# Results of Incomplete Gamma and Related Functions: Difference between revisions

DLMF Formula Maple Mathematica Symbolic
Maple
Symbolic
Mathematica
Numeric
Maple
Numeric
Mathematica
8.2.E1 ${\displaystyle{\displaystyle\gamma\left(a,z\right)=\int_{0}^{z}t^{a-1}e^{-t}% \mathrm{d}t}}$ GAMMA(a)-GAMMA(a, z)= int((t)^(a - 1)* exp(- t), t = 0..z) Gamma[a, 0, z]= Integrate[(t)^(a - 1)* Exp[- t], {t, 0, z}] Failure Successful Skip -
8.2.E2 ${\displaystyle{\displaystyle\Gamma\left(a,z\right)=\int_{z}^{\infty}t^{a-1}e^{% -t}\mathrm{d}t}}$ GAMMA(a, z)= int((t)^(a - 1)* exp(- t), t = z..infinity) Gamma[a, z]= Integrate[(t)^(a - 1)* Exp[- t], {t, z, Infinity}] Failure Failure Skip Successful
8.2.E3 ${\displaystyle{\displaystyle\gamma\left(a,z\right)+\Gamma\left(a,z\right)=% \Gamma\left(a\right)}}$ GAMMA(a)-GAMMA(a, z)+ GAMMA(a, z)= GAMMA(a) Gamma[a, 0, z]+ Gamma[a, z]= Gamma[a] Successful Successful - -
8.2#Ex1 ${\displaystyle{\displaystyle P\left(a,z\right)=\frac{\gamma\left(a,z\right)}{% \Gamma\left(a\right)}}}$ (GAMMA(a)-GAMMA(a, z))/GAMMA(a)=(GAMMA(a)-GAMMA(a, z))/(GAMMA(a)) GammaRegularized[a, 0, z]=Divide[Gamma[a, 0, z],Gamma[a]] Successful Successful - -
8.2#Ex2 ${\displaystyle{\displaystyle Q\left(a,z\right)=\frac{\Gamma\left(a,z\right)}{% \Gamma\left(a\right)}}}$ GAMMA(a, z)/GAMMA(a)=(GAMMA(a, z))/(GAMMA(a)) GammaRegularized[a, z]=Divide[Gamma[a, z],Gamma[a]] Successful Successful - -
8.2.E5 ${\displaystyle{\displaystyle P\left(a,z\right)+Q\left(a,z\right)=1}}$ (GAMMA(a)-GAMMA(a, z))/GAMMA(a)+ GAMMA(a, z)/GAMMA(a)= 1 GammaRegularized[a, 0, z]+ GammaRegularized[a, z]= 1 Successful Successful - -
8.2.E6 ${\displaystyle{\displaystyle\gamma^{*}\left(a,z\right)=z^{-a}P\left(a,z\right)}}$ (z)^(-(a))*(GAMMA(a)-GAMMA(a, z))/GAMMA(a)= (z)^(- a)* (GAMMA(a)-GAMMA(a, z))/GAMMA(a) Error Successful Error - -
8.2.E6 ${\displaystyle{\displaystyle z^{-a}P\left(a,z\right)=\frac{z^{-a}}{\Gamma\left% (a\right)}\gamma\left(a,z\right)}}$ (z)^(- a)* (GAMMA(a)-GAMMA(a, z))/GAMMA(a)=((z)^(- a))/(GAMMA(a))*GAMMA(a)-GAMMA(a, z) (z)^(- a)* GammaRegularized[a, 0, z]=Divide[(z)^(- a),Gamma[a]]*Gamma[a, 0, z] Failure Successful
Fail
.3504429851+.4826856014*I <- {a = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}
-.4474572306+.2704599710*I <- {a = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}
23.62700226+82.69161801*I <- {a = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
3.420707652-13.57627439*I <- {a = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
-
8.2.E7 ${\displaystyle{\displaystyle\gamma^{*}\left(a,z\right)=\frac{1}{\Gamma\left(a% \right)}\int_{0}^{1}t^{a-1}e^{-zt}\mathrm{d}t}}$ (z)^(-(a))*(GAMMA(a)-GAMMA(a, z))/GAMMA(a)=(1)/(GAMMA(a))*int((t)^(a - 1)* exp(- z*t), t = 0..1) Error Failure Error Skip -
8.2.E8 ${\displaystyle{\displaystyle\gamma\left(a,ze^{2\pi mi}\right)=e^{2\pi mia}% \gamma\left(a,z\right)}}$ GAMMA(a)-GAMMA(a, z*exp(2*Pi*m*I))= exp(2*Pi*m*I*a)*GAMMA(a)-GAMMA(a, z) Gamma[a, 0, z*Exp[2*Pi*m*I]]= Exp[2*Pi*m*I*a]*Gamma[a, 0, z] Failure Failure Successful Successful
8.2.E9 ${\displaystyle{\displaystyle\Gamma\left(a,ze^{2\pi mi}\right)=e^{2\pi mia}% \Gamma\left(a,z\right)+(1-e^{2\pi mia})\Gamma\left(a\right)}}$ GAMMA(a, z*exp(2*Pi*m*I))= exp(2*Pi*m*I*a)*GAMMA(a, z)+(1 - exp(2*Pi*m*I*a))* GAMMA(a) Gamma[a, z*Exp[2*Pi*m*I]]= Exp[2*Pi*m*I*a]*Gamma[a, z]+(1 - Exp[2*Pi*m*I*a])* Gamma[a] Failure Failure
Fail
-.2249049111-.4410511843e-1*I <- {a = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 1}
-.2248750758-.4411585330e-1*I <- {a = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 2}
-.2248750795-.4411584875e-1*I <- {a = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 3}
