Incomplete Gamma and Related Functions - 8.8 Recurrence Relations and Derivatives

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
8.8.E1 γ ⁡ ( a + 1 , z ) = a ⁢ γ ⁡ ( a , z ) - z a ⁢ e - z incomplete-gamma 𝑎 1 𝑧 𝑎 incomplete-gamma 𝑎 𝑧 superscript 𝑧 𝑎 superscript 𝑒 𝑧 {\displaystyle{\displaystyle\gamma\left(a+1,z\right)=a\gamma\left(a,z\right)-z% ^{a}e^{-z}}}
\incgamma@{a+1}{z} = a\incgamma@{a}{z}-z^{a}e^{-z}		 ℜ ⁡ ( a + 1 ) > 0 , ℜ ⁡ a > 0 formulae-sequence 𝑎 1 0 𝑎 0 {\displaystyle{\displaystyle\Re(a+1)>0,\Re a>0}} 
GAMMA(a + 1)-GAMMA(a + 1, z) = a*GAMMA(a)-GAMMA(a, z)- (z)^(a)* exp(- z)

Gamma[a + 1, 0, z] == a*Gamma[a, 0, z]- (z)^(a)* Exp[- z]
Failure Successful
Failed [21 / 21]
Result: -.2676693395+.995081412e-1*I
Test Values: {a = 1.5, z = 1/2*3^(1/2)+1/2*I}


Result: -.820738205+.231239721*I
Test Values: {a = 1.5, z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
 Successful [Tested: 21]
8.8.E2 Γ ⁡ ( a + 1 , z ) = a ⁢ Γ ⁡ ( a , z ) + z a ⁢ e - z incomplete-Gamma 𝑎 1 𝑧 𝑎 incomplete-Gamma 𝑎 𝑧 superscript 𝑧 𝑎 superscript 𝑒 𝑧 {\displaystyle{\displaystyle\Gamma\left(a+1,z\right)=a\Gamma\left(a,z\right)+z% ^{a}e^{-z}}}
\incGamma@{a+1}{z} = a\incGamma@{a}{z}+z^{a}e^{-z}		 

GAMMA(a + 1, z) = a*GAMMA(a, z)+ (z)^(a)* exp(- z)

Gamma[a + 1, z] == a*Gamma[a, z]+ (z)^(a)* Exp[- z]
Failure Successful Successful [Tested: 42] Successful [Tested: 42]
8.8.E3 w ⁢ ( a + 2 , z ) - ( a + 1 + z ) ⁢ w ⁢ ( a + 1 , z ) + a ⁢ z ⁢ w ⁢ ( a , z ) = 0 𝑤 𝑎 2 𝑧 𝑎 1 𝑧 𝑤 𝑎 1 𝑧 𝑎 𝑧 𝑤 𝑎 𝑧 0 {\displaystyle{\displaystyle w(a+2,z)-(a+1+z)w(a+1,z)+azw(a,z)=0}}
w(a+2,z)-(a+1+z)w(a+1,z)+azw(a,z) = 0		 

w(a + 2 , z)-(a + 1 + z)*w(a + 1 , z)+ azw(a , z) = 0
 
w[a + 2 , z]-(a + 1 + z)*w[a + 1 , z]+ azw[a , z] == 0
 Skipped - no semantic math Skipped - no semantic math - -
8.8.E4 z ⁢ γ * ⁡ ( a + 1 , z ) = γ * ⁡ ( a , z ) - e - z Γ ⁡ ( a + 1 ) 𝑧 incomplete-gamma-star 𝑎 1 𝑧 incomplete-gamma-star 𝑎 𝑧 superscript 𝑒 𝑧 Euler-Gamma 𝑎 1 {\displaystyle{\displaystyle z\gamma^{*}\left(a+1,z\right)=\gamma^{*}\left(a,z% \right)-\frac{e^{-z}}{\Gamma\left(a+1\right)}}}
z\scincgamma@{a+1}{z} = \scincgamma@{a}{z}-\frac{e^{-z}}{\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}} 
z*(z)^(-(a + 1))*(GAMMA(a + 1)-GAMMA(a + 1, z))/GAMMA(a + 1) = (z)^(-(a))*(GAMMA(a)-GAMMA(a, z))/GAMMA(a)-(exp(- z))/(GAMMA(a + 1))

