Gamma Function - 5.4 Special Values and Extrema

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
5.4#Ex1 Γ ⁡ ( 1 ) = 1 Euler-Gamma 1 1 {\displaystyle{\displaystyle\Gamma\left(1\right)=1}}
\EulerGamma@{1} = 1		 

GAMMA(1) = 1

Gamma[1] == 1
Successful Successful - Successful [Tested: 1]
5.4#Ex2 n ! = Γ ⁡ ( n + 1 ) 𝑛 Euler-Gamma 𝑛 1 {\displaystyle{\displaystyle n!=\Gamma\left(n+1\right)}}
n! = \EulerGamma@{n+1}		 ℜ ⁡ ( n + 1 ) > 0 𝑛 1 0 {\displaystyle{\displaystyle\Re(n+1)>0}} 
factorial(n) = GAMMA(n + 1)

(n)! == Gamma[n + 1]
Successful Successful - Successful [Tested: 3]
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}}}
|\EulerGamma@{iy}| = \left(\frac{\pi}{y\sinh@{\pi y}}\right)^{1/2}		 ℜ ⁡ ( i ⁢ y ) > 0 imaginary-unit 𝑦 0 {\displaystyle{\displaystyle\Re(\mathrm{i}y)>0}} 
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 Error Skip - No test values generated
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}}}
\EulerGamma@{\tfrac{1}{2}+\iunit y}\EulerGamma@{\tfrac{1}{2}-\iunit y} = \left|\EulerGamma@{\tfrac{1}{2}+\iunit y}\right|^{2}		 ℜ ⁡ ( 1 2 + i ⁢ y ) > 0 , ℜ ⁡ ( 1 2 - i ⁢ y ) > 0 formulae-sequence 1 2 imaginary-unit 𝑦 0 1 2 imaginary-unit 𝑦 0 {\displaystyle{\displaystyle\Re(\tfrac{1}{2}+\mathrm{i}y)>0,\Re(\tfrac{1}{2}-% \mathrm{i}y)>0}} 
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 [Tested: 6] Successful [Tested: 6]
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)}}}
\left|\EulerGamma@{\tfrac{1}{2}+\iunit y}\right|^{2} = \frac{\pi}{\cosh@{\pi y}}		 ℜ ⁡ ( 1 2 + i ⁢ y ) > 0 , ℜ ⁡ ( 1 2 - i ⁢ y ) > 0 formulae-sequence 1 2 imaginary-unit 𝑦 0 1 2 imaginary-unit 𝑦 0 {\displaystyle{\displaystyle\Re(\tfrac{1}{2}+\mathrm{i}y)>0,\Re(\tfrac{1}{2}-% \mathrm{i}y)>0}} 
(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 [Tested: 6] Successful [Tested: 6]
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)}}}
\EulerGamma@{\tfrac{1}{4}+\iunit y}\EulerGamma@{\tfrac{3}{4}-\iunit y} = \frac{\pi\sqrt{2}}{\cosh@{\pi y}+\iunit\sinh@{\pi y}}		 ℜ ⁡ ( 1 4 + i ⁢ y ) > 0 , ℜ ⁡ ( 3 4 - i ⁢ y ) > 0 formulae-sequence 1 4 imaginary-unit 𝑦 0 3 4 imaginary-unit 𝑦 0 {\displaystyle{\displaystyle\Re(\tfrac{1}{4}+\mathrm{i}y)>0,\Re(\tfrac{3}{4}-% \mathrm{i}y)>0}} 
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]]
Successful Successful - Successful [Tested: 6]
5.4.E6 Γ ⁡ ( 1 2 ) = π 1 / 2 Euler-Gamma 1 2 superscript 𝜋 1 2 {\displaystyle{\displaystyle\Gamma\left(\tfrac{1}{2}\right)=\pi^{1/2}\\ }}
\EulerGamma@{\tfrac{1}{2}} = \pi^{1/2}\\		 

GAMMA((1)/(2)) = (Pi)^(1/2)

Gamma[Divide[1,2]] == (Pi)^(1/2)
Successful Successful - Successful [Tested: 1]
5.4.E6 π 1 / 2 = 1.77245 38509 05516 02729 ⁢ … superscript 𝜋 1 2 1.77245 38509 05516 02729 … {\displaystyle{\displaystyle\pi^{1/2}\\ =1.77245\;38509\;05516\;02729\;\dots}}
\pi^{1/2}\\ = 1.77245\;38509\;05516\;02729\;\dots		 

(Pi)^(1/2) = 1.77245385090551602729

(Pi)^(1/2) == 1.77245385090551602729
Successful Successful - Successful [Tested: 1]
5.4.E7 Γ ⁡ ( 1 3 ) = 2.67893 85347 07747 63365 ⁢ … Euler-Gamma 1 3 2.67893 85347 07747 63365 … {\displaystyle{\displaystyle\Gamma\left(\tfrac{1}{3}\right)=2.67893\;85347\;07% 747\;63365\;\dots}}
\EulerGamma@{\tfrac{1}{3}} = 2.67893\;85347\;07747\;63365\;\dots		 

