8.4

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
8.4.E1	${\displaystyle{\displaystyle\gamma\left(\tfrac{1}{2},z^{2}\right)=2\int_{0}^{z}e^{-t^{2}}\mathrm{d}t}}$ \incgamma@{\tfrac{1}{2}}{z^{2}} = 2\int_{0}^{z}e^{-t^{2}}\diff{t}		GAMMA((1)/(2))-GAMMA((1)/(2), (z)^(2)) = 2*int(exp(- (t)^(2)), t = 0..z)	Gamma[Divide[1,2], 0, (z)^(2)] == 2*Integrate[Exp[- (t)^(2)], {t, 0, z}, GenerateConditions->None]	Failure	Failure	Failed [2 / 7] Result: 3.465949776-3.038201708*I Test Values: {z = -1/2+1/2*I*3^(1/2)} Result: 3.197911286+.8974462698*I Test Values: {z = -1/2*3^(1/2)-1/2*I}	Failed [2 / 7] Result: Complex[3.4659497742269214, -3.038201707267986] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} Result: Complex[3.197911285535813, 0.8974462701863266] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}
8.4.E1	${\displaystyle{\displaystyle 2\int_{0}^{z}e^{-t^{2}}\mathrm{d}t=\sqrt{\pi}\operatorname{erf}\left(z\right)}}$ 2\int_{0}^{z}e^{-t^{2}}\diff{t} = \sqrt{\pi}\erf@{z}		2*int(exp(- (t)^(2)), t = 0..z) = sqrt(Pi)*erf(z)	2*Integrate[Exp[- (t)^(2)], {t, 0, z}, GenerateConditions->None] == Sqrt[Pi]*Erf[z]	Successful	Successful	Skip - symbolical successful subtest	Successful [Tested: 7]
8.4.E2	${\displaystyle{\displaystyle\gamma^{*}\left(a,0\right)=\frac{1}{\Gamma\left(a+1\right)}}}$ \scincgamma@{a}{0} = \frac{1}{\EulerGamma@{a+1}}	${\displaystyle{\displaystyle\Re(a+1)>0,\Re a>0}}$	(0)^(-(a))*(GAMMA(a)-GAMMA(a, 0))/GAMMA(a) = (1)/(GAMMA(a + 1))	Error	Failure	Missing Macro Error	Failed [3 / 3] Result: -.7522527782 Test Values: {a = 1.5} Result: -1.128379167 Test Values: {a = .5} ... skip entries to safe data	-
8.4.E3	${\displaystyle{\displaystyle\gamma^{*}\left(\tfrac{1}{2},-z^{2}\right)=\frac{2e^{z^{2}}}{z\sqrt{\pi}}F\left(z\right)}}$ \scincgamma@{\tfrac{1}{2}}{-z^{2}} = \frac{2e^{z^{2}}}{z\sqrt{\pi}}\DawsonsintF@{z}		(- (z)^(2))^(-((1)/(2)))*(GAMMA((1)/(2))-GAMMA((1)/(2), - (z)^(2)))/GAMMA((1)/(2)) = (2*exp((z)^(2)))/(z*sqrt(Pi))*dawson(z)	Error	Successful	Missing Macro Error	-	-
8.4.E4	${\displaystyle{\displaystyle\Gamma\left(0,z\right)=\int_{z}^{\infty}t^{-1}e^{-t}\mathrm{d}t}}$ \incGamma@{0}{z} = \int_{z}^{\infty}t^{-1}e^{-t}\diff{t}		GAMMA(0, z) = int((t)^(- 1)* exp(- t), t = z..infinity)	Gamma[0, z] == Integrate[(t)^(- 1)* Exp[- t], {t, z, Infinity}, GenerateConditions->None]	Successful	Successful	Skip - symbolical successful subtest	Successful [Tested: 7]
8.4.E4	${\displaystyle{\displaystyle\int_{z}^{\infty}t^{-1}e^{-t}\mathrm{d}t=E_{1}\left(z\right)}}$ \int_{z}^{\infty}t^{-1}e^{-t}\diff{t} = \expintE@{z}		int((t)^(- 1)* exp(- t), t = z..infinity) = Ei(z)	Integrate[(t)^(- 1)* Exp[- t], {t, z, Infinity}, GenerateConditions->None] == ExpIntegralE[1, z]	Failure	Failure	Failed [7 / 7] Result: -1.393548628-1.498247032*I Test Values: {z = 1/2*3^(1/2)+1/2*I} Result: -.8944744989-3.773814377*I Test Values: {z = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Successful [Tested: 7]
8.4.E5	${\displaystyle{\displaystyle\Gamma\left(1,z\right)=e^{-z}}}$ \incGamma@{1}{z} = e^{-z}		GAMMA(1, z) = exp(- z)	Gamma[1, z] == Exp[- z]	Successful	Successful	-	Successful [Tested: 7]