Error
Failure Missing Macro Error Successful [Tested: 21] -
8.8.E5 P ⁡ ( a + 1 , z ) = P ⁡ ( a , z ) - z a ⁢ e - z Γ ⁡ ( a + 1 ) incomplete-gamma-P 𝑎 1 𝑧 incomplete-gamma-P 𝑎 𝑧 superscript 𝑧 𝑎 superscript 𝑒 𝑧 Euler-Gamma 𝑎 1 {\displaystyle{\displaystyle P\left(a+1,z\right)=P\left(a,z\right)-\frac{z^{a}% e^{-z}}{\Gamma\left(a+1\right)}}}
\normincGammaP@{a+1}{z} = \normincGammaP@{a}{z}-\frac{z^{a}e^{-z}}{\EulerGamma@{a+1}}		 ℜ ⁡ ( a + 1 ) > 0 𝑎 1 0 {\displaystyle{\displaystyle\Re(a+1)>0}} 
(GAMMA(a + 1)-GAMMA(a + 1, z))/GAMMA(a + 1) = (GAMMA(a)-GAMMA(a, z))/GAMMA(a)-((z)^(a)* exp(- z))/(GAMMA(a + 1))

GammaRegularized[a + 1, 0, z] == GammaRegularized[a, 0, z]-Divide[(z)^(a)* Exp[- z],Gamma[a + 1]]
Failure Successful Successful [Tested: 28] Successful [Tested: 28]
8.8.E6 Q ⁡ ( a + 1 , z ) = Q ⁡ ( a , z ) + z a ⁢ e - z Γ ⁡ ( a + 1 ) incomplete-gamma-Q 𝑎 1 𝑧 incomplete-gamma-Q 𝑎 𝑧 superscript 𝑧 𝑎 superscript 𝑒 𝑧 Euler-Gamma 𝑎 1 {\displaystyle{\displaystyle Q\left(a+1,z\right)=Q\left(a,z\right)+\frac{z^{a}% e^{-z}}{\Gamma\left(a+1\right)}}}
\normincGammaQ@{a+1}{z} = \normincGammaQ@{a}{z}+\frac{z^{a}e^{-z}}{\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}} 
GAMMA(a + 1, z)/GAMMA(a + 1) = GAMMA(a, z)/GAMMA(a)+((z)^(a)* exp(- z))/(GAMMA(a + 1))

GammaRegularized[a + 1, z] == GammaRegularized[a, z]+Divide[(z)^(a)* Exp[- z],Gamma[a + 1]]
Failure Successful Successful [Tested: 28] Successful [Tested: 21]
8.8.E7 γ ⁡ ( a + n , z ) = ( a ) n ⁢ γ ⁡ ( a , z ) - z a ⁢ e - z ⁢ ∑ k = 0 n - 1 Γ ⁡ ( a + n ) Γ ⁡ ( a + k + 1 ) ⁢ z k incomplete-gamma 𝑎 𝑛 𝑧 Pochhammer 𝑎 𝑛 incomplete-gamma 𝑎 𝑧 superscript 𝑧 𝑎 superscript 𝑒 𝑧 superscript subscript 𝑘 0 𝑛 1 Euler-Gamma 𝑎 𝑛 Euler-Gamma 𝑎 𝑘 1 superscript 𝑧 𝑘 {\displaystyle{\displaystyle\gamma\left(a+n,z\right)={\left(a\right)_{n}}% \gamma\left(a,z\right)-z^{a}e^{-z}\sum_{k=0}^{n-1}\frac{\Gamma\left(a+n\right)% }{\Gamma\left(a+k+1\right)}z^{k}}}
\incgamma@{a+n}{z} = \Pochhammersym{a}{n}\incgamma@{a}{z}-z^{a}e^{-z}\sum_{k=0}^{n-1}\frac{\EulerGamma@{a+n}}{\EulerGamma@{a+k+1}}z^{k}		 ℜ ⁡ ( a + n ) > 0 , ℜ ⁡ ( a + k + 1 ) > 0 , ℜ ⁡ a > 0 formulae-sequence 𝑎 𝑛 0 formulae-sequence 𝑎 𝑘 1 0 𝑎 0 {\displaystyle{\displaystyle\Re(a+n)>0,\Re(a+k+1)>0,\Re a>0}} 
GAMMA(a + n)-GAMMA(a + n, z) = pochhammer(a, n)*GAMMA(a)-GAMMA(a, z)- (z)^(a)* exp(- z)*sum((GAMMA(a + n))/(GAMMA(a + k + 1))*(z)^(k), k = 0..n - 1)

Gamma[a + n, 0, z] == Pochhammer[a, n]*Gamma[a, 0, z]- (z)^(a)* Exp[- z]*Sum[Divide[Gamma[a + n],Gamma[a + k + 1]]*(z)^(k), {k, 0, n - 1}, GenerateConditions->None]
Failure Successful
Failed [63 / 63]
Result: -.2676693391+.995081412e-1*I
Test Values: {a = 1.5, z = 1/2*3^(1/2)+1/2*I, n = 1}


