failure Failure Successful Successful
5.6.E5 Γ ⁡ ( x + 1 ) Γ ⁡ ( x + s ) ≤ exp ⁡ ( ( 1 - s ) ⁢ ψ ⁡ ( x + 1 2 ⁢ ( s + 1 ) ) ) Euler-Gamma 𝑥 1 Euler-Gamma 𝑥 𝑠 1 𝑠 digamma 𝑥 1 2 𝑠 1 {\displaystyle{\displaystyle\frac{\Gamma\left(x+1\right)}{\Gamma\left(x+s% \right)}<=\exp\left((1-s)\psi\left(x+\tfrac{1}{2}(s+1)\right)\right)}} (GAMMA(x + 1))/(GAMMA(x + s))< = exp((1 - s)* Psi(x +(1)/(2)*(s + 1))) Divide[Gamma[x + 1],Gamma[x + s]]< = Exp[(1 - s)* PolyGamma[x +Divide[1,2]*(s + 1)]] Failure Failure Successful Successful
5.6.E6 | Γ ⁡ ( x + i ⁢ y ) | ≤ | Γ ⁡ ( x ) | Euler-Gamma 𝑥 imaginary-unit 𝑦 Euler-Gamma 𝑥 {\displaystyle{\displaystyle|\Gamma\left(x+\mathrm{i}y\right)|<=|\Gamma\left(x% \right)|}} abs(GAMMA(x + I*y))< =abs(GAMMA(x)) Abs[Gamma[x + I*y]]< =Abs[Gamma[x]] Failure Failure Successful Successful
5.6.E7 | Γ ⁡ ( x + i ⁢ y ) | ≥ ( sech ⁡ ( π ⁢ y ) ) 1 / 2 ⁢ Γ ⁡ ( x ) Euler-Gamma 𝑥 imaginary-unit 𝑦 superscript 𝜋 𝑦 1 2 Euler-Gamma 𝑥 {\displaystyle{\displaystyle|\Gamma\left(x+\mathrm{i}y\right)|>=(\operatorname% {sech}\left(\pi y\right))^{1/2}\Gamma\left(x\right)}} abs(GAMMA(x + I*y))> =(sech(Pi*y))^(1/ 2)* GAMMA(x) Abs[Gamma[x + I*y]]> =(Sech[Pi*y])^(1/ 2)* Gamma[x] Failure Failure Skip Successful
5.6.E8 | Γ ⁡ ( z + a ) Γ ⁡ ( z + b ) | ≤ 1 | z | b - a Euler-Gamma 𝑧 𝑎 Euler-Gamma 𝑧 𝑏 1 superscript 𝑧 𝑏 𝑎 {\displaystyle{\displaystyle\left|\frac{\Gamma\left(z+a\right)}{\Gamma\left(z+% b\right)}\right|<=\frac{1}{|z|^{b-a}}}} abs((GAMMA(z + a))/(GAMMA(z + b)))< =(1)/((abs(z))^(b - a)) Abs[Divide[Gamma[z + a],Gamma[z + b]]]< =Divide[1,(Abs[z])^(b - a)] Failure Failure Error Successful
5.6.E9 | Γ ⁡ ( z ) | ≤ ( 2 ⁢ π ) 1 / 2 ⁢ | z | x - ( 1 / 2 ) ⁢ e - π ⁢ | y | / 2 ⁢ exp ⁡ ( 1 6 ⁢ | z | - 1 ) Euler-Gamma 𝑧 superscript 2 𝜋 1 2 superscript 𝑧 𝑥 1 2 superscript 𝑒 𝜋 𝑦 2 1 6 superscript 𝑧 1 {\displaystyle{\displaystyle|\Gamma\left(z\right)|<=(2\pi)^{1/2}|z|^{x-(1/2)}e% ^{-\pi|y|/2}\exp\left(\tfrac{1}{6}|z|^{-1}\right)}} abs(GAMMA(z))< =(2*Pi)^(1/ 2)*(abs(z))^(x -(1/ 2))* exp(- Pi*abs(y)/ 2)*exp((1)/(6)*(abs(z))^(- 1)) Abs[Gamma[z]]< =(2*Pi)^(1/ 2)*(Abs[z])^(x -(1/ 2))* Exp[- Pi*Abs[y]/ 2]*Exp[Divide[1,6]*(Abs[z])^(- 1)] Failure Failure
Fail
.3896047846 <= .1665021267 <- {z = 2^(1/2)+I*2^(1/2), x = 1, y = 2}
.3896047846 <= .3461239156e-1 <- {z = 2^(1/2)+I*2^(1/2), x = 1, y = 3}
.3896047846 <= .3330042534 <- {z = 2^(1/2)+I*2^(1/2), x = 2, y = 2}
.3896047846 <= .6922478312e-1 <- {z = 2^(1/2)+I*2^(1/2), x = 2, y = 3}
... skip entries to safe data
 Successful
5.7.E1 1 Γ ⁡ ( z ) = ∑ k = 1 ∞ c k ⁢ z k 1 Euler-Gamma 𝑧 superscript subscript 𝑘 1 subscript 𝑐 𝑘 superscript 𝑧 𝑘 {\displaystyle{\displaystyle\frac{1}{\Gamma\left(z\right)}=\sum_{k=1}^{\infty}% c_{k}z^{k}}} (1)/(GAMMA(z))= sum(c[k]*(z)^(k), k = 1..infinity) Divide[1,Gamma[z]]= Sum[Subscript[c, k]*(z)^(k), {k, 1, Infinity}] Failure Failure Skip Skip
5.7.E3 ln ⁡ Γ ⁡ ( 1 + z ) = - ln ⁡ ( 1 + z ) + z ⁢ ( 1 - γ ) + ∑ k = 2 ∞ ( - 1 ) k ⁢ ( ζ ⁡ ( k ) - 1 ) ⁢ z k k Euler-Gamma 1 𝑧 1 𝑧 𝑧 1 superscript subscript 𝑘 2 superscript 1 𝑘 Riemann-zeta 𝑘 1 superscript 𝑧 𝑘 𝑘 {\displaystyle{\displaystyle\ln\Gamma\left(1+z\right)=-\ln\left(1+z\right)+z(1% -\gamma)+\sum_{k=2}^{\infty}(-1)^{k}(\zeta\left(k\right)-1)\frac{z^{k}}{k}}} ln(GAMMA(1 + z))= - ln(1 + z)+ z*(1 - gamma)+ sum((- 1)^(k)*(Zeta(k)- 1)*((z)^(k))/(k), k = 2..infinity) Log[Gamma[1 + z]]= - Log[1 + z]+ z*(1 - EulerGamma)+ Sum[(- 1)^(k)*(Zeta[k]- 1)*Divide[(z)^(k),k], {k, 2, Infinity}] Failure Successful Skip -
5.7.E4 ψ ⁡ ( 1 + z ) = - γ + ∑ k = 2 ∞ ( - 1 ) k ⁢ ζ ⁡ ( k ) ⁢ z k - 1 digamma 1 𝑧 superscript subscript 𝑘 2 superscript 1 𝑘 Riemann-zeta 𝑘 superscript 𝑧 𝑘 1 {\displaystyle{\displaystyle\psi\left(1+z\right)=-\gamma+\sum_{k=2}^{\infty}(-% 1)^{k}\zeta\left(k\right)z^{k-1}}} Psi(1 + z)= - gamma + sum((- 1)^(k)* Zeta(k)*(z)^(k - 1), k = 2..infinity) PolyGamma[1 + z]= - EulerGamma + Sum[(- 1)^(k)* Zeta[k]*(z)^(k - 1), {k, 2, Infinity}] Failure Successful Skip -
5.7.E5 ψ ⁡ ( 1 + z ) = 1 2 ⁢ z - π 2 ⁢ cot ⁡ ( π ⁢ z ) + 1 z 2 - 1 + 1 - γ - ∑ k = 1 ∞ ( ζ ⁡ ( 2 ⁢ k + 1 ) - 1 ) ⁢ z 2 ⁢ k digamma 1 𝑧 1 2 𝑧 𝜋 2 𝜋 𝑧 1 superscript 𝑧 2 1 1 superscript subscript 𝑘 1 Riemann-zeta 2 𝑘 1 1 superscript 𝑧 2 𝑘 {\displaystyle{\displaystyle\psi\left(1+z\right)=\frac{1}{2z}-\frac{\pi}{2}% \cot\left(\pi z\right)+\frac{1}{z^{2}-1}+1-\gamma-\sum_{k=1}^{\infty}(\zeta% \left(2k+1\right)-1)z^{2k}}} Psi(1 + z)=(1)/(2*z)-(Pi)/(2)*cot(Pi*z)+(1)/((z)^(2)- 1)+ 1 - gamma - sum((Zeta(2*k + 1)- 1)* (z)^(2*k), k = 1..infinity) PolyGamma[1 + z]=Divide[1,2*z]-Divide[Pi,2]*Cot[Pi*z]+Divide[1,(z)^(2)- 1]+ 1 - EulerGamma - Sum[(Zeta[2*k + 1]- 1)* (z)^(2*k), {k, 1, Infinity}] Failure Successful Skip -