8.4.E6	${\displaystyle{\displaystyle\Gamma\left(\tfrac{1}{2},z^{2}\right)=2\int_{z}^{\infty}e^{-t^{2}}\mathrm{d}t}}$ \incGamma@{\tfrac{1}{2}}{z^{2}} = 2\int_{z}^{\infty}e^{-t^{2}}\diff{t}		GAMMA((1)/(2), (z)^(2)) = 2*int(exp(- (t)^(2)), t = z..infinity)	Gamma[Divide[1,2], (z)^(2)] == 2*Integrate[Exp[- (t)^(2)], {t, z, Infinity}, GenerateConditions->None]	Failure	Failure	Failed [2 / 7] Result: -3.465949776+3.038201708*I Test Values: {z = -1/2+1/2*I*3^(1/2)} Result: -3.197911286-.8974462698*I Test Values: {z = -1/2*3^(1/2)-1/2*I}	Failed [2 / 7] Result: Complex[-3.4659497742269214, 3.038201707267986] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} Result: Complex[-3.1979112855358127, -0.8974462701863266] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}
8.4.E6	${\displaystyle{\displaystyle 2\int_{z}^{\infty}e^{-t^{2}}\mathrm{d}t=\sqrt{\pi}\operatorname{erfc}\left(z\right)}}$ 2\int_{z}^{\infty}e^{-t^{2}}\diff{t} = \sqrt{\pi}\erfc@{z}		2*int(exp(- (t)^(2)), t = z..infinity) = sqrt(Pi)*erfc(z)	2*Integrate[Exp[- (t)^(2)], {t, z, Infinity}, GenerateConditions->None] == Sqrt[Pi]*Erfc[z]	Successful	Successful	Skip - symbolical successful subtest	Successful [Tested: 7]
8.4.E7	${\displaystyle{\displaystyle\gamma\left(n+1,z\right)=n!(1-e^{-z}e_{n}(z))}}$ \incgamma@{n+1}{z} = n!(1-e^{-z}e_{n}(z))	${\displaystyle{\displaystyle\Re(n+1)>0}}$	GAMMA(n + 1)-GAMMA(n + 1, z) = factorial(n)*(1 - exp(- z)*exp(1)[n]*(z))	Gamma[n + 1, 0, z] == (n)!*(1 - Exp[- z]*Subscript[E, n]*(z))	Failure	Failure	Error	Failed [21 / 21] Result: Plus[Complex[-0.7896317094254578, 0.19173078621885742], Times[Complex[0.42050290937849244, 0.009925196319850484], Subscript[2.718281828459045, 1]]] Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Plus[Complex[0.06153297742196945, 0.16461464559793018], Times[-2.0, Plus[1.0, Times[Complex[-0.42050290937849244, -0.009925196319850484], Subscript[2.718281828459045, 2]]]]] Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
8.4.E8	${\displaystyle{\displaystyle\Gamma\left(n+1,z\right)=n!e^{-z}e_{n}(z)}}$ \incGamma@{n+1}{z} = n!e^{-z}e_{n}(z)		GAMMA(n + 1, z) = factorial(n)*exp(- z)*exp(1)[n]*(z)	Gamma[n + 1, z] == (n)!*Exp[- z]*Subscript[E, n]*(z)	Failure	Failure	Error	Failed [21 / 21] Result: Plus[Complex[0.7896317094254578, -0.19173078621885742], Times[Complex[-0.42050290937849244, -0.009925196319850484], Subscript[2.718281828459045, 1]]] Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Plus[Complex[1.9384670225780305, -0.16461464559793018], Times[Complex[-0.8410058187569849, -0.019850392639700967], Subscript[2.718281828459045, 2]]] Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
8.4.E9	${\displaystyle{\displaystyle P\left(n+1,z\right)=1-e^{-z}e_{n}(z)}}$ \normincGammaP@{n+1}{z} = 1-e^{-z}e_{n}(z)		(GAMMA(n + 1)-GAMMA(n + 1, z))/GAMMA(n + 1) = 1 - exp(- z)*exp(1)[n]*(z)	GammaRegularized[n + 1, 0, z] == 1 - Exp[- z]*Subscript[E, n]*(z)	Failure	Failure	Error	Failed [21 / 21] Result: Plus[Complex[-0.7896317094254579, 0.1917307862188573], Times[Complex[0.42050290937849244, 0.009925196319850484], Subscript[2.718281828459045, 1]]] Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Plus[Complex[-0.9692335112890152, 0.08230732279896512], Times[Complex[0.42050290937849244, 0.009925196319850484], Subscript[2.718281828459045, 2]]] Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