Result: -1.472181365+.5472947763*I
Test Values: {a = 1.5, z = 1/2*3^(1/2)+1/2*I, n = 2}

... skip entries to safe data
 Successful [Tested: 63]
8.8.E8 γ ⁡ ( a , z ) = Γ ⁡ ( a ) Γ ⁡ ( a - n ) ⁢ γ ⁡ ( a - n , z ) - z a - 1 ⁢ e - z ⁢ ∑ k = 0 n - 1 Γ ⁡ ( a ) Γ ⁡ ( a - k ) ⁢ z - k incomplete-gamma 𝑎 𝑧 Euler-Gamma 𝑎 Euler-Gamma 𝑎 𝑛 incomplete-gamma 𝑎 𝑛 𝑧 superscript 𝑧 𝑎 1 superscript 𝑒 𝑧 superscript subscript 𝑘 0 𝑛 1 Euler-Gamma 𝑎 Euler-Gamma 𝑎 𝑘 superscript 𝑧 𝑘 {\displaystyle{\displaystyle\gamma\left(a,z\right)=\frac{\Gamma\left(a\right)}% {\Gamma\left(a-n\right)}\gamma\left(a-n,z\right)-z^{a-1}e^{-z}\sum_{k=0}^{n-1}% \frac{\Gamma\left(a\right)}{\Gamma\left(a-k\right)}z^{-k}}}
\incgamma@{a}{z} = \frac{\EulerGamma@{a}}{\EulerGamma@{a-n}}\incgamma@{a-n}{z}-z^{a-1}e^{-z}\sum_{k=0}^{n-1}\frac{\EulerGamma@{a}}{\EulerGamma@{a-k}}z^{-k}		 ℜ ⁡ a > 0 , ℜ ⁡ ( a - n ) > 0 , ℜ ⁡ ( a - k ) > 0 formulae-sequence 𝑎 0 formulae-sequence 𝑎 𝑛 0 𝑎 𝑘 0 {\displaystyle{\displaystyle\Re a>0,\Re(a-n)>0,\Re(a-k)>0}} 
GAMMA(a)-GAMMA(a, z) = (GAMMA(a))/(GAMMA(a - n))*GAMMA(a - n)-GAMMA(a - n, z)- (z)^(a - 1)* exp(- z)*sum((GAMMA(a))/(GAMMA(a - k))*(z)^(- k), k = 0..n - 1)

Gamma[a, 0, z] == Divide[Gamma[a],Gamma[a - n]]*Gamma[a - n, 0, z]- (z)^(a - 1)* Exp[- z]*Sum[Divide[Gamma[a],Gamma[a - k]]*(z)^(- k), {k, 0, n - 1}, GenerateConditions->None]
Failure Successful
Failed [7 / 14]
Result: .1265952281-.9976912441e-1*I
Test Values: {a = 1.5, z = 1/2*3^(1/2)+1/2*I, n = 1}


Result: .19739482e-1-.7595504274*I
Test Values: {a = 1.5, z = -1/2+1/2*I*3^(1/2), n = 1}

... skip entries to safe data
 Successful [Tested: 14]
8.8.E9 Γ ⁡ ( a + n , z ) = ( a ) n ⁢ Γ ⁡ ( a , z ) + z a ⁢ e - z ⁢ ∑ k = 0 n - 1 Γ ⁡ ( a + n ) Γ ⁡ ( a + k + 1 ) ⁢ z k incomplete-Gamma 𝑎 𝑛 𝑧 Pochhammer 𝑎 𝑛 incomplete-Gamma 𝑎 𝑧 superscript 𝑧 𝑎 superscript 𝑒 𝑧 superscript subscript 𝑘 0 𝑛 1 Euler-Gamma 𝑎 𝑛 Euler-Gamma 𝑎 𝑘 1 superscript 𝑧 𝑘 {\displaystyle{\displaystyle\Gamma\left(a+n,z\right)={\left(a\right)_{n}}% \Gamma\left(a,z\right)+z^{a}e^{-z}\sum_{k=0}^{n-1}\frac{\Gamma\left(a+n\right)% }{\Gamma\left(a+k+1\right)}z^{k}}}
\incGamma@{a+n}{z} = \Pochhammersym{a}{n}\incGamma@{a}{z}+z^{a}e^{-z}\sum_{k=0}^{n-1}\frac{\EulerGamma@{a+n}}{\EulerGamma@{a+k+1}}z^{k}		 ℜ ⁡ ( a + n ) > 0 , ℜ ⁡ ( a + k + 1 ) > 0 formulae-sequence 𝑎 𝑛 0 𝑎 𝑘 1 0 {\displaystyle{\displaystyle\Re(a+n)>0,\Re(a+k+1)>0}} 
GAMMA(a + n, z) = pochhammer(a, n)*GAMMA(a, z)+ (z)^(a)* exp(- z)*sum((GAMMA(a + n))/(GAMMA(a + k + 1))*(z)^(k), k = 0..n - 1)

