Incomplete Gamma and Related Functions - 8.4 Special Values

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
8.4.E1 γ ⁡ ( 1 2 , z 2 ) = 2 ⁢ ∫ 0 z e - t 2 ⁢ d t incomplete-gamma 1 2 superscript 𝑧 2 2 superscript subscript 0 𝑧 superscript 𝑒 superscript 𝑡 2 𝑡 {\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 2 ⁢ ∫ 0 z e - t 2 ⁢ d t = π ⁢ erf ⁡ ( z ) 2 superscript subscript 0 𝑧 superscript 𝑒 superscript 𝑡 2 𝑡 𝜋 error-function 𝑧 {\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 γ * ⁡ ( a , 0 ) = 1 Γ ⁡ ( a + 1 ) incomplete-gamma-star 𝑎 0 1 Euler-Gamma 𝑎 1 {\displaystyle{\displaystyle\gamma^{*}\left(a,0\right)=\frac{1}{\Gamma\left(a+% 1\right)}}}
\scincgamma@{a}{0} = \frac{1}{\EulerGamma@{a+1}}		 ℜ ⁡ ( a + 1 ) > 0 , ℜ ⁡ a > 0 formulae-sequence 𝑎 1 0 𝑎 0 {\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 γ * ⁡ ( 1 2 , - z 2 ) = 2 ⁢ e z 2 z ⁢ π ⁢ F ⁡ ( z ) incomplete-gamma-star 1 2 superscript 𝑧 2 2 superscript 𝑒 superscript 𝑧 2 𝑧 𝜋 Dawsons-integral 𝑧 {\displaystyle{\displaystyle\gamma^{*}\left(\tfrac{1}{2},-z^{2}\right)=\frac{2% e^{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 Γ ⁡ ( 0 , z ) = ∫ z ∞ t - 1 ⁢ e - t ⁢ d t incomplete-Gamma 0 𝑧 superscript subscript 𝑧 superscript 𝑡 1 superscript 𝑒 𝑡 𝑡 {\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 ∫ z ∞ t - 1 ⁢ e - t ⁢ d t = E 1 ⁡ ( z ) superscript subscript 𝑧 superscript 𝑡 1 superscript 𝑒 𝑡 𝑡 exponential-integral 𝑧 {\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 Γ ⁡ ( 1 , z ) = e - z incomplete-Gamma 1 𝑧 superscript 𝑒 𝑧 {\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 Γ ⁡ ( 1 2 , z 2 ) = 2 ⁢ ∫ z ∞ e - t 2 ⁢ d t incomplete-Gamma 1 2 superscript 𝑧 2 2 superscript subscript 𝑧 superscript 𝑒 superscript 𝑡 2 𝑡 {\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 2 ⁢ ∫ z ∞ e - t 2 ⁢ d t = π ⁢ erfc ⁡ ( z ) 2 superscript subscript 𝑧 superscript 𝑒 superscript 𝑡 2 𝑡 𝜋 complementary-error-function 𝑧 {\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 γ ⁡ ( n + 1 , z ) = n ! ⁢ ( 1 - e - z ⁢ e n ⁢ ( z ) ) incomplete-gamma 𝑛 1 𝑧 𝑛 1 superscript 𝑒 𝑧 subscript 𝑒 𝑛 𝑧 {\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))		 ℜ ⁡ ( n + 1 ) > 0 𝑛 1 0 {\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 Γ ⁡ ( n + 1 , z ) = n ! ⁢ e - z ⁢ e n ⁢ ( z ) incomplete-Gamma 𝑛 1 𝑧 𝑛 superscript 𝑒 𝑧 subscript 𝑒 𝑛 𝑧 {\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 P ⁡ ( n + 1 , z ) = 1 - e - z ⁢ e n ⁢ ( z ) incomplete-gamma-P 𝑛 1 𝑧 1 superscript 𝑒 𝑧 subscript 𝑒 𝑛 𝑧 {\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 Q ⁡ ( n + 1 , z ) = e - z ⁢ e n ⁢ ( z ) incomplete-gamma-Q 𝑛 1 𝑧 superscript 𝑒 𝑧 subscript 𝑒 𝑛 𝑧 {\displaystyle{\displaystyle Q\left(n+1,z\right)=e^{-z}e_{n}(z)}}
\normincGammaQ@{n+1}{z} = e^{-z}e_{n}(z)		 ℜ ⁡ ( n + 1 ) > 0 𝑛 1 0 {\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 γ * ⁡ ( - n , z ) = z n incomplete-gamma-star 𝑛 𝑧 superscript 𝑧 𝑛 {\displaystyle{\displaystyle\gamma^{*}\left(-n,z\right)=z^{n}}}
\scincgamma@{-n}{z} = z^{n}		 ℜ ⁡ ( - n ) > 0 𝑛 0 {\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 Γ ⁡ ( 1 - n , z ) = z 1 - n ⁢ E n ⁡ ( z ) incomplete-Gamma 1 𝑛 𝑧 superscript 𝑧 1 𝑛 exponential-integral-En 𝑛 𝑧 {\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 Q ⁡ ( n + 1 2 , z 2 ) = erfc ⁡ ( z ) + e - z 2 π ⁢ ∑ k = 1 n z 2 ⁢ k - 1 ( 1 2 ) k incomplete-gamma-Q 𝑛 1 2 superscript 𝑧 2 complementary-error-function 𝑧 superscript 𝑒 superscript 𝑧 2 𝜋 superscript subscript 𝑘 1 𝑛 superscript 𝑧 2 𝑘 1 Pochhammer 1 2 𝑘 {\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}}		 ℜ ⁡ ( n + 1 2 ) > 0 𝑛 1 2 0 {\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 Γ ⁡ ( - n , z ) = ( - 1 ) n n ! ⁢ ( E 1 ⁡ ( z ) - e - z ⁢ ∑ k = 0 n - 1 ( - 1 ) k ⁢ k ! z k + 1 ) incomplete-Gamma 𝑛 𝑧 superscript 1 𝑛 𝑛 exponential-integral 𝑧 superscript 𝑒 𝑧 superscript subscript 𝑘 0 𝑛 1 superscript 1 𝑘 𝑘 superscript 𝑧 𝑘 1 {\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]
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