-1.005323136+.3326243216*I <- {a = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2), m = 1}
... skip entries to safe data
Fail
Complex[-0.22490491118791595, -0.04410511845656586] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[m, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-0.2248750764783257, -0.044115852492705915] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[m, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-0.22487507925834865, -0.04411584909968558] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[m, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-1.0053231382729926, 0.33262432134470665] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[m, 1], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
8.2.E10 ${\displaystyle{\displaystyle e^{-\pi ia}\Gamma\left(a,ze^{\pi i}\right)-e^{\pi ia% }\Gamma\left(a,ze^{-\pi i}\right)=-\frac{2\pi i}{\Gamma\left(1-a\right)}}}$ exp(- Pi*I*a)*GAMMA(a, z*exp(Pi*I))- exp(Pi*I*a)*GAMMA(a, z*exp(- Pi*I))= -(2*Pi*I)/(GAMMA(1 - a)) Exp[- Pi*I*a]*Gamma[a, z*Exp[Pi*I]]- Exp[Pi*I*a]*Gamma[a, z*Exp[- Pi*I]]= -Divide[2*Pi*I,Gamma[1 - a]] Failure Failure
Fail
-7167.292469-174.9289096*I <- {a = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}
2.16987973+12.77160007*I <- {a = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}
8.705606105-17.43270949*I <- {a = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
-4.50134822-89.91653387*I <- {a = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[-7167.2924809060105, -174.9289096706231] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[2.169879706441371, 12.771600034859095] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[8.70560609871773, -17.43270953363519] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-4.501348191090425, -89.91653394957189] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
8.2.E11 ${\displaystyle{\displaystyle\Gamma\left(a,ze^{+\pi i}\right)=\Gamma\left(a% \right)(1-z^{a}e^{+\pi ia}\gamma^{*}\left(a,-z\right))}}$ GAMMA(a, z*exp(+ Pi*I))= GAMMA(a)*(1 - (z)^(a)* exp(+ Pi*I*a)*(- z)^(-(a))*(GAMMA(a)-GAMMA(a, - z))/GAMMA(a)) Error Failure Error
Fail
20.46249972+81.80630504*I <- {a = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}
-1.005323138+.3326243220*I <- {a = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
1095.761010-111.2868863*I <- {a = 2^(1/2)-I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}
-1231.554386+1108.053849*I <- {a = 2^(1/2)-I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
-
8.2.E11 ${\displaystyle{\displaystyle\Gamma\left(a,ze^{-\pi i}\right)=\Gamma\left(a% \right)(1-z^{a}e^{-\pi ia}\gamma^{*}\left(a,-z\right))}}$ GAMMA(a, z*exp(- Pi*I))= GAMMA(a)*(1 - (z)^(a)* exp(- Pi*I*a)*(- z)^(-(a))*(GAMMA(a)-GAMMA(a, - z))/GAMMA(a)) Error Failure Error
Fail
1095.761010+111.2868863*I <- {a = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}
-1231.554386-1108.053849*I <- {a = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
20.46249972-81.80630504*I <- {a = 2^(1/2)-I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}
-1.005323138-.3326243220*I <- {a = 2^(1/2)-I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
... skip entries to safe data
-
8.2.E12 ${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(% 1+\frac{1-a}{z}\right)\frac{\mathrm{d}w}{\mathrm{d}z}=0}}$ diff(w, [z$(2)])+(1 +(1 - a)/(z))* diff(w, z)= 0 D[w, {z, 2}]+(1 +Divide[1 - a,z])* D[w, z]= 0 Successful Successful - - 8.2.E13 ${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}-\left(% 1+\frac{1-a}{z}\right)\frac{\mathrm{d}w}{\mathrm{d}z}+\frac{1-a}{z^{2}}w=0}}$ diff(w, [z$(2)])-(1 +(1 - a)/(z))* diff(w, z)+(1 - a)/((z)^(2))*w = 0 D[w, {z, 2}]-(1 +Divide[1 - a,z])* D[w, z]+Divide[1 - a,(z)^(2)]*w = 0 Failure Failure