Gamma[a + n, z] == Pochhammer[a, n]*Gamma[a, z]+ (z)^(a)* Exp[- z]*Sum[Divide[Gamma[a + n],Gamma[a + k + 1]]*(z)^(k), {k, 0, n - 1}, GenerateConditions->None]
Successful Successful -
Failed [7 / 105]
Result: Indeterminate
Test Values: {Rule[a, -2], Rule[n, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Indeterminate
Test Values: {Rule[a, -2], Rule[n, 3], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
8.8.E10 Γ ⁡ ( a , z ) = Γ ⁡ ( a ) Γ ⁡ ( a - n ) ⁢ Γ ⁡ ( a - n , z ) + z a - 1 ⁢ e - z ⁢ ∑ k = 0 n - 1 Γ ⁡ ( a ) Γ ⁡ ( a - k ) ⁢ z - k incomplete-Gamma 𝑎 𝑧 Euler-Gamma 𝑎 Euler-Gamma 𝑎 𝑛 incomplete-Gamma 𝑎 𝑛 𝑧 superscript 𝑧 𝑎 1 superscript 𝑒 𝑧 superscript subscript 𝑘 0 𝑛 1 Euler-Gamma 𝑎 Euler-Gamma 𝑎 𝑘 superscript 𝑧 𝑘 {\displaystyle{\displaystyle\Gamma\left(a,z\right)=\frac{\Gamma\left(a\right)}% {\Gamma\left(a-n\right)}\Gamma\left(a-n,z\right)+z^{a-1}e^{-z}\sum_{k=0}^{n-1}% \frac{\Gamma\left(a\right)}{\Gamma\left(a-k\right)}z^{-k}}}
\incGamma@{a}{z} = \frac{\EulerGamma@{a}}{\EulerGamma@{a-n}}\incGamma@{a-n}{z}+z^{a-1}e^{-z}\sum_{k=0}^{n-1}\frac{\EulerGamma@{a}}{\EulerGamma@{a-k}}z^{-k}		 ℜ ⁡ a > 0 , ℜ ⁡ ( a - n ) > 0 , ℜ ⁡ ( a - k ) > 0 formulae-sequence 𝑎 0 formulae-sequence 𝑎 𝑛 0 𝑎 𝑘 0 {\displaystyle{\displaystyle\Re a>0,\Re(a-n)>0,\Re(a-k)>0}} 
GAMMA(a, z) = (GAMMA(a))/(GAMMA(a - n))*GAMMA(a - n, z)+ (z)^(a - 1)* exp(- z)*sum((GAMMA(a))/(GAMMA(a - k))*(z)^(- k), k = 0..n - 1)

Gamma[a, z] == Divide[Gamma[a],Gamma[a - n]]*Gamma[a - n, z]+ (z)^(a - 1)* Exp[- z]*Sum[Divide[Gamma[a],Gamma[a - k]]*(z)^(- k), {k, 0, n - 1}, GenerateConditions->None]
Failure Successful Successful [Tested: 14] Successful [Tested: 14]
8.8.E11 P ⁡ ( a + n , z ) = P ⁡ ( a , z ) - z a ⁢ e - z ⁢ ∑ k = 0 n - 1 z k Γ ⁡ ( a + k + 1 ) incomplete-gamma-P 𝑎 𝑛 𝑧 incomplete-gamma-P 𝑎 𝑧 superscript 𝑧 𝑎 superscript 𝑒 𝑧 superscript subscript 𝑘 0 𝑛 1 superscript 𝑧 𝑘 Euler-Gamma 𝑎 𝑘 1 {\displaystyle{\displaystyle P\left(a+n,z\right)=P\left(a,z\right)-z^{a}e^{-z}% \sum_{k=0}^{n-1}\frac{z^{k}}{\Gamma\left(a+k+1\right)}}}
\normincGammaP@{a+n}{z} = \normincGammaP@{a}{z}-z^{a}e^{-z}\sum_{k=0}^{n-1}\frac{z^{k}}{\EulerGamma@{a+k+1}}		 ℜ ⁡ ( a + k + 1 ) > 0 𝑎 𝑘 1 0 {\displaystyle{\displaystyle\Re(a+k+1)>0}} 
(GAMMA(a + n)-GAMMA(a + n, z))/GAMMA(a + n) = (GAMMA(a)-GAMMA(a, z))/GAMMA(a)- (z)^(a)* exp(- z)*sum(((z)^(k))/(GAMMA(a + k + 1)), k = 0..n - 1)