5.7.E6 ψ ⁡ ( z ) = - γ - 1 z + ∑ k = 1 ∞ z k ⁢ ( k + z ) digamma 𝑧 1 𝑧 superscript subscript 𝑘 1 𝑧 𝑘 𝑘 𝑧 {\displaystyle{\displaystyle\psi\left(z\right)=-\gamma-\frac{1}{z}+\sum_{k=1}^% {\infty}\frac{z}{k(k+z)}}} Psi(z)= - gamma -(1)/(z)+ sum((z)/(k*(k + z)), k = 1..infinity) PolyGamma[z]= - EulerGamma -Divide[1,z]+ Sum[Divide[z,k*(k + z)], {k, 1, Infinity}] Successful Successful - -
5.7.E6 - γ - 1 z + ∑ k = 1 ∞ z k ⁢ ( k + z ) = - γ + ∑ k = 0 ∞ ( 1 k + 1 - 1 k + z ) 1 𝑧 superscript subscript 𝑘 1 𝑧 𝑘 𝑘 𝑧 superscript subscript 𝑘 0 1 𝑘 1 1 𝑘 𝑧 {\displaystyle{\displaystyle-\gamma-\frac{1}{z}+\sum_{k=1}^{\infty}\frac{z}{k(% k+z)}=-\gamma+\sum_{k=0}^{\infty}\left(\frac{1}{k+1}-\frac{1}{k+z}\right)}} - gamma -(1)/(z)+ sum((z)/(k*(k + z)), k = 1..infinity)= - gamma + sum((1)/(k + 1)-(1)/(k + z), k = 0..infinity) - EulerGamma -Divide[1,z]+ Sum[Divide[z,k*(k + z)], {k, 1, Infinity}]= - EulerGamma + Sum[Divide[1,k + 1]-Divide[1,k + z], {k, 0, Infinity}] Successful Successful - -
5.7.E7 ψ ⁡ ( z + 1 2 ) - ψ ⁡ ( z 2 ) = 2 ⁢ ∑ k = 0 ∞ ( - 1 ) k k + z digamma 𝑧 1 2 digamma 𝑧 2 2 superscript subscript 𝑘 0 superscript 1 𝑘 𝑘 𝑧 {\displaystyle{\displaystyle\psi\left(\frac{z+1}{2}\right)-\psi\left(\frac{z}{% 2}\right)=2\sum_{k=0}^{\infty}\frac{(-1)^{k}}{k+z}}} Psi((z + 1)/(2))- Psi((z)/(2))= 2*sum(((- 1)^(k))/(k + z), k = 0..infinity) PolyGamma[Divide[z + 1,2]]- PolyGamma[Divide[z,2]]= 2*Sum[Divide[(- 1)^(k),k + z], {k, 0, Infinity}] Successful Successful - -
5.7.E8 ℑ ⁡ ψ ⁡ ( 1 + i ⁢ y ) = ∑ k = 1 ∞ y k 2 + y 2 digamma 1 imaginary-unit 𝑦 superscript subscript 𝑘 1 𝑦 superscript 𝑘 2 superscript 𝑦 2 {\displaystyle{\displaystyle\Im\psi\left(1+\mathrm{i}y\right)=\sum_{k=1}^{% \infty}\frac{y}{k^{2}+y^{2}}}} Im(Psi(1 + I*y))= sum((y)/((k)^(2)+ (y)^(2)), k = 1..infinity) Im[PolyGamma[1 + I*y]]= Sum[Divide[y,(k)^(2)+ (y)^(2)], {k, 1, Infinity}] Failure Failure Skip Successful
5.8.E2 1 Γ ⁡ ( z ) = z ⁢ e γ ⁢ z ⁢ ∏ k = 1 ∞ ( 1 + z k ) ⁢ e - z / k 1 Euler-Gamma 𝑧 𝑧 superscript 𝑒 𝑧 superscript subscript product 𝑘 1 1 𝑧 𝑘 superscript 𝑒 𝑧 𝑘 {\displaystyle{\displaystyle\frac{1}{\Gamma\left(z\right)}=ze^{\gamma z}\prod_% {k=1}^{\infty}\left(1+\frac{z}{k}\right)e^{-z/k}}} (1)/(GAMMA(z))= z*exp(gamma*z)*product((1 +(z)/(k))* exp(- z/ k), k = 1..infinity) Divide[1,Gamma[z]]= z*Exp[EulerGamma*z]*Product[(1 +Divide[z,k])* Exp[- z/ k], {k, 1, Infinity}] Successful Failure - Successful
5.8.E3 | Γ ⁡ ( x ) Γ ⁡ ( x + i ⁢ y ) | 2 = ∏ k = 0 ∞ ( 1 + y 2 ( x + k ) 2 ) superscript Euler-Gamma 𝑥 Euler-Gamma 𝑥 imaginary-unit 𝑦 2 superscript subscript product 𝑘 0 1 superscript 𝑦 2 superscript 𝑥 𝑘 2 {\displaystyle{\displaystyle\left|\frac{\Gamma\left(x\right)}{\Gamma\left(x+% \mathrm{i}y\right)}\right|^{2}=\prod_{k=0}^{\infty}\left(1+\frac{y^{2}}{(x+k)^% {2}}\right)}} (abs((GAMMA(x))/(GAMMA(x + I*y))))^(2)= product(1 +((y)^(2))/((x + k)^(2)), k = 0..infinity) (Abs[Divide[Gamma[x],Gamma[x + I*y]]])^(2)= Product[1 +Divide[(y)^(2),(x + k)^(2)], {k, 0, Infinity}] Failure Failure Skip Successful
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}} (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}] Failure Failure Skip Successful
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}} (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 +)}] Error Failure - 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!}}} 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}]+ Sum[Divide[(- 1)^(k),(z + k)* (k)!], {k, 0, Infinity}] Failure Successful Skip -
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}} 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}]), {t, 0, Infinity}] Failure Failure Skip Skip
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}} 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}] Successful Failure - Successful
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}} 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}] Successful Failure - Successful
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}} 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}] Successful Failure - Successful
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}} 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}] Successful Failure - Successful
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(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}] Failure Failure Skip Skip
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}} 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}] Failure Failure Skip Successful
