Gamma Function - 5.5 Functional Relations

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
5.5.E1	${\displaystyle{\displaystyle\Gamma\left(z+1\right)=z\Gamma\left(z\right)}}$ \EulerGamma@{z+1} = z\EulerGamma@{z}	${\displaystyle{\displaystyle\Re(z+1)>0,\Re z>0}}$	GAMMA(z + 1) = z*GAMMA(z)	Gamma[z + 1] == z*Gamma[z]	Successful	Successful	-	Successful [Tested: 5]
5.5.E2	${\displaystyle{\displaystyle\psi\left(z+1\right)=\psi\left(z\right)+\frac{1}{z}}}$ \digamma@{z+1} = \digamma@{z}+\frac{1}{z}		Psi(z + 1) = Psi(z)+(1)/(z)	PolyGamma[z + 1] == PolyGamma[z]+Divide[1,z]	Successful	Successful	-	Successful [Tested: 7]
5.5.E3	${\displaystyle{\displaystyle\Gamma\left(z\right)\Gamma\left(1-z\right)=\pi/\sin\left(\pi z\right)}}$ \EulerGamma@{z}\EulerGamma@{1-z} = \pi/\sin@{\pi z}	${\displaystyle{\displaystyle\Re z>0,\Re(1-z)>0}}$	GAMMA(z)*GAMMA(1 - z) = Pi/sin(Pi*z)	Gamma[z]*Gamma[1 - z] == Pi/Sin[Pi*z]	Successful	Successful	-	Successful [Tested: 1]
5.5.E4	${\displaystyle{\displaystyle\psi\left(z\right)-\psi\left(1-z\right)=-\pi/\tan\left(\pi z\right)}}$ \digamma@{z}-\digamma@{1-z} = -\pi/\tan@{\pi z}		Psi(z)- Psi(1 - z) = - Pi/tan(Pi*z)	PolyGamma[z]- PolyGamma[1 - z] == - Pi/Tan[Pi*z]	Successful	Successful	-	Successful [Tested: 1]
5.5.E5	${\displaystyle{\displaystyle\Gamma\left(2z\right)=\pi^{-1/2}2^{2z-1}\Gamma\left(z\right)\Gamma\left(z+\tfrac{1}{2}\right)}}$ \EulerGamma@{2z} = \pi^{-1/2}2^{2z-1}\EulerGamma@{z}\EulerGamma@{z+\tfrac{1}{2}}	${\displaystyle{\displaystyle\Re(2z)>0,\Re z>0,\Re(z+\tfrac{1}{2})>0}}$	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	-	Successful [Tested: 5]
5.5.E6	${\displaystyle{\displaystyle\Gamma\left(nz\right)=(2\pi)^{(1-n)/2}n^{nz-(1/2)}\prod_{k=0}^{n-1}\Gamma\left(z+\frac{k}{n}\right)}}$ \EulerGamma@{nz} = (2\pi)^{(1-n)/2}n^{nz-(1/2)}\prod_{k=0}^{n-1}\EulerGamma@{z+\frac{k}{n}}		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}, GenerateConditions->None]	Failure	Successful	Successful [Tested: 15]	Successful [Tested: 15]
5.5.E7	${\displaystyle{\displaystyle\prod_{k=1}^{n-1}\Gamma\left(\frac{k}{n}\right)=(2\pi)^{(n-1)/2}n^{-1/2}}}$ \prod_{k=1}^{n-1}\EulerGamma@{\frac{k}{n}} = (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}, GenerateConditions->None] == (2*Pi)^((n - 1)/2)* (n)^(- 1/2)	Failure	Failure	Successful [Tested: 3]	Failed [3 / 3] Result: Indeterminate Test Values: {Rule[n, 1]} Result: Indeterminate Test Values: {Rule[n, 2]} ... skip entries to safe data
5.5.E8	${\displaystyle{\displaystyle\psi\left(2z\right)=\tfrac{1}{2}\left(\psi\left(z\right)+\psi\left(z+\tfrac{1}{2}\right)\right)+\ln 2}}$ \digamma@{2z} = \tfrac{1}{2}\left(\digamma@{z}+\digamma@{z+\tfrac{1}{2}}\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	-	Successful [Tested: 7]
5.5.E9	${\displaystyle{\displaystyle\psi\left(nz\right)=\frac{1}{n}\sum_{k=0}^{n-1}\psi\left(z+\frac{k}{n}\right)+\ln n}}$ \digamma@{nz} = \frac{1}{n}\sum_{k=0}^{n-1}\digamma@{z+\frac{k}{n}}+\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}, GenerateConditions->None]+ Log[n]	Failure	Successful	Successful [Tested: 21]	Successful [Tested: 21]