GammaRegularized[a + n, 0, z] == GammaRegularized[a, 0, z]- (z)^(a)* Exp[- z]*Sum[Divide[(z)^(k),Gamma[a + k + 1]], {k, 0, n - 1}, GenerateConditions->None]
Successful Successful -
Failed [21 / 126]
Result: Indeterminate
Test Values: {Rule[a, -2], Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Indeterminate
Test Values: {Rule[a, -2], Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
8.8.E12 Q ⁡ ( a + n , z ) = Q ⁡ ( a , z ) + z a ⁢ e - z ⁢ ∑ k = 0 n - 1 z k Γ ⁡ ( a + k + 1 ) incomplete-gamma-Q 𝑎 𝑛 𝑧 incomplete-gamma-Q 𝑎 𝑧 superscript 𝑧 𝑎 superscript 𝑒 𝑧 superscript subscript 𝑘 0 𝑛 1 superscript 𝑧 𝑘 Euler-Gamma 𝑎 𝑘 1 {\displaystyle{\displaystyle Q\left(a+n,z\right)=Q\left(a,z\right)+z^{a}e^{-z}% \sum_{k=0}^{n-1}\frac{z^{k}}{\Gamma\left(a+k+1\right)}}}
\normincGammaQ@{a+n}{z} = \normincGammaQ@{a}{z}+z^{a}e^{-z}\sum_{k=0}^{n-1}\frac{z^{k}}{\EulerGamma@{a+k+1}}		 ℜ ⁡ ( a + k + 1 ) > 0 , ℜ ⁡ ( a + n ) > 0 , ℜ ⁡ a > 0 formulae-sequence 𝑎 𝑘 1 0 formulae-sequence 𝑎 𝑛 0 𝑎 0 {\displaystyle{\displaystyle\Re(a+k+1)>0,\Re(a+n)>0,\Re a>0}} 
GAMMA(a + n, z)/GAMMA(a + n) = GAMMA(a, z)/GAMMA(a)+ (z)^(a)* exp(- z)*sum(((z)^(k))/(GAMMA(a + k + 1)), k = 0..n - 1)

GammaRegularized[a + n, z] == GammaRegularized[a, z]+ (z)^(a)* Exp[- z]*Sum[Divide[(z)^(k),Gamma[a + k + 1]], {k, 0, n - 1}, GenerateConditions->None]
Successful Successful - Successful [Tested: 63]
8.8.E13 d d z ⁡ γ ⁡ ( a , z ) = - d d z ⁡ Γ ⁡ ( a , z ) derivative 𝑧 incomplete-gamma 𝑎 𝑧 derivative 𝑧 incomplete-Gamma 𝑎 𝑧 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\gamma\left(a,z% \right)=-\frac{\mathrm{d}}{\mathrm{d}z}\Gamma\left(a,z\right)}}
\deriv{}{z}\incgamma@{a}{z} = -\deriv{}{z}\incGamma@{a}{z}		 ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle\Re a>0}} 
diff(GAMMA(a)-GAMMA(a, z), z) = - diff(GAMMA(a, z), z)

D[Gamma[a, 0, z], z] == - D[Gamma[a, z], z]
Successful Successful - Successful [Tested: 21]
8.8.E13 - d d z ⁡ Γ ⁡ ( a , z ) = z a - 1 ⁢ e - z derivative 𝑧 incomplete-Gamma 𝑎 𝑧 superscript 𝑧 𝑎 1 superscript 𝑒 𝑧 {\displaystyle{\displaystyle-\frac{\mathrm{d}}{\mathrm{d}z}\Gamma\left(a,z% \right)=z^{a-1}e^{-z}}}
-\deriv{}{z}\incGamma@{a}{z} = z^{a-1}e^{-z}		 ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle\Re a>0}} 
- diff(GAMMA(a, z), z) = (z)^(a - 1)* exp(- z)

- D[Gamma[a, z], z] == (z)^(a - 1)* Exp[- z]
Successful Successful - Successful [Tested: 21]
8.8.E15 d n d z n ⁡ ( z - a ⁢ γ ⁡ ( a , z ) ) = ( - 1 ) n ⁢ z - a - n ⁢ γ ⁡ ( a + n , z ) derivative 𝑧 𝑛 superscript 𝑧 𝑎 incomplete-gamma 𝑎 𝑧 superscript 1 𝑛 superscript 𝑧 𝑎 𝑛 incomplete-gamma 𝑎 𝑛 𝑧 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}(z^{-a}% \gamma\left(a,z\right))=(-1)^{n}z^{-a-n}\gamma\left(a+n,z\right)}}
\deriv[n]{}{z}(z^{-a}\incgamma@{a}{z}) = (-1)^{n}z^{-a-n}\incgamma@{a+n}{z}		 ℜ ⁡ a > 0 , ℜ ⁡ ( a + n ) > 0 formulae-sequence 𝑎 0 𝑎 𝑛 0 {\displaystyle{\displaystyle\Re a>0,\Re(a+n)>0}} 
diff((z)^(- a)* GAMMA(a)-GAMMA(a, z), [z$(n)]) = (- 1)^(n)* (z)^(- a - n)* GAMMA(a + n)-GAMMA(a + n, z)