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}} 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}] Failure Failure Skip Error
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}}} 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}] Failure Failure Skip Successful
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)}}} 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}] Failure Failure Skip Skip
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}} 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}] Failure Failure Skip Successful
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((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}]= Integrate[Divide[1 - (t)^(z - 1),1 - t], {t, 0, 1}] Failure Failure Skip Skip
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}} 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}] Failure Failure Skip Skip
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}} gamma = - int(exp(- t)*ln(t), t = 0..infinity) EulerGamma = - Integrate[Exp[- t]*Log[t], {t, 0, Infinity}] Successful Successful - -
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(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}]= Integrate[(Divide[1,1 + t]- Exp[- t])*Divide[1,t], {t, 0, Infinity}] Successful Successful - -
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(((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}]= Integrate[(1 - Exp[- t])*Divide[1,t], {t, 0, 1}]- Integrate[Exp[- t]*Divide[1,t], {t, 1, Infinity}] Successful Successful - -
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((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}]- Integrate[Exp[- t]*Divide[1,t], {t, 1, Infinity}]= Integrate[Divide[Exp[- t],1 - Exp[- t]]-Divide[Exp[- t],t], {t, 0, Infinity}] Successful Successful - -
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}} subs( temp=z, diff( GAMMA(temp), temp$(n) ) )= int((ln(t))^(n)* exp(- t)*(t)^(z - 1), t = 0..infinity) (D[Gamma[temp], {temp, n}]/.temp-> z)= Integrate[(Log[t])^(n)* Exp[- t]*(t)^(z - 1), {t, 0, Infinity}] Successful Failure - Error
5.11.E5 g k = 2 ⁢ ( 1 2 ) k ⁢ a 2 ⁢ k subscript 𝑔 𝑘 2 Pochhammer 1 2 𝑘 subscript 𝑎 2 𝑘 {\displaystyle{\displaystyle g_{k}=\sqrt{2}{\left(\tfrac{1}{2}\right)_{k}}a_{2% k}}} g[k]=sqrt(2)*pochhammer((1)/(2), k)*a[2*k] Subscript[g, k]=Sqrt[2]*Pochhammer[Divide[1,2], k]*Subscript[a, 2*k] Failure Failure
Fail
.4142135625+.4142135625*I <- {a[2*k] = 2^(1/2)+I*2^(1/2), g[k] = 2^(1/2)+I*2^(1/2), k = 1}
-.8578643792e-1-.8578643792e-1*I <- {a[2*k] = 2^(1/2)+I*2^(1/2), g[k] = 2^(1/2)+I*2^(1/2), k = 2}
-2.335786436-2.335786436*I <- {a[2*k] = 2^(1/2)+I*2^(1/2), g[k] = 2^(1/2)+I*2^(1/2), k = 3}
.4142135625-2.414213561*I <- {a[2*k] = 2^(1/2)+I*2^(1/2), g[k] = 2^(1/2)-I*2^(1/2), k = 1}
... skip entries to safe data
 Successful
5.11.E10 Γ ⁡ ( z ) = e - z ⁢ z z ⁢ ( 2 ⁢ π z ) 1 / 2 ⁢ ( ∑ k = 0 K - 1 g k z k + R K ⁢ ( z ) ) Euler-Gamma 𝑧 superscript 𝑒 𝑧 superscript 𝑧 𝑧 superscript 2 𝜋 𝑧 1 2 superscript subscript 𝑘 0 𝐾 1 subscript 𝑔 𝑘 superscript 𝑧 𝑘 subscript 𝑅 𝐾 𝑧 {\displaystyle{\displaystyle\Gamma\left(z\right)=e^{-z}z^{z}\left(\frac{2\pi}{% z}\right)^{1/2}\left(\sum_{k=0}^{K-1}\frac{g_{k}}{z^{k}}+R_{K}(z)\right)}} GAMMA(z)= exp(- z)*(z)^(z)*((2*Pi)/(z))^(1/ 2)*(sum((g[k])/((z)^(k)), k = 0..K - 1)+ R[K]*(z)) Gamma[z]= Exp[- z]*(z)^(z)*(Divide[2*Pi,z])^(1/ 2)*(Sum[Divide[Subscript[g, k],(z)^(k)], {k, 0, K - 1}]+ Subscript[R, K]*(z)) Failure Failure Skip Skip
5.11#Ex10 G 2 ⁢ ( a , b ) = 1 12 ⁢ ( a - b 2 ) ⁢ ( 3 ⁢ ( a + b - 1 ) 2 - ( a - b + 1 ) ) subscript 𝐺 2 𝑎 𝑏 1 12 binomial 𝑎 𝑏 2 3 superscript 𝑎 𝑏 1 2 𝑎 𝑏 1 {\displaystyle{\displaystyle G_{2}(a,b)=\frac{1}{12}\genfrac{(}{)}{0.0pt}{}{a-% b}{2}(3(a+b-1)^{2}-(a-b+1))}} G[2]*(a , b)=(1)/(12)*binomial(a - b,2)*(3*(a + b - 1)^(2)-(a - b + 1)) Subscript[G, 2]*(a , b)=Divide[1,12]*Binomial[a - b,2]*(3*(a + b - 1)^(2)-(a - b + 1)) Failure Failure Error Error
5.11#Ex12 H 1 ⁢ ( a , b ) = - 1 12 ⁢ ( a - b 2 ) ⁢ ( a - b + 1 ) subscript 𝐻 1 𝑎 𝑏 1 12 binomial 𝑎 𝑏 2 𝑎 𝑏 1 {\displaystyle{\displaystyle H_{1}(a,b)=-\frac{1}{12}\genfrac{(}{)}{0.0pt}{}{a% -b}{2}(a-b+1)}} H[1]*(a , b)= -(1)/(12)*binomial(a - b,2)*(a - b + 1) Subscript[H, 1]*(a , b)= -Divide[1,12]*Binomial[a - b,2]*(a - b + 1) Failure Failure Error Error
5.11#Ex13 H 2 ⁢ ( a , b ) = 1 240 ⁢ ( a - b 4 ) ⁢ ( 2 ⁢ ( a - b + 1 ) + 5 ⁢ ( a - b + 1 ) 2 ) subscript 𝐻 2 𝑎 𝑏 1 240 binomial 𝑎 𝑏 4 2 𝑎 𝑏 1 5 superscript 𝑎 𝑏 1 2 {\displaystyle{\displaystyle H_{2}(a,b)=\frac{1}{240}\genfrac{(}{)}{0.0pt}{}{a% -b}{4}(2(a-b+1)+5(a-b+1)^{2})}} H[2]*(a , b)=(1)/(240)*binomial(a - b,4)*(2*(a - b + 1)+ 5*(a - b + 1)^(2)) Subscript[H, 2]*(a , b)=Divide[1,240]*Binomial[a - b,4]*(2*(a - b + 1)+ 5*(a - b + 1)^(2)) Failure Failure Error Error
