Gamma Function - 5.8 Infinite Products

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
5.8.E2	${\displaystyle{\displaystyle\frac{1}{\Gamma\left(z\right)}=ze^{\gamma z}\prod_{k=1}^{\infty}\left(1+\frac{z}{k}\right)e^{-z/k}}}$ \frac{1}{\EulerGamma@{z}} = ze^{\EulerConstant z}\prod_{k=1}^{\infty}\left(1+\frac{z}{k}\right)e^{-z/k}	${\displaystyle{\displaystyle\Re z>0}}$	(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}, GenerateConditions->None]	Successful	Successful	-	Successful [Tested: 5]
5.8.E3	${\displaystyle{\displaystyle\left|\frac{\Gamma\left(x\right)}{\Gamma\left(x+\mathrm{i}y\right)}\right|^{2}=\prod_{k=0}^{\infty}\left(1+\frac{y^{2}}{(x+k)^{2}}\right)}}$ \left|\frac{\EulerGamma@{x}}{\EulerGamma@{x+\iunit y}}\right|^{2} = \prod_{k=0}^{\infty}\left(1+\frac{y^{2}}{(x+k)^{2}}\right)	${\displaystyle{\displaystyle\Re x>0,\Re(x+\mathrm{i}y)>0}}$	(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}, GenerateConditions->None]	Failure	Failure	Failed [18 / 18] Result: -6.243891895+0.*I Test Values: {x = 1.5, y = -1.5, x = 1} Result: -6.243891895+0.*I Test Values: {x = 1.5, y = 1.5, x = 1} Result: -.210463144+0.*I Test Values: {x = 1.5, y = -.5, x = 1} Result: -.210463144+0.*I Test Values: {x = 1.5, y = .5, x = 1} ... skip entries to safe data	Successful [Tested: 18]