D[(z)^(- a)* Gamma[a, 0, z], {z, n}] == (- 1)^(n)* (z)^(- a - n)* Gamma[a + n, 0, z]
Failure Failure
Failed [63 / 63]
Result: 1.615357258-.2793504168*I
Test Values: {a = 1.5, z = 1/2*3^(1/2)+1/2*I, n = 1}


Result: 3.050292670-.1918135000*I
Test Values: {a = 1.5, z = 1/2*3^(1/2)+1/2*I, n = 2}

... skip entries to safe data
Failed [63 / 63]
Result: Plus[Complex[0.20573036539123668, -0.07193062032175179], Inactive[Sum][Times[Power[Complex[0.8660254037844387, 0.49999999999999994], Plus[-1.5, Times[-1.0, K[1.0]]]], Binomial[1.0, K[1.0]], D[Complex[0.3508882473022298, 0.19901628242832037]
Test Values: {Complex[0.8660254037844387, 0.49999999999999994], Plus[1.0, Times[-1.0, K[1.0]]]}], FactorialPower[-1.5, K[1.0]]], {K[1.0], 0.0, 1.0}]], {Rule[a, 1.5], Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Plus[Complex[-0.13665910465469025, 0.05369371428345661], Inactive[Sum][Times[Power[Complex[0.8660254037844387, 0.49999999999999994], Plus[-1.5, Times[-1.0, K[1.0]]]], Binomial[2.0, K[1.0]], D[Complex[0.3508882473022298, 0.19901628242832037]
Test Values: {Complex[0.8660254037844387, 0.49999999999999994], Plus[2.0, Times[-1.0, K[1.0]]]}], FactorialPower[-1.5, K[1.0]]], {K[1.0], 0.0, 2.0}]], {Rule[a, 1.5], Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
8.8.E16 d n d z n ⁡ ( z - a ⁢ Γ ⁡ ( a , z ) ) = ( - 1 ) n ⁢ z - a - n ⁢ Γ ⁡ ( a + n , z ) derivative 𝑧 𝑛 superscript 𝑧 𝑎 incomplete-Gamma 𝑎 𝑧 superscript 1 𝑛 superscript 𝑧 𝑎 𝑛 incomplete-Gamma 𝑎 𝑛 𝑧 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}(z^{-a}% \Gamma\left(a,z\right))=(-1)^{n}z^{-a-n}\Gamma\left(a+n,z\right)}}
\deriv[n]{}{z}(z^{-a}\incGamma@{a}{z}) = (-1)^{n}z^{-a-n}\incGamma@{a+n}{z}		 

diff((z)^(- a)* GAMMA(a, z), [z$(n)]) = (- 1)^(n)* (z)^(- a - n)* GAMMA(a + n, z)