Fail
-.6464466093-.3535533907*I <- {a = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}
.6464466093+.3535533907*I <- {a = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}
-.6464466093-.3535533907*I <- {a = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
.6464466093+.3535533907*I <- {a = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[-0.6464466094067263, -0.35355339059327373] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.6464466094067263, 0.35355339059327373] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-0.6464466094067263, -0.35355339059327373] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[0.6464466094067263, 0.35355339059327373] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
8.2.E14 ${\displaystyle{\displaystyle z\frac{{\mathrm{d}}^{2}\gamma^{*}}{{\mathrm{d}z}^% {2}}+(a+1+z)\frac{\mathrm{d}\gamma^{*}}{\mathrm{d}z}+a\gamma^{*}=0}}$ z*diff((+)^(-(z))*(GAMMA(z)-GAMMA(z, +))/GAMMA(z), [(a + 1 + z)*\$(2)])*diff((+)^(-(z))*(GAMMA(z)-GAMMA(z, +))/GAMMA(z), a)*(0)^(-(=))*(GAMMA(=)-GAMMA(=, 0))/GAMMA(=) Error Error Error - -
8.4.E1 ${\displaystyle{\displaystyle\gamma\left(\tfrac{1}{2},z^{2}\right)=2\int_{0}^{z% }e^{-t^{2}}\mathrm{d}t}}$ 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}] Failure Failure Skip
Fail
Complex[3.581461769189045, -0.9710415344467407] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[3.581461769189045, 0.9710415344467407] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
8.4.E1 ${\displaystyle{\displaystyle 2\int_{0}^{z}e^{-t^{2}}\mathrm{d}t=\sqrt{\pi}% \operatorname{erf}\left(z\right)}}$ 2*int(exp(- (t)^(2)), t = 0..z)=sqrt(Pi)*erf(z) 2*Integrate[Exp[- (t)^(2)], {t, 0, z}]=Sqrt[Pi]*Erf[z] Successful Successful - -
8.4.E2 ${\displaystyle{\displaystyle\gamma^{*}\left(a,0\right)=\frac{1}{\Gamma\left(a+% 1\right)}}}$ (0)^(-(a))*(GAMMA(a)-GAMMA(a, 0))/GAMMA(a)=(1)/(GAMMA(a + 1)) Error Failure Error
Fail
-.6493698774+1.106937485*I <- {a = 2^(1/2)+I*2^(1/2)}
-.6493698774-1.106937485*I <- {a = 2^(1/2)-I*2^(1/2)}
4.564263782+2.639434666*I <- {a = -2^(1/2)-I*2^(1/2)}
4.564263782-2.639434666*I <- {a = -2^(1/2)+I*2^(1/2)}
-
8.4.E3 ${\displaystyle{\displaystyle\gamma^{*}\left(\tfrac{1}{2},-z^{2}\right)=\frac{2% e^{z^{2}}}{z\sqrt{\pi}}F\left(z\right)}}$ (- (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 Error - -
8.4.E4 ${\displaystyle{\displaystyle\Gamma\left(0,z\right)=\int_{z}^{\infty}t^{-1}e^{-% t}\mathrm{d}t}}$ GAMMA(0, z)= int((t)^(- 1)* exp(- t), t = z..infinity) Gamma[0, z]= Integrate[(t)^(- 1)* Exp[- t], {t, z, Infinity}] Successful Failure - Successful
8.4.E4 ${\displaystyle{\displaystyle\int_{z}^{\infty}t^{-1}e^{-t}\mathrm{d}t=E_{1}% \left(z\right)}}$ int((t)^(- 1)* exp(- t), t = z..infinity)= Ei(z) Integrate[(t)^(- 1)* Exp[- t], {t, z, Infinity}]= -ExpIntegralEi[-(z)] Failure Failure Skip Successful
8.4.E5 ${\displaystyle{\displaystyle\Gamma\left(1,z\right)=e^{-z}}}$ GAMMA(1, z)= exp(- z) Gamma[1, z]= Exp[- z] Successful Successful - -
8.4.E6 ${\displaystyle{\displaystyle\Gamma\left(\tfrac{1}{2},z^{2}\right)=2\int_{z}^{% \infty}e^{-t^{2}}\mathrm{d}t}}$ 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}] Failure Failure Skip
Fail
Complex[-3.581461769189044, 0.9710415344467407] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-3.581461769189044, -0.9710415344467407] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