5.12.E1 B ⁡ ( a , b ) = ∫ 0 1 t a - 1 ⁢ ( 1 - t ) b - 1 ⁢ d t Euler-Beta 𝑎 𝑏 superscript subscript 0 1 superscript 𝑡 𝑎 1 superscript 1 𝑡 𝑏 1 𝑡 {\displaystyle{\displaystyle\mathrm{B}\left(a,b\right)=\int_{0}^{1}t^{a-1}(1-t% )^{b-1}\mathrm{d}t}} Beta(a, b)= int((t)^(a - 1)*(1 - t)^(b - 1), t = 0..1) Beta[a, b]= Integrate[(t)^(a - 1)*(1 - t)^(b - 1), {t, 0, 1}] Failure Failure Skip Successful
5.12.E1 ∫ 0 1 t a - 1 ⁢ ( 1 - t ) b - 1 ⁢ d t = Γ ⁡ ( a ) ⁢ Γ ⁡ ( b ) Γ ⁡ ( a + b ) superscript subscript 0 1 superscript 𝑡 𝑎 1 superscript 1 𝑡 𝑏 1 𝑡 Euler-Gamma 𝑎 Euler-Gamma 𝑏 Euler-Gamma 𝑎 𝑏 {\displaystyle{\displaystyle\int_{0}^{1}t^{a-1}(1-t)^{b-1}\mathrm{d}t=\frac{% \Gamma\left(a\right)\Gamma\left(b\right)}{\Gamma\left(a+b\right)}}} int((t)^(a - 1)*(1 - t)^(b - 1), t = 0..1)=(GAMMA(a)*GAMMA(b))/(GAMMA(a + b)) Integrate[(t)^(a - 1)*(1 - t)^(b - 1), {t, 0, 1}]=Divide[Gamma[a]*Gamma[b],Gamma[a + b]] Successful Failure - Successful
5.12.E2 ∫ 0 π / 2 sin 2 ⁢ a - 1 ⁡ θ ⁢ cos 2 ⁢ b - 1 ⁡ θ ⁢ d θ = 1 2 ⁢ B ⁡ ( a , b ) superscript subscript 0 𝜋 2 2 𝑎 1 𝜃 2 𝑏 1 𝜃 𝜃 1 2 Euler-Beta 𝑎 𝑏 {\displaystyle{\displaystyle\int_{0}^{\pi/2}{\sin^{2a-1}}\theta{\cos^{2b-1}}% \theta\mathrm{d}\theta=\tfrac{1}{2}\mathrm{B}\left(a,b\right)}} int((sin(theta))^(2*a - 1)* (cos(theta))^(2*b - 1), theta = 0..Pi/ 2)=(1)/(2)*Beta(a, b) Integrate[(Sin[\[Theta]])^(2*a - 1)* (Cos[\[Theta]])^(2*b - 1), {\[Theta], 0, Pi/ 2}]=Divide[1,2]*Beta[a, b] Failure Failure Skip Successful
5.12.E3 ∫ 0 ∞ t a - 1 ⁢ d t ( 1 + t ) a + b = B ⁡ ( a , b ) superscript subscript 0 superscript 𝑡 𝑎 1 𝑡 superscript 1 𝑡 𝑎 𝑏 Euler-Beta 𝑎 𝑏 {\displaystyle{\displaystyle\int_{0}^{\infty}\frac{t^{a-1}\mathrm{d}t}{(1+t)^{% a+b}}=\mathrm{B}\left(a,b\right)}} int(((t)^(a - 1))/((1 + t)^(a + b)), t = 0..infinity)= Beta(a, b) Integrate[Divide[(t)^(a - 1),(1 + t)^(a + b)], {t, 0, Infinity}]= Beta[a, b] Failure Failure Skip Successful
5.12.E4 ∫ 0 1 t a - 1 ⁢ ( 1 - t ) b - 1 ( t + z ) a + b ⁢ d t = B ⁡ ( a , b ) ⁢ ( 1 + z ) - a ⁢ z - b superscript subscript 0 1 superscript 𝑡 𝑎 1 superscript 1 𝑡 𝑏 1 superscript 𝑡 𝑧 𝑎 𝑏 𝑡 Euler-Beta 𝑎 𝑏 superscript 1 𝑧 𝑎 superscript 𝑧 𝑏 {\displaystyle{\displaystyle\int_{0}^{1}\frac{t^{a-1}(1-t)^{b-1}}{(t+z)^{a+b}}% \mathrm{d}t=\mathrm{B}\left(a,b\right)(1+z)^{-a}z^{-b}}} int(((t)^(a - 1)*(1 - t)^(b - 1))/((t + z)^(a + b)), t = 0..1)= Beta(a, b)*(1 + z)^(- a)* (z)^(- b) Integrate[Divide[(t)^(a - 1)*(1 - t)^(b - 1),(t + z)^(a + b)], {t, 0, 1}]= Beta[a, b]*(1 + z)^(- a)* (z)^(- b) Failure Failure Skip Successful
5.12.E5 ∫ 0 π / 2 ( cos ⁡ t ) a - 1 ⁢ cos ⁡ ( b ⁢ t ) ⁢ d t = π 2 a ⁢ 1 a ⁢ B ⁡ ( 1 2 ⁢ ( a + b + 1 ) , 1 2 ⁢ ( a - b + 1 ) ) superscript subscript 0 𝜋 2 superscript 𝑡 𝑎 1 𝑏 𝑡 𝑡 𝜋 superscript 2 𝑎 1 𝑎 Euler-Beta 1 2 𝑎 𝑏 1 1 2 𝑎 𝑏 1 {\displaystyle{\displaystyle\int_{0}^{\pi/2}(\cos t)^{a-1}\cos\left(bt\right)% \mathrm{d}t=\frac{\pi}{2^{a}}\frac{1}{a\mathrm{B}\left(\frac{1}{2}(a+b+1),% \frac{1}{2}(a-b+1)\right)}}} int((cos(t))^(a - 1)* cos(b*t), t = 0..Pi/ 2)=(Pi)/((2)^(a))*(1)/(a*Beta((1)/(2)*(a + b + 1), (1)/(2)*(a - b + 1))) Integrate[(Cos[t])^(a - 1)* Cos[b*t], {t, 0, Pi/ 2}]=Divide[Pi,(2)^(a)]*Divide[1,a*Beta[Divide[1,2]*(a + b + 1), Divide[1,2]*(a - b + 1)]] Failure Failure Skip Error
5.12.E6 ∫ 0 π ( sin ⁡ t ) a - 1 ⁢ e i ⁢ b ⁢ t ⁢ d t = π 2 a - 1 ⁢ e i ⁢ π ⁢ b / 2 a ⁢ B ⁡ ( 1 2 ⁢ ( a + b + 1 ) , 1 2 ⁢ ( a - b + 1 ) ) superscript subscript 0 𝜋 superscript 𝑡 𝑎 1 superscript 𝑒 𝑖 𝑏 𝑡 𝑡 𝜋 superscript 2 𝑎 1 superscript 𝑒 𝑖 𝜋 𝑏 2 𝑎 Euler-Beta 1 2 𝑎 𝑏 1 1 2 𝑎 𝑏 1 {\displaystyle{\displaystyle\int_{0}^{\pi}(\sin t)^{a-1}e^{ibt}\mathrm{d}t=% \frac{\pi}{2^{a-1}}\frac{e^{i\pi b/2}}{a\mathrm{B}\left(\frac{1}{2}(a+b+1),% \frac{1}{2}(a-b+1)\right)}}} int((sin(t))^(a - 1)* exp(I*b*t), t = 0..Pi)=(Pi)/((2)^(a - 1))*(exp(I*Pi*b/ 2))/(a*Beta((1)/(2)*(a + b + 1), (1)/(2)*(a - b + 1))) Integrate[(Sin[t])^(a - 1)* Exp[I*b*t], {t, 0, Pi}]=Divide[Pi,(2)^(a - 1)]*Divide[Exp[I*Pi*b/ 2],a*Beta[Divide[1,2]*(a + b + 1), Divide[1,2]*(a - b + 1)]] Failure Failure Skip Skip
5.12.E7 ∫ 0 ∞ cosh ⁡ ( 2 ⁢ b ⁢ t ) ( cosh ⁡ t ) 2 ⁢ a ⁢ d t = 4 a - 1 ⁢ B ⁡ ( a + b , a - b ) superscript subscript 0 2 𝑏 𝑡 superscript 𝑡 2 𝑎 𝑡 superscript 4 𝑎 1 Euler-Beta 𝑎 𝑏 𝑎 𝑏 {\displaystyle{\displaystyle\int_{0}^{\infty}\frac{\cosh\left(2bt\right)}{(% \cosh t)^{2a}}\mathrm{d}t=4^{a-1}\mathrm{B}\left(a+b,a-b\right)}} int((cosh(2*b*t))/((cosh(t))^(2*a)), t = 0..infinity)= (4)^(a - 1)* Beta(a + b, a - b) Integrate[Divide[Cosh[2*b*t],(Cosh[t])^(2*a)], {t, 0, Infinity}]= (4)^(a - 1)* Beta[a + b, a - b] Failure Failure Skip Skip