GAMMA((1)/(3)) = 2.67893853470774763365

Gamma[Divide[1,3]] == 2.67893853470774763365
Failure Successful Successful [Tested: 0] Successful [Tested: 1]
5.4.E8 Γ ⁡ ( 2 3 ) = 1.35411 79394 26400 41694 ⁢ … Euler-Gamma 2 3 1.35411 79394 26400 41694 … {\displaystyle{\displaystyle\Gamma\left(\tfrac{2}{3}\right)=1.35411\;79394\;26% 400\;41694\;\dots}}
\EulerGamma@{\tfrac{2}{3}} = 1.35411\;79394\;26400\;41694\;\dots		 

GAMMA((2)/(3)) = 1.35411793942640041694

Gamma[Divide[2,3]] == 1.35411793942640041694
Successful Successful - Successful [Tested: 1]
5.4.E9 Γ ⁡ ( 1 4 ) = 3.62560 99082 21908 31193 ⁢ … Euler-Gamma 1 4 3.62560 99082 21908 31193 … {\displaystyle{\displaystyle\Gamma\left(\tfrac{1}{4}\right)=3.62560\;99082\;21% 908\;31193\;\dots}}
\EulerGamma@{\tfrac{1}{4}} = 3.62560\;99082\;21908\;31193\;\dots		 

GAMMA((1)/(4)) = 3.62560990822190831193

Gamma[Divide[1,4]] == 3.62560990822190831193
Failure Successful Successful [Tested: 0] Successful [Tested: 1]
5.4.E10 Γ ⁡ ( 3 4 ) = 1.22541 67024 65177 64512 ⁢ … Euler-Gamma 3 4 1.22541 67024 65177 64512 … {\displaystyle{\displaystyle\Gamma\left(\tfrac{3}{4}\right)=1.22541\;67024\;65% 177\;64512\;\dots}}
\EulerGamma@{\tfrac{3}{4}} = 1.22541\;67024\;65177\;64512\;\dots		 

GAMMA((3)/(4)) = 1.22541670246517764512

Gamma[Divide[3,4]] == 1.22541670246517764512
Successful Successful - Successful [Tested: 1]
5.4.E11 Γ ′ ⁡ ( 1 ) = - γ diffop Euler-Gamma 1 1 {\displaystyle{\displaystyle\Gamma'\left(1\right)=-\gamma}}
\EulerGamma'@{1} = -\EulerConstant		 

subs( temp=1, diff( GAMMA(temp), temp$(1) ) ) = - gamma

(D[Gamma[temp], {temp, 1}]/.temp-> 1) == - EulerGamma
Successful Successful - Successful [Tested: 1]
5.4#Ex3 ψ ⁡ ( 1 ) = - γ digamma 1 {\displaystyle{\displaystyle\psi\left(1\right)=-\gamma}}
\digamma@{1} = -\EulerConstant		 

Psi(1) = - gamma

PolyGamma[1] == - EulerGamma
Successful Successful - Successful [Tested: 1]
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}}}
\digamma'@{1} = \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 - Successful [Tested: 1]
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}}
\digamma@{\tfrac{1}{2}} = -\EulerConstant-2\ln@@{2}		 

Psi((1)/(2)) = - gamma - 2*ln(2)

PolyGamma[Divide[1,2]] == - EulerGamma - 2*Log[2]
Successful Successful - Successful [Tested: 1]
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}}}
\digamma'@{\tfrac{1}{2}} = \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 - Successful [Tested: 1]
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}}
\digamma@{n+1} = \sum_{k=1}^{n}\frac{1}{k}-\EulerConstant		 

Psi(n + 1) = sum((1)/(k), k = 1..n)- gamma

PolyGamma[n + 1] == Sum[Divide[1,k], {k, 1, n}, GenerateConditions->None]- EulerGamma
Successful Successful - Successful [Tested: 3]
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)}}
\imagpart@@{\digamma@{iy}} = \frac{1}{2y}+\frac{\pi}{2}\coth@{\pi y}		 

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 [Tested: 6] Successful [Tested: 6]
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)}}
\imagpart@@{\digamma@{\tfrac{1}{2}+iy}} = \frac{\pi}{2}\tanh@{\pi y}		 

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 [Tested: 6] Successful [Tested: 6]
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)}}
\imagpart@@{\digamma@{1+iy}} = -\frac{1}{2y}+\frac{\pi}{2}\coth@{\pi y}		 

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 [Tested: 6] Successful [Tested: 6]
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)}}
\digamma@{\frac{p}{q}} = -\EulerConstant-\ln@@{q}-\frac{\pi}{2}\cot@{\frac{\pi p}{q}}+\frac{1}{2}\sum_{k=1}^{q-1}\cos@{\frac{2\pi kp}{q}}\ln@{2-2\cos@{\frac{2\pi k}{q}}}		 

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}, GenerateConditions->None]
Failure Failure Skipped - Because timed out Skipped - Because timed out
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