8.4.E10	${\displaystyle{\displaystyle Q\left(n+1,z\right)=e^{-z}e_{n}(z)}}$ \normincGammaQ@{n+1}{z} = e^{-z}e_{n}(z)	${\displaystyle{\displaystyle\Re(n+1)>0}}$	GAMMA(n + 1, z)/GAMMA(n + 1) = exp(- z)*exp(1)[n]*(z)	GammaRegularized[n + 1, z] == Exp[- z]*Subscript[E, n]*(z)	Failure	Failure	Error	Failed [21 / 21] Result: Plus[Complex[0.7896317094254579, -0.1917307862188573], Times[Complex[-0.42050290937849244, -0.009925196319850484], Subscript[2.718281828459045, 1]]] Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Plus[Complex[0.9692335112890152, -0.08230732279896512], Times[Complex[-0.42050290937849244, -0.009925196319850484], Subscript[2.718281828459045, 2]]] Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
8.4.E12	${\displaystyle{\displaystyle\gamma^{*}\left(-n,z\right)=z^{n}}}$ \scincgamma@{-n}{z} = z^{n}	${\displaystyle{\displaystyle\Re(-n)>0}}$	(z)^(-(- n))*(GAMMA(- n)-GAMMA(- n, z))/GAMMA(- n) = (z)^(n)	Error	Failure	Missing Macro Error	Error	-
8.4.E13	${\displaystyle{\displaystyle\Gamma\left(1-n,z\right)=z^{1-n}E_{n}\left(z\right)}}$ \incGamma@{1-n}{z} = z^{1-n}\genexpintE{n}@{z}		GAMMA(1 - n, z) = (z)^(1 - n)* Ei(n, z)	Gamma[1 - n, z] == (z)^(1 - n)* ExpIntegralE[n, z]	Successful	Successful	-	Successful [Tested: 21]
8.4.E14	${\displaystyle{\displaystyle Q\left(n+\tfrac{1}{2},z^{2}\right)=\operatorname{erfc}\left(z\right)+\frac{e^{-z^{2}}}{\sqrt{\pi}}\sum_{k=1}^{n}\frac{z^{2k-1}}{{\left(\tfrac{1}{2}\right)_{k}}}}}$ \normincGammaQ@{n+\tfrac{1}{2}}{z^{2}} = \erfc@{z}+\frac{e^{-z^{2}}}{\sqrt{\pi}}\sum_{k=1}^{n}\frac{z^{2k-1}}{\Pochhammersym{\tfrac{1}{2}}{k}}	${\displaystyle{\displaystyle\Re(n+\tfrac{1}{2})>0}}$	GAMMA(n +(1)/(2), (z)^(2))/GAMMA(n +(1)/(2)) = erfc(z)+(exp(- (z)^(2)))/(sqrt(Pi))*sum(((z)^(2*k - 1))/(pochhammer((1)/(2), k)), k = 1..n)	GammaRegularized[n +Divide[1,2], (z)^(2)] == Erfc[z]+Divide[Exp[- (z)^(2)],Sqrt[Pi]]*Sum[Divide[(z)^(2*k - 1),Pochhammer[Divide[1,2], k]], {k, 1, n}, GenerateConditions->None]	Failure	Failure	Failed [6 / 21] Result: 1.704415567+1.043704337*I Test Values: {z = -1/2+1/2*I*3^(1/2), n = 1} Result: .97393781e-1-.8458491548*I Test Values: {z = -1/2+1/2*I*3^(1/2), n = 2} ... skip entries to safe data	Failed [6 / 21] Result: Complex[1.7044155650581054, 1.0437043365740406] Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} Result: Complex[0.09739377924871273, -0.8458491528064774] Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
8.4.E15	${\displaystyle{\displaystyle\Gamma\left(-n,z\right)=\frac{(-1)^{n}}{n!}\left(E_{1}\left(z\right)-e^{-z}\sum_{k=0}^{n-1}\frac{(-1)^{k}k!}{z^{k+1}}\right)}}$ \incGamma@{-n}{z} = \frac{(-1)^{n}}{n!}\left(\expintE@{z}-e^{-z}\sum_{k=0}^{n-1}\frac{(-1)^{k}k!}{z^{k+1}}\right)		GAMMA(- n, z) = ((- 1)^(n))/(factorial(n))*(Ei(z)- exp(- z)*sum(((- 1)^(k)* factorial(k))/((z)^(k + 1)), k = 0..n - 1))	Gamma[- n, z] == Divide[(- 1)^(n),(n)!]*(ExpIntegralE[1, z]- Exp[- z]*Sum[Divide[(- 1)^(k)* (k)!,(z)^(k + 1)], {k, 0, n - 1}, GenerateConditions->None])	Failure	Failure	Manual Skip!	Successful [Tested: 21]