D[(z)^(- a)* Gamma[a, z], {z, n}] == (- 1)^(n)* (z)^(- a - n)* Gamma[a + n, z]
Failure Failure Error
Failed [111 / 126]
Result: Plus[Complex[0.14584260074790834, -0.14889469354125948], Times[Complex[0.9659258262890683, 0.25881904510252074], DifferenceRoot[Function[{, }
Test Values: {Equal[Plus[Times[-1, , Plus[-1, Times[-1, ], 1], Plus[Times[-1, ], 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], []], Times[-1, Plus[-1, Times[-1, ], 1], Plus[1, Times[3, ], Times[3, Power[, 2]], Power[, 3], Times[-2, -1.5], Times[-4, , -1.5], Times[-2, Power[, 2], -1.5], Power[-1.5, 2], Times[, Power[-1.5, 2]], Times[-1, 1], Times[-2, , 1], Times[-1, Power[, 2], 1], Times[-1.5, 1], Times[, -1.5, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Times[2, , Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[2, Power[, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, , -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, 1, Power[E, Times[Complex[0, Rational[1, 6]], <syntaxhighlight lang=mathematica>Result: Plus[Complex[-0.19291890162425956, 0.2582696599924231], Times[Complex[1.9318516525781366, -0.5176380902050415], DifferenceRoot[Function[{, }
Test Values: {Equal[Plus[Times[-1, , Plus[-1, Times[-1, ], 2], Plus[Times[-1, ], 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], []], Times[-1, Plus[-1, Times[-1, ], 2], Plus[1, Times[3, ], Times[3, Power[, 2]], Power[, 3], Times[-2, -1.5], Times[-4, , -1.5], Times[-2, Power[, 2], -1.5], Power[-1.5, 2], Times[, Power[-1.5, 2]], Times[-1, 2], Times[-2, , 2], Times[-1, Power[, 2], 2], Times[-1.5, 2], Times[, -1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Times[2, , Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[2, Power[, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, , -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, 2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-2, , 2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], [Plus[1, ]]], Times[Plus[1, ], Plus[-1, Times[-1, ], -1.5, 2], Plus[5, Times[6, ], Times[2, Power[, 2]], Times[-3, -1.5], Times[-2, , -1.5], Times[-3, 2], Times[-2, , 2], Times[-1.5, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Times[, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, 2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], [Plus[2, ]]], Times[Plus[1, ], Plus[2, ], Plus[-2, Times[-1, ], -1.5, 2], Plus[-1, Times[-1, ], -1.5, 2], [Plus[3, ]]]], 0], Equal[[0], 0], Equal[[1], Times[Binomial[Times[-1, -1.5], 2], Gamma[-1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], Equal[[2], Times[Binomial[Times[-1, -1.5], 2], Plus[Times[Power[E, Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], 2, Power[Plus[-1, -1.5, 2], -1], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], -1.5]], Gamma[-1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]]}]][3.0]]], {Rule[a, -1.5], Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
8.8.E17 d n d z n ⁡ ( e z ⁢ γ ⁡ ( a , z ) ) = ( - 1 ) n ⁢ ( 1 - a ) n ⁢ e z ⁢ γ ⁡ ( a - n , z ) derivative 𝑧 𝑛 superscript 𝑒 𝑧 incomplete-gamma 𝑎 𝑧 superscript 1 𝑛 Pochhammer 1 𝑎 𝑛 superscript 𝑒 𝑧 incomplete-gamma 𝑎 𝑛 𝑧 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}(e^{z}% \gamma\left(a,z\right))=(-1)^{n}{\left(1-a\right)_{n}}e^{z}\gamma\left(a-n,z% \right)}}
\deriv[n]{}{z}(e^{z}\incgamma@{a}{z}) = (-1)^{n}\Pochhammersym{1-a}{n}e^{z}\incgamma@{a-n}{z}		 ℜ ⁡ a > 0 , ℜ ⁡ ( a - n ) > 0 formulae-sequence 𝑎 0 𝑎 𝑛 0 {\displaystyle{\displaystyle\Re a>0,\Re(a-n)>0}} 
diff(exp(z)*GAMMA(a)-GAMMA(a, z), [z$(n)]) = (- 1)^(n)* pochhammer(1 - a, n)*exp(z)*GAMMA(a - n)-GAMMA(a - n, z)

D[Exp[z]*Gamma[a, 0, z], {z, n}] == (- 1)^(n)* Pochhammer[1 - a, n]*Exp[z]*Gamma[a - n, 0, z]
Failure Failure
Failed [14 / 14]
Result: .6619339064-.2987854069*I
Test Values: {a = 1.5, z = 1/2*3^(1/2)+1/2*I, n = 1}


Result: 1.661215891-1.222029870*I
Test Values: {a = 1.5, z = -1/2+1/2*I*3^(1/2), n = 1}

... skip entries to safe data
Failed [14 / 14]
Result: Plus[Complex[-1.471179750131411, -1.0739918488339026], Inactive[Sum][Times[Complex[2.0864022336812553, 1.1398067350757155], Binomial[1.0, K[1.0]], D[Complex[0.3508882473022298, 0.19901628242832037]
Test Values: {Complex[0.8660254037844387, 0.49999999999999994], Plus[1.0, Times[-1.0, K[1.0]]]}]], {K[1.0], 0.0, 1.0}]], {Rule[a, 1.5], Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Plus[Complex[0.01045242262446705, -0.698806597134537], Inactive[Sum][Times[Complex[0.3929465558343552, 0.4620307840711054], Binomial[1.0, K[1.0]], D[Complex[-0.7552494829576352, 0.46247944264186114]
Test Values: {Complex[-0.4999999999999998, 0.8660254037844387], Plus[1.0, Times[-1.0, K[1.0]]]}]], {K[1.0], 0.0, 1.0}]], {Rule[a, 1.5], Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
8.8.E18 d n d z n ⁡ ( z a ⁢ e z ⁢ γ * ⁡ ( a , z ) ) = z a - n ⁢ e z ⁢ γ * ⁡ ( a - n , z ) derivative 𝑧 𝑛 superscript 𝑧 𝑎 superscript 𝑒 𝑧 incomplete-gamma-star 𝑎 𝑧 superscript 𝑧 𝑎 𝑛 superscript 𝑒 𝑧 incomplete-gamma-star 𝑎 𝑛 𝑧 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}(z^{a}e^% {z}\gamma^{*}\left(a,z\right))=z^{a-n}e^{z}\gamma^{*}\left(a-n,z\right)}}
\deriv[n]{}{z}(z^{a}e^{z}\scincgamma@{a}{z}) = z^{a-n}e^{z}\scincgamma@{a-n}{z}		 ℜ ⁡ a > 0 , ℜ ⁡ ( a - n ) > 0 formulae-sequence 𝑎 0 𝑎 𝑛 0 {\displaystyle{\displaystyle\Re a>0,\Re(a-n)>0}} 
diff((z)^(a)* exp(z)*(z)^(-(a))*(GAMMA(a)-GAMMA(a, z))/GAMMA(a), [z$(n)]) = (z)^(a - n)* exp(z)*(z)^(-(a - n))*(GAMMA(a - n)-GAMMA(a - n, z))/GAMMA(a - n)