5.12.E11 1 e 2 ⁢ π ⁢ i ⁢ a - 1 ⁢ ∫ ∞ ( 0 + ) t a - 1 ⁢ ( 1 + t ) - a - b ⁢ d t = B ⁡ ( a , b ) 1 superscript 𝑒 2 𝜋 𝑖 𝑎 1 superscript subscript limit-from 0 superscript 𝑡 𝑎 1 superscript 1 𝑡 𝑎 𝑏 𝑡 Euler-Beta 𝑎 𝑏 {\displaystyle{\displaystyle\frac{1}{e^{2\pi ia}-1}\int_{\infty}^{(0+)}t^{a-1}% (1+t)^{-a-b}\mathrm{d}t=\mathrm{B}\left(a,b\right)}} (1)/(exp(2*Pi*I*a)- 1)*int((t)^(a - 1)*(1 + t)^(- a - b), t = infinity..(0 +))= Beta(a, b) Divide[1,Exp[2*Pi*I*a]- 1]*Integrate[(t)^(a - 1)*(1 + t)^(- a - b), {t, Infinity, (0 +)}]= Beta[a, b] Error Failure - Error
5.12.E12 ∫ P ( 1 + , 0 + , 1 - , 0 - ) t a - 1 ⁢ ( 1 - t ) b - 1 ⁢ d t = - 4 ⁢ e π ⁢ i ⁢ ( a + b ) ⁢ sin ⁡ ( π ⁢ a ) ⁢ sin ⁡ ( π ⁢ b ) ⁢ B ⁡ ( a , b ) superscript subscript 𝑃 limit-from 1 limit-from 0 limit-from 1 limit-from 0 superscript 𝑡 𝑎 1 superscript 1 𝑡 𝑏 1 𝑡 4 superscript 𝑒 𝜋 𝑖 𝑎 𝑏 𝜋 𝑎 𝜋 𝑏 Euler-Beta 𝑎 𝑏 {\displaystyle{\displaystyle\int_{P}^{(1+,0+,1-,0-)}t^{a-1}(1-t)^{b-1}\mathrm{% d}t=-4e^{\pi i(a+b)}\sin\left(\pi a\right)\sin\left(\pi b\right)\mathrm{B}% \left(a,b\right)}} int((t)^(a - 1)*(1 - t)^(b - 1), t = P..(1 + , 0 + , 1 - , 0 -))= - 4*exp(Pi*I*(a + b))*sin(Pi*a)*sin(Pi*b)*Beta(a, b) Integrate[(t)^(a - 1)*(1 - t)^(b - 1), {t, P, (1 + , 0 + , 1 - , 0 -)}]= - 4*Exp[Pi*I*(a + b)]*Sin[Pi*a]*Sin[Pi*b]*Beta[a, b] Error Failure - Error
5.13.E3 1 2 ⁢ π ⁢ ∫ - ∞ ∞ Γ ⁡ ( a + i ⁢ t ) ⁢ Γ ⁡ ( b + i ⁢ t ) ⁢ Γ ⁡ ( c - i ⁢ t ) ⁢ Γ ⁡ ( d - i ⁢ t ) ⁢ d t = Γ ⁡ ( a + c ) ⁢ Γ ⁡ ( a + d ) ⁢ Γ ⁡ ( b + c ) ⁢ Γ ⁡ ( b + d ) Γ ⁡ ( a + b + c + d ) 1 2 𝜋 superscript subscript Euler-Gamma 𝑎 𝑖 𝑡 Euler-Gamma 𝑏 𝑖 𝑡 Euler-Gamma 𝑐 𝑖 𝑡 Euler-Gamma 𝑑 𝑖 𝑡 𝑡 Euler-Gamma 𝑎 𝑐 Euler-Gamma 𝑎 𝑑 Euler-Gamma 𝑏 𝑐 Euler-Gamma 𝑏 𝑑 Euler-Gamma 𝑎 𝑏 𝑐 𝑑 {\displaystyle{\displaystyle\frac{1}{2\pi}\int_{-\infty}^{\infty}\Gamma\left(a% +it\right)\Gamma\left(b+it\right)\Gamma\left(c-it\right)\Gamma\left(d-it\right% )\mathrm{d}t=\frac{\Gamma\left(a+c\right)\Gamma\left(a+d\right)\Gamma\left(b+c% \right)\Gamma\left(b+d\right)}{\Gamma\left(a+b+c+d\right)}}} (1)/(2*Pi)*int(GAMMA(a + I*t)*GAMMA(b + I*t)*GAMMA(c - I*t)*GAMMA(d - I*t), t = - infinity..infinity)=(GAMMA(a + c)*GAMMA(a + d)*GAMMA(b + c)*GAMMA(b + d))/(GAMMA(a + b + c + d)) Divide[1,2*Pi]*Integrate[Gamma[a + I*t]*Gamma[b + I*t]*Gamma[c - I*t]*Gamma[d - I*t], {t, - Infinity, Infinity}]=Divide[Gamma[a + c]*Gamma[a + d]*Gamma[b + c]*Gamma[b + d],Gamma[a + b + c + d]] Error Failure - Successful
5.13.E4 ∫ - ∞ ∞ d t Γ ⁡ ( a + t ) ⁢ Γ ⁡ ( b + t ) ⁢ Γ ⁡ ( c - t ) ⁢ Γ ⁡ ( d - t ) = Γ ⁡ ( a + b + c + d - 3 ) Γ ⁡ ( a + c - 1 ) ⁢ Γ ⁡ ( a + d - 1 ) ⁢ Γ ⁡ ( b + c - 1 ) ⁢ Γ ⁡ ( b + d - 1 ) superscript subscript 𝑡 Euler-Gamma 𝑎 𝑡 Euler-Gamma 𝑏 𝑡 Euler-Gamma 𝑐 𝑡 Euler-Gamma 𝑑 𝑡 Euler-Gamma 𝑎 𝑏 𝑐 𝑑 3 Euler-Gamma 𝑎 𝑐 1 Euler-Gamma 𝑎 𝑑 1 Euler-Gamma 𝑏 𝑐 1 Euler-Gamma 𝑏 𝑑 1 {\displaystyle{\displaystyle\int_{-\infty}^{\infty}\frac{\mathrm{d}t}{\Gamma% \left(a+t\right)\Gamma\left(b+t\right)\Gamma\left(c-t\right)\Gamma\left(d-t% \right)}=\frac{\Gamma\left(a+b+c+d-3\right)}{\Gamma\left(a+c-1\right)\Gamma% \left(a+d-1\right)\Gamma\left(b+c-1\right)\Gamma\left(b+d-1\right)}}} int((1)/(GAMMA(a + t)*GAMMA(b + t)*GAMMA(c - t)*GAMMA(d - t)), t = - infinity..infinity)=(GAMMA(a + b + c + d - 3))/(GAMMA(a + c - 1)*GAMMA(a + d - 1)*GAMMA(b + c - 1)*GAMMA(b + d - 1)) Integrate[Divide[1,Gamma[a + t]*Gamma[b + t]*Gamma[c - t]*Gamma[d - t]], {t, - Infinity, Infinity}]=Divide[Gamma[a + b + c + d - 3],Gamma[a + c - 1]*Gamma[a + d - 1]*Gamma[b + c - 1]*Gamma[b + d - 1]] Failure Failure Skip Error
5.13.E5 1 4 ⁢ π ⁢ ∫ - ∞ ∞ ∏ k = 1 4 Γ ⁡ ( a k + i ⁢ t ) ⁢ Γ ⁡ ( a k - i ⁢ t ) Γ ⁡ ( 2 ⁢ i ⁢ t ) ⁢ Γ ⁡ ( - 2 ⁢ i ⁢ t ) ⁢ d t = ∏ 1 ≤ j < k ≤ 4 Γ ⁡ ( a j + a k ) Γ ⁡ ( a 1 + a 2 + a 3 + a 4 ) 1 4 𝜋 superscript subscript superscript subscript product 𝑘 1 4 Euler-Gamma subscript 𝑎 𝑘 𝑖 𝑡 Euler-Gamma subscript 𝑎 𝑘 𝑖 𝑡 Euler-Gamma 2 𝑖 𝑡 Euler-Gamma 2 𝑖 𝑡 𝑡 subscript product 1 𝑗 𝑘 4 Euler-Gamma subscript 𝑎 𝑗 subscript 𝑎 𝑘 Euler-Gamma subscript 𝑎 1 subscript 𝑎 2 subscript 𝑎 3 subscript 𝑎 4 {\displaystyle{\displaystyle\frac{1}{4\pi}\int_{-\infty}^{\infty}\frac{\prod_{% k=1}^{4}\Gamma\left(a_{k}+it\right)\Gamma\left(a_{k}-it\right)}{\Gamma\left(2% it\right)\Gamma\left(-2it\right)}\mathrm{d}t=\frac{\prod_{1\leq j<k\leq 4}% \Gamma\left(a_{j}+a_{k}\right)}{\Gamma\left(a_{1}+a_{2}+a_{3}+a_{4}\right)}}} (1)/(4*Pi)*int((product(GAMMA(a[k]+ I*t)*GAMMA(a[k]- I*t), k = 1..4))/(GAMMA(2*I*t)*GAMMA(- 2*I*t)), t = - infinity..infinity)=(product(product(GAMMA(a[j]+ a[k]), k = j + 1..4), j = 1..k - 1))/(GAMMA(a[1]+ a[2]+ a[3]+ a[4])) Divide[1,4*Pi]*Integrate[Divide[Product[Gamma[Subscript[a, k]+ I*t]*Gamma[Subscript[a, k]- I*t], {k, 1, 4}],Gamma[2*I*t]*Gamma[- 2*I*t]], {t, - Infinity, Infinity}]=Divide[Product[Product[Gamma[Subscript[a, j]+ Subscript[a, k]], {k, j + 1, 4}], {j, 1, k - 1}],Gamma[Subscript[a, 1]+ Subscript[a, 2]+ Subscript[a, 3]+ Subscript[a, 4]]] Failure Failure Skip Error
