Incomplete Gamma and Related Functions - 8.22 Mathematical Applications

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
8.22.E1	${\displaystyle{\displaystyle\frac{\Gamma\left(p\right)}{2\pi}z^{1-p}E_{p}\left(z\right)=\frac{\Gamma\left(p\right)}{2\pi}\Gamma\left(1-p,z\right)}}$ \frac{\EulerGamma@{p}}{2\pi}z^{1-p}\genexpintE{p}@{z} = \frac{\EulerGamma@{p}}{2\pi}\incGamma@{1-p}{z}	${\displaystyle{\displaystyle\Re p>0,\Re(n+p)>0}}$	(GAMMA(p))/(2*Pi)*(z)^(1 - p)* Ei(p, z) = (GAMMA(p))/(2*Pi)*GAMMA(1 - p, z)	Divide[Gamma[p],2*Pi]*(z)^(1 - p)* ExpIntegralE[p, z] == Divide[Gamma[p],2*Pi]*Gamma[1 - p, z]	Successful	Successful	-	Successful [Tested: 35]
8.22.E3	${\displaystyle{\displaystyle\zeta_{x}(s)=\sum_{k=1}^{\infty}k^{-s}P\left(s,kx\right)}}$ \zeta_{x}(s) = \sum_{k=1}^{\infty}k^{-s}\normincGammaP@{s}{kx}	${\displaystyle{\displaystyle\Re s>1}}$	((1)/(GAMMA(s))*int(((t)^(s - 1))/(exp(t)- 1), t = 0..x)) = sum((k)^(- s)* (GAMMA(s)-GAMMA(s, k*x))/GAMMA(s), k = 1..infinity)	(Divide[1,Gamma[s]]*Integrate[Divide[(t)^(s - 1),Exp[t]- 1], {t, 0, x}, GenerateConditions->None]) == Sum[(k)^(- s)* GammaRegularized[s, 0, k*x], {k, 1, Infinity}, GenerateConditions->None]	Failure	Aborted	Manual Skip!	Skipped - Because timed out