Error
Failure Missing Macro Error Successful [Tested: 14] -
8.8.E19 d n d z n ⁡ ( e z ⁢ Γ ⁡ ( a , z ) ) = ( - 1 ) n ⁢ ( 1 - a ) n ⁢ e z ⁢ Γ ⁡ ( a - n , z ) derivative 𝑧 𝑛 superscript 𝑒 𝑧 incomplete-Gamma 𝑎 𝑧 superscript 1 𝑛 Pochhammer 1 𝑎 𝑛 superscript 𝑒 𝑧 incomplete-Gamma 𝑎 𝑛 𝑧 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}(e^{z}% \Gamma\left(a,z\right))=(-1)^{n}{\left(1-a\right)_{n}}e^{z}\Gamma\left(a-n,z% \right)}}
\deriv[n]{}{z}(e^{z}\incGamma@{a}{z}) = (-1)^{n}\Pochhammersym{1-a}{n}e^{z}\incGamma@{a-n}{z}		 

diff(exp(z)*GAMMA(a, z), [z$(n)]) = (- 1)^(n)* pochhammer(1 - a, n)*exp(z)*GAMMA(a - n, z)

D[Exp[z]*Gamma[a, z], {z, n}] == (- 1)^(n)* Pochhammer[1 - a, n]*Exp[z]*Gamma[a - n, z]
Failure Failure Error
Failed [96 / 126]
Result: Plus[Complex[0.06772606154573046, -0.6693082179083164], Times[Complex[2.0864022336812553, 1.1398067350757155], DifferenceRoot[Function[{, }
Test Values: {Equal[Plus[Times[-1, , Plus[-1, Times[-1, ], 1], Plus[Times[-1, ], 1], []], Times[Plus[-1, Times[-1, ], 1], Plus[-1, Times[-2, ], Times[-2, Power[, 2]], -1.5, Times[, -1.5], Times[, 1], Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, , Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], [Plus[1, ]]], Times[-1, Plus[1, ], Plus[1, Times[2, ], Power[, 2], Times[-1, -1.5], Times[-1, , -1.5], Times[-1, 1], Times[-1, , 1], Times[-1.5, 1], Times[3, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[2, , Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, 1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], [Plus[2, ]]], Times[Plus[1, ], Plus[2, ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], [Plus[3, ]]]], 0], Equal[[0], 0], Equal[[1<syntaxhighlight lang=mathematica>Result: Plus[Complex[0.714773674302028, 1.7455063478143567], Times[Complex[4.172804467362511, 2.279613470151431], DifferenceRoot[Function[{, }
Test Values: {Equal[Plus[Times[-1, , Plus[-1, Times[-1, ], 2], Plus[Times[-1, ], 2], []], Times[Plus[-1, Times[-1, ], 2], Plus[-1, Times[-2, ], Times[-2, Power[, 2]], -1.5, Times[, -1.5], Times[, 2], Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, , Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], [Plus[1, ]]], Times[-1, Plus[1, ], Plus[1, Times[2, ], Power[, 2], Times[-1, -1.5], Times[-1, , -1.5], Times[-1, 2], Times[-1, , 2], Times[-1.5, 2], Times[3, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[2, , Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, 2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], [Plus[2, ]]], Times[Plus[1, ], Plus[2, ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], [Plus[3, ]]]], 0], Equal[[0], 0], Equal[[1], Times[Power[Factorial[2], -1], Gamma[-1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], Equal[[2], Times[Power[Factorial[2], -1], Plus[Times[-1, Power[E, Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], 2, Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Plus[-1, -1.5]]], Gamma[-1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]]}]][3.0]]], {Rule[a, -1.5], Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
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