5.15.E1 ψ ′ ⁡ ( z ) = ∑ k = 0 ∞ 1 ( k + z ) 2 diffop digamma 1 𝑧 superscript subscript 𝑘 0 1 superscript 𝑘 𝑧 2 {\displaystyle{\displaystyle\psi'\left(z\right)=\sum_{k=0}^{\infty}\frac{1}{(k% +z)^{2}}}} subs( temp=z, diff( Psi(temp), temp$(1) ) )= sum((1)/((k + z)^(2)), k = 0..infinity) (D[PolyGamma[temp], {temp, 1}]/.temp-> z)= Sum[Divide[1,(k + z)^(2)], {k, 0, Infinity}] Successful Successful - -
5.15.E2 ψ ( n ) ⁡ ( 1 ) = ( - 1 ) n + 1 ⁢ n ! ⁢ ζ ⁡ ( n + 1 ) polygamma 𝑛 1 superscript 1 𝑛 1 𝑛 Riemann-zeta 𝑛 1 {\displaystyle{\displaystyle\psi^{(n)}\left(1\right)=(-1)^{n+1}n!\zeta\left(n+% 1\right)}} Psi(n, 1)=(- 1)^(n + 1)* factorial(n)*Zeta(n + 1) PolyGamma[n, 1]=(- 1)^(n + 1)* (n)!*Zeta[n + 1] Failure Failure Successful Successful
5.15.E3 ψ ( n ) ⁡ ( 1 2 ) = ( - 1 ) n + 1 ⁢ n ! ⁢ ( 2 n + 1 - 1 ) ⁢ ζ ⁡ ( n + 1 ) polygamma 𝑛 1 2 superscript 1 𝑛 1 𝑛 superscript 2 𝑛 1 1 Riemann-zeta 𝑛 1 {\displaystyle{\displaystyle\psi^{(n)}\left(\tfrac{1}{2}\right)=(-1)^{n+1}n!(2% ^{n+1}-1)\zeta\left(n+1\right)}} Psi(n, (1)/(2))=(- 1)^(n + 1)* factorial(n)*((2)^(n + 1)- 1)* Zeta(n + 1) PolyGamma[n, Divide[1,2]]=(- 1)^(n + 1)* (n)!*((2)^(n + 1)- 1)* Zeta[n + 1] Failure Failure Successful Successful
5.15.E4 ψ ′ ⁡ ( n - 1 2 ) = 1 2 ⁢ π 2 - 4 ⁢ ∑ k = 1 n - 1 1 ( 2 ⁢ k - 1 ) 2 diffop digamma 1 𝑛 1 2 1 2 superscript 𝜋 2 4 superscript subscript 𝑘 1 𝑛 1 1 superscript 2 𝑘 1 2 {\displaystyle{\displaystyle\psi'\left(n-\tfrac{1}{2}\right)=\tfrac{1}{2}\pi^{% 2}-4\sum_{k=1}^{n-1}\frac{1}{(2k-1)^{2}}}} subs( temp=n -(1)/(2), diff( Psi(temp), temp$(1) ) )=(1)/(2)*(Pi)^(2)- 4*sum((1)/((2*k - 1)^(2)), k = 1..n - 1) (D[PolyGamma[temp], {temp, 1}]/.temp-> n -Divide[1,2])=Divide[1,2]*(Pi)^(2)- 4*Sum[Divide[1,(2*k - 1)^(2)], {k, 1, n - 1}] Successful Successful - -
5.15.E5 ψ ( n ) ⁡ ( z + 1 ) = ψ ( n ) ⁡ ( z ) + ( - 1 ) n ⁢ n ! ⁢ z - n - 1 digamma 𝑛 𝑧 1 digamma 𝑛 𝑧 superscript 1 𝑛 𝑛 superscript 𝑧 𝑛 1 {\displaystyle{\displaystyle{\psi^{(n)}}\left(z+1\right)={\psi^{(n)}}\left(z% \right)+(-1)^{n}n!z^{-n-1}}} subs( temp=z + 1, diff( Psi(temp), temp$(n) ) )= subs( temp=z, diff( Psi(temp), temp$(n) ) )+(- 1)^(n)* factorial(n)*(z)^(- n - 1) (D[PolyGamma[temp], {temp, n}]/.temp-> z + 1)= (D[PolyGamma[temp], {temp, n}]/.temp-> z)+(- 1)^(n)* (n)!*(z)^(- n - 1) Failure Failure Successful Successful
5.15.E6 ψ ( n ) ⁡ ( 1 - z ) + ( - 1 ) n - 1 ⁢ ψ ( n ) ⁡ ( z ) = ( - 1 ) n ⁢ π ⁢ d n d z n ⁡ cot ⁡ ( π ⁢ z ) digamma 𝑛 1 𝑧 superscript 1 𝑛 1 digamma 𝑛 𝑧 superscript 1 𝑛 𝜋 derivative 𝑧 𝑛 𝜋 𝑧 {\displaystyle{\displaystyle{\psi^{(n)}}\left(1-z\right)+(-1)^{n-1}{\psi^{(n)}% }\left(z\right)=(-1)^{n}\pi\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\cot\left% (\pi z\right)}} subs( temp=1 - z, diff( Psi(temp), temp$(n) ) )+(- 1)^(n - 1)* subs( temp=z, diff( Psi(temp), temp$(n) ) )=(- 1)^(n)* Pi*diff(cot(Pi*z), [z$(n)]) (D[PolyGamma[temp], {temp, n}]/.temp-> 1 - z)+(- 1)^(n - 1)* (D[PolyGamma[temp], {temp, n}]/.temp-> z)=(- 1)^(n)* Pi*D[Cot[Pi*z], {z, n}] Failure Failure Successful Successful
5.15.E7 ψ ( n ) ⁡ ( m ⁢ z ) = 1 m n + 1 ⁢ ∑ k = 0 m - 1 ψ ( n ) ⁡ ( z + k m ) digamma 𝑛 𝑚 𝑧 1 superscript 𝑚 𝑛 1 superscript subscript 𝑘 0 𝑚 1 digamma 𝑛 𝑧 𝑘 𝑚 {\displaystyle{\displaystyle{\psi^{(n)}}\left(mz\right)=\frac{1}{m^{n+1}}\sum_% {k=0}^{m-1}{\psi^{(n)}}\left(z+\frac{k}{m}\right)}} subs( temp=m*z, diff( Psi(temp), temp$(n) ) )=(1)/((m)^(n + 1))*sum(subs( temp=z +(k)/(m), diff( Psi(temp), temp$(n) ) ), k = 0..m - 1) (D[PolyGamma[temp], {temp, n}]/.temp-> m*z)=Divide[1,(m)^(n + 1)]*Sum[D[PolyGamma[temp], {temp, n}]/.temp-> z +Divide[k,m], {k, 0, m - 1}] Failure Failure Skip Successful
5.16.E1 ∑ k = 1 ∞ ( - 1 ) k ⁢ ψ ′ ⁡ ( k ) = - π 2 8 superscript subscript 𝑘 1 superscript 1 𝑘 diffop digamma 1 𝑘 superscript 𝜋 2 8 {\displaystyle{\displaystyle\sum_{k=1}^{\infty}(-1)^{k}\psi'\left(k\right)=-% \frac{\pi^{2}}{8}}} sum((- 1)^(k)* subs( temp=k, diff( Psi(temp), temp$(1) ) ), k = 1..infinity)= -((Pi)^(2))/(8) Sum[(- 1)^(k)* (D[PolyGamma[temp], {temp, 1}]/.temp-> k), {k, 1, Infinity}]= -Divide[(Pi)^(2),8] Failure Successful Skip -
