8.5

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
8.5.E1	${\displaystyle{\displaystyle\gamma\left(a,z\right)=a^{-1}z^{a}e^{-z}M\left(1,1+a,z\right)}}$ \incgamma@{a}{z} = a^{-1}z^{a}e^{-z}\KummerconfhyperM@{1}{1+a}{z}	${\displaystyle{\displaystyle\Re a>0}}$	GAMMA(a)-GAMMA(a, z) = (a)^(- 1)* (z)^(a)* exp(- z)*KummerM(1, 1 + a, z)	Gamma[a, 0, z] == (a)^(- 1)* (z)^(a)* Exp[- z]*Hypergeometric1F1[1, 1 + a, z]	Successful	Successful	-	Successful [Tested: 7]
8.5.E1	${\displaystyle{\displaystyle a^{-1}z^{a}e^{-z}M\left(1,1+a,z\right)=a^{-1}z^{a}M\left(a,1+a,-z\right)}}$ a^{-1}z^{a}e^{-z}\KummerconfhyperM@{1}{1+a}{z} = a^{-1}z^{a}\KummerconfhyperM@{a}{1+a}{-z}	${\displaystyle{\displaystyle\Re a>0}}$	(a)^(- 1)* (z)^(a)* exp(- z)*KummerM(1, 1 + a, z) = (a)^(- 1)* (z)^(a)* KummerM(a, 1 + a, - z)	(a)^(- 1)* (z)^(a)* Exp[- z]*Hypergeometric1F1[1, 1 + a, z] == (a)^(- 1)* (z)^(a)* Hypergeometric1F1[a, 1 + a, - z]	Successful	Successful	-	Successful [Tested: 7]
8.5.E2	${\displaystyle{\displaystyle\gamma^{*}\left(a,z\right)=e^{-z}{\mathbf{M}}\left(1,1+a,z\right)}}$ \scincgamma@{a}{z} = e^{-z}\OlverconfhyperM@{1}{1+a}{z}	${\displaystyle{\displaystyle\Re a>0}}$	(z)^(-(a))*(GAMMA(a)-GAMMA(a, z))/GAMMA(a) = exp(- z)*KummerM(1, 1 + a, z)/GAMMA(1 + a)	Error	Successful	Missing Macro Error	-	-
8.5.E2	${\displaystyle{\displaystyle e^{-z}{\mathbf{M}}\left(1,1+a,z\right)={\mathbf{M}}\left(a,1+a,-z\right)}}$ e^{-z}\OlverconfhyperM@{1}{1+a}{z} = \OlverconfhyperM@{a}{1+a}{-z}	${\displaystyle{\displaystyle\Re a>0}}$	exp(- z)*KummerM(1, 1 + a, z)/GAMMA(1 + a) = KummerM(a, 1 + a, - z)/GAMMA(1 + a)	Exp[- z]*Hypergeometric1F1Regularized[1, 1 + a, z] == Hypergeometric1F1Regularized[a, 1 + a, - z]	Successful	Successful	-	Failed [7 / 42] Result: Indeterminate Test Values: {Rule[a, -2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Indeterminate Test Values: {Rule[a, -2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
8.5.E3	${\displaystyle{\displaystyle\Gamma\left(a,z\right)=e^{-z}U\left(1-a,1-a,z\right)}}$ \incGamma@{a}{z} = e^{-z}\KummerconfhyperU@{1-a}{1-a}{z}		GAMMA(a, z) = exp(- z)*KummerU(1 - a, 1 - a, z)	Gamma[a, z] == Exp[- z]*HypergeometricU[1 - a, 1 - a, z]	Successful	Successful	-	Successful [Tested: 42]
8.5.E3	${\displaystyle{\displaystyle e^{-z}U\left(1-a,1-a,z\right)=z^{a}e^{-z}U\left(1,1+a,z\right)}}$ e^{-z}\KummerconfhyperU@{1-a}{1-a}{z} = z^{a}e^{-z}\KummerconfhyperU@{1}{1+a}{z}		exp(- z)*KummerU(1 - a, 1 - a, z) = (z)^(a)* exp(- z)*KummerU(1, 1 + a, z)	Exp[- z]*HypergeometricU[1 - a, 1 - a, z] == (z)^(a)* Exp[- z]*HypergeometricU[1, 1 + a, z]	Successful	Successful	-	Successful [Tested: 42]
8.5.E4	${\displaystyle{\displaystyle\gamma\left(a,z\right)=a^{-1}z^{\frac{1}{2}a-\frac{1}{2}}e^{-\frac{1}{2}z}M_{\frac{1}{2}a-\frac{1}{2},\frac{1}{2}a}\left(z\right)}}$ \incgamma@{a}{z} = a^{-1}z^{\frac{1}{2}a-\frac{1}{2}}e^{-\frac{1}{2}z}\WhittakerconfhyperM{\frac{1}{2}a-\frac{1}{2}}{\frac{1}{2}a}@{z}	${\displaystyle{\displaystyle\Re a>0}}$	GAMMA(a)-GAMMA(a, z) = (a)^(- 1)* (z)^((1)/(2)*a -(1)/(2))* exp(-(1)/(2)*z)*WhittakerM((1)/(2)*a -(1)/(2), (1)/(2)*a, z)	Gamma[a, 0, z] == (a)^(- 1)* (z)^(Divide[1,2]*a -Divide[1,2])* Exp[-Divide[1,2]*z]*WhittakerM[Divide[1,2]*a -Divide[1,2], Divide[1,2]*a, z]	Successful	Successful	-	Successful [Tested: 21]
8.5.E5	${\displaystyle{\displaystyle\Gamma\left(a,z\right)=e^{-\frac{1}{2}z}z^{\frac{1}{2}a-\frac{1}{2}}W_{\frac{1}{2}a-\frac{1}{2},\frac{1}{2}a}\left(z\right)}}$ \incGamma@{a}{z} = e^{-\frac{1}{2}z}z^{\frac{1}{2}a-\frac{1}{2}}\WhittakerconfhyperW{\frac{1}{2}a-\frac{1}{2}}{\frac{1}{2}a}@{z}		GAMMA(a, z) = exp(-(1)/(2)*z)*(z)^((1)/(2)*a -(1)/(2))* WhittakerW((1)/(2)*a -(1)/(2), (1)/(2)*a, z)	Gamma[a, z] == Exp[-Divide[1,2]*z]*(z)^(Divide[1,2]*a -Divide[1,2])* WhittakerW[Divide[1,2]*a -Divide[1,2], Divide[1,2]*a, z]	Successful	Successful	-	Successful [Tested: 42]