5.16.E2 ∑ k = 1 ∞ 1 k ⁢ ψ ′ ⁡ ( k + 1 ) = ζ ⁡ ( 3 ) superscript subscript 𝑘 1 1 𝑘 diffop digamma 1 𝑘 1 Riemann-zeta 3 {\displaystyle{\displaystyle\sum_{k=1}^{\infty}\frac{1}{k}\psi'\left(k+1\right% )=\zeta\left(3\right)}} sum((1)/(k)*subs( temp=k + 1, diff( Psi(temp), temp$(1) ) ), k = 1..infinity)= Zeta(3) Sum[Divide[1,k]*(D[PolyGamma[temp], {temp, 1}]/.temp-> k + 1), {k, 1, Infinity}]= Zeta[3] Failure Successful Skip -
5.16.E2 ζ ⁡ ( 3 ) = - 1 2 ⁢ ψ ′′ ⁡ ( 1 ) Riemann-zeta 3 1 2 diffop digamma 2 1 {\displaystyle{\displaystyle\zeta\left(3\right)=-\frac{1}{2}\psi''\left(1% \right)}} Zeta(3)= -(1)/(2)*subs( temp=1, diff( Psi(temp), temp$(2) ) ) Zeta[3]= -Divide[1,2]*(D[PolyGamma[temp], {temp, 2}]/.temp-> 1) Successful Successful - -
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)}} Error BarnesG[z + 1]= Gamma[z]*BarnesG[z] Error Failure - Successful
5.17#Ex2 G ⁡ ( 1 ) = 1 Barnes-Gamma 1 1 {\displaystyle{\displaystyle G\left(1\right)=1}} Error BarnesG[1]= 1 Error Successful - -
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)}} 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}] Error Successful - -
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}} 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}] Error Failure - Successful
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 = 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}]-(Divide[1,2]*(n)^(2)+Divide[1,2]*n +Divide[1,12])* Log[n]+Divide[1,4]*(n)^(2), n -> Infinity] Failure Failure Skip
Fail
Complex[1.165459085339311, 1.4142135623730951] <- {Rule[C, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[1.165459085339311, -1.4142135623730951] <- {Rule[C, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.6629680394068793, -1.4142135623730951] <- {Rule[C, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.6629680394068793, 1.4142135623730951] <- {Rule[C, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
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}}}} 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$(1) ) ))/(2*(Pi)^(2)) Limit[Sum[k*Log[k], {k, 1, n}]-(Divide[1,2]*(n)^(2)+Divide[1,2]*n +Divide[1,12])* Log[n]+Divide[1,4]*(n)^(2), n -> Infinity]=Divide[EulerGamma + Log[2*Pi],12]-Divide[D[Zeta[temp], {temp, 1}]/.temp-> 2,2*(Pi)^(2)] Failure Successful Skip -
5.17.E7 γ + ln ⁡ ( 2 ⁢ π ) 12 - ζ ′ ⁡ ( 2 ) 2 ⁢ π 2 = 1 12 - ζ ′ ⁡ ( - 1 ) 2 𝜋 12 diffop Riemann-zeta 1 2 2 superscript 𝜋 2 1 12 diffop Riemann-zeta 1 1 {\displaystyle{\displaystyle\frac{\gamma+\ln\left(2\pi\right)}{12}-\frac{\zeta% '\left(2\right)}{2\pi^{2}}=\frac{1}{12}-\zeta'\left(-1\right)}} (gamma + ln(2*Pi))/(12)-(subs( temp=2, diff( Zeta(temp), temp$(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 Skip -
5.19.E3 S = ψ ⁡ ( 1 2 ) - 2 ⁢ ψ ⁡ ( 2 3 ) - γ 𝑆 digamma 1 2 2 digamma 2 3 {\displaystyle{\displaystyle S=\psi\left(\tfrac{1}{2}\right)-2\psi\left(\tfrac% {2}{3}\right)-\gamma}} S = Psi((1)/(2))- 2*Psi((2)/(3))- gamma S = PolyGamma[Divide[1,2]]- 2*PolyGamma[Divide[2,3]]- EulerGamma Failure Failure
Fail
1.318470421+1.414213562*I <- {S = 2^(1/2)+I*2^(1/2)}
1.318470421-1.414213562*I <- {S = 2^(1/2)-I*2^(1/2)}
-1.509956703-1.414213562*I <- {S = -2^(1/2)-I*2^(1/2)}
-1.509956703+1.414213562*I <- {S = -2^(1/2)+I*2^(1/2)}
Fail
Complex[1.3184704217228749, 1.4142135623730951] <- {Rule[S, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[1.3184704217228749, -1.4142135623730951] <- {Rule[S, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.5099567030233154, -1.4142135623730951] <- {Rule[S, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.5099567030233154, 1.4142135623730951] <- {Rule[S, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
5.19.E3 ψ ⁡ ( 1 2 ) - 2 ⁢ ψ ⁡ ( 2 3 ) - γ = 3 ⁢ ln ⁡ 3 - 2 ⁢ ln ⁡ 2 - 1 3 ⁢ π ⁢ 3 digamma 1 2 2 digamma 2 3 3 3 2 2 1 3 𝜋 3 {\displaystyle{\displaystyle\psi\left(\tfrac{1}{2}\right)-2\psi\left(\tfrac{2}% {3}\right)-\gamma=3\ln 3-2\ln 2-\tfrac{1}{3}\pi\sqrt{3}}} Psi((1)/(2))- 2*Psi((2)/(3))- gamma = 3*ln(3)- 2*ln(2)-(1)/(3)*Pi*sqrt(3) PolyGamma[Divide[1,2]]- 2*PolyGamma[Divide[2,3]]- EulerGamma = 3*Log[3]- 2*Log[2]-Divide[1,3]*Pi*Sqrt[3] Successful Successful - -
5.19#Ex3 V = π 1 2 ⁢ n ⁢ r n Γ ⁡ ( 1 2 ⁢ n + 1 ) 𝑉 superscript 𝜋 1 2 𝑛 superscript 𝑟 𝑛 Euler-Gamma 1 2 𝑛 1 {\displaystyle{\displaystyle V=\frac{\pi^{\frac{1}{2}n}r^{n}}{\Gamma\left(% \frac{1}{2}n+1\right)}}} V =((Pi)^((1)/(2)*n)* (r)^(n))/(GAMMA((1)/(2)*n + 1)) V =Divide[(Pi)^(Divide[1,2]*n)* (r)^(n),Gamma[Divide[1,2]*n + 1]] Failure Failure
Fail
-1.414213562-1.414213562*I <- {V = 2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), n = 1}
1.414213562-11.15215705*I <- {V = 2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), n = 2}
25.10958922-22.28116210*I <- {V = 2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), n = 3}
-1.414213562+4.242640686*I <- {V = 2^(1/2)+I*2^(1/2), r = 2^(1/2)-I*2^(1/2), n = 1}
... skip entries to safe data
Fail
Complex[-1.4142135623730951, -1.4142135623730951] <- {Rule[n, 1], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[1.4142135623730951, -11.152157051986077] <- {Rule[n, 2], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[25.10958923255105, -22.281162107804857] <- {Rule[n, 3], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-1.4142135623730951, -4.242640687119286] <- {Rule[n, 1], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
5.19#Ex4 S = 2 ⁢ π 1 2 ⁢ n ⁢ r n - 1 Γ ⁡ ( 1 2 ⁢ n ) 𝑆 2 superscript 𝜋 1 2 𝑛 superscript 𝑟 𝑛 1 Euler-Gamma 1 2 𝑛 {\displaystyle{\displaystyle S=\frac{2\pi^{\frac{1}{2}n}r^{n-1}}{\Gamma\left(% \frac{1}{2}n\right)}}} S =(2*(Pi)^((1)/(2)*n)* (r)^(n - 1))/(GAMMA((1)/(2)*n)) S =Divide[2*(Pi)^(Divide[1,2]*n)* (r)^(n - 1),Gamma[Divide[1,2]*n]] Failure Failure
Fail
-.585786438+1.414213562*I <- {S = 2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), n = 1}
-7.471552313-7.471552313*I <- {S = 2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), n = 2}
1.414213562-48.85126889*I <- {S = 2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), n = 3}
-.585786438+1.414213562*I <- {S = 2^(1/2)+I*2^(1/2), r = 2^(1/2)-I*2^(1/2), n = 1}
... skip entries to safe data
Fail
Complex[-0.5857864376269049, 1.4142135623730951] <- {Rule[n, 1], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[S, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-7.471552313943637, -7.471552313943637] <- {Rule[n, 2], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[S, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[1.4142135623730951, -48.8512688950636] <- {Rule[n, 3], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[S, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-0.5857864376269049, -1.4142135623730951] <- {Rule[n, 1], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[S, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
5.19#Ex4 2 ⁢ π 1 2 ⁢ n ⁢ r n - 1 Γ ⁡ ( 1 2 ⁢ n ) = n r ⁢ V 2 superscript 𝜋 1 2 𝑛 superscript 𝑟 𝑛 1 Euler-Gamma 1 2 𝑛 𝑛 𝑟 𝑉 {\displaystyle{\displaystyle\frac{2\pi^{\frac{1}{2}n}r^{n-1}}{\Gamma\left(% \frac{1}{2}n\right)}=\frac{n}{r}V}} (2*(Pi)^((1)/(2)*n)* (r)^(n - 1))/(GAMMA((1)/(2)*n))=(n)/(r)*V Divide[2*(Pi)^(Divide[1,2]*n)* (r)^(n - 1),Gamma[Divide[1,2]*n]]=Divide[n,r]*V Failure Failure
Fail
1.000000000 <- {V = 2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), n = 1}
6.885765875+8.885765875*I <- {V = 2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), n = 2}
-3.+50.26548245*I <- {V = 2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), n = 3}
2.000000000-1.000000000*I <- {V = 2^(1/2)+I*2^(1/2), r = 2^(1/2)-I*2^(1/2), n = 1}
... skip entries to safe data
Fail
1.0 <- {Rule[n, 1], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[6.885765876316732, 8.885765876316732] <- {Rule[n, 2], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-3.0, 50.26548245743669] <- {Rule[n, 3], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[2.0, 1.0] <- {Rule[n, 1], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
5.20.E1 W = 1 2 ⁢ ∑ ℓ = 1 n x ℓ 2 - ∑ 1 ≤ ℓ < j ≤ n ln ⁡ | x ℓ - x j | 𝑊 1 2 superscript subscript ℓ 1 𝑛 superscript subscript 𝑥 ℓ 2 subscript 1 ℓ 𝑗 𝑛 subscript 𝑥 ℓ subscript 𝑥 𝑗 {\displaystyle{\displaystyle W=\frac{1}{2}\sum_{\ell=1}^{n}x_{\ell}^{2}-\sum_{% 1\leq\ell<j\leq n}\ln|x_{\ell}-x_{j}|}} W =sum(x(x[ell])^(2), ell = 1..n)- sum(sum(ln(abs(x[ell]- x[j])), j = ell + 1..n), ell = 1..j - 1) W =Sum[x(Subscript[x, \[ScriptL]])^(2), {\[ScriptL], 1, n}]- Sum[Sum[Log[Abs[Subscript[x, \[ScriptL]]- Subscript[x, j]]], {j, [ScriptL] + 1, n}], {\[ScriptL], 1, j - 1}] Failure Failure Skip Error
5.20.E4 W = - ∑ 1 ≤ ℓ < j ≤ n ln ⁡ | e i ⁢ θ ℓ - e i ⁢ θ j | 𝑊 subscript 1 ℓ 𝑗 𝑛 superscript 𝑒 𝑖 subscript 𝜃 ℓ superscript 𝑒 𝑖 subscript 𝜃 𝑗 {\displaystyle{\displaystyle W=-\sum_{1\leq\ell<j\leq n}\ln|e^{i\theta_{\ell}}% -e^{i\theta_{j}}|}} W = - sum(sum(ln(abs(exp(I*theta[ell])- exp(I*theta[j]))), j = ell + 1..n), ell = 1..j - 1) W = - Sum[Sum[Log[Abs[Exp[I*Subscript[\[Theta], \[ScriptL]]]- Exp[I*Subscript[\[Theta], j]]]], {j, [ScriptL] + 1, n}], {\[ScriptL], 1, j - 1}] Failure Failure Skip Error
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