8.4.E6 ${\displaystyle{\displaystyle 2\int_{z}^{\infty}e^{-t^{2}}\mathrm{d}t=\sqrt{\pi% }\operatorname{erfc}\left(z\right)}}$ 2*int(exp(- (t)^(2)), t = z..infinity)=sqrt(Pi)*erfc(z) 2*Integrate[Exp[- (t)^(2)], {t, z, Infinity}]=Sqrt[Pi]*Erfc[z] Successful Successful - -
8.4.E7 ${\displaystyle{\displaystyle\gamma\left(n+1,z\right)=n!(1-e^{-z}e_{n}(z))}}$ 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 Successful
8.4.E8 ${\displaystyle{\displaystyle\Gamma\left(n+1,z\right)=n!e^{-z}e_{n}(z)}}$ 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 Successful
8.4.E9 ${\displaystyle{\displaystyle P\left(n+1,z\right)=1-e^{-z}e_{n}(z)}}$ (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 Successful
8.4.E10 ${\displaystyle{\displaystyle Q\left(n+1,z\right)=e^{-z}e_{n}(z)}}$ 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 Successful
8.4.E12 ${\displaystyle{\displaystyle\gamma^{*}\left(-n,z\right)=z^{n}}}$ (z)^(-(- n))*(GAMMA(- n)-GAMMA(- n, z))/GAMMA(- n)= (z)^(n) Error Failure Error
Fail
Float(undefined)+Float(undefined)*I <- {z = 2^(1/2)+I*2^(1/2), n = 1}
Float(undefined)+Float(undefined)*I <- {z = 2^(1/2)+I*2^(1/2), n = 2}
Float(undefined)+Float(undefined)*I <- {z = 2^(1/2)+I*2^(1/2), n = 3}
Float(undefined)+Float(undefined)*I <- {z = 2^(1/2)-I*2^(1/2), n = 1}
... skip entries to safe data
-
8.4.E13 ${\displaystyle{\displaystyle\Gamma\left(1-n,z\right)=z^{1-n}E_{n}\left(z\right% )}}$ GAMMA(1 - n, z)= (z)^(1 - n)* Ei(n, z) Gamma[1 - n, z]= (z)^(1 - n)* ExpIntegralE[n, z] Successful Successful - -
8.4.E14 ${\displaystyle{\displaystyle Q\left(n+\tfrac{1}{2},z^{2}\right)=\operatorname{% erfc}\left(z\right)+\frac{e^{-z^{2}}}{\sqrt{\pi}}\sum_{k=1}^{n}\frac{z^{2k-1}}% {{\left(\tfrac{1}{2}\right)_{k}}}}}$ 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}] Failure Failure Skip
Fail
Complex[-6.522116143801526, 0.8770870118427658] <- {Rule[n, 1], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-7.400077458243353, -11.126893574158686] <- {Rule[n, 2], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[11.80629147935897, -12.531631677265604] <- {Rule[n, 3], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-6.522116143801526, -0.8770870118427658] <- {Rule[n, 1], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
8.4.E15 ${\displaystyle{\displaystyle\Gamma\left(-n,z\right)=\frac{(-1)^{n}}{n!}\left(E% _{1}\left(z\right)-e^{-z}\sum_{k=0}^{n-1}\frac{(-1)^{k}k!}{z^{k+1}}\right)}}$ 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)!]*(-ExpIntegralEi[-(z)]- Exp[- z]*Sum[Divide[(- 1)^(k)* (k)!,(z)^(k + 1)], {k, 0, n - 1}]) Failure Failure Skip
Fail
Complex[1.3877787807814457*^-17, 3.141592653589793] <- {Rule[n, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[1.734723475976807*^-18, -1.5707963267948966] <- {Rule[n, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-4.3368086899420177*^-19, 0.5235987755982987] <- {Rule[n, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[1.3877787807814457*^-17, -3.141592653589793] <- {Rule[n, 1], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
8.5.E1 ${\displaystyle{\displaystyle\gamma\left(a,z\right)=a^{-1}z^{a}e^{-z}M\left(1,1% +a,z\right)}}$ 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 - -
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)* 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 - -
8.5.E2 ${\displaystyle{\displaystyle\gamma^{*}\left(a,z\right)=e^{-z}{\mathbf{M}}\left% (1,1+a,z\right)}}$ (z)^(-(a))*(GAMMA(a)-GAMMA(a, z))/GAMMA(a)= exp(- z)*KummerM(1, 1 + a, z)/GAMMA(1 + a) Error Successful Error - -
8.5.E2 ${\displaystyle{\displaystyle e^{-z}{\mathbf{M}}\left(1,1+a,z\right)={\mathbf{M% }}\left(a,1+a,-z\right)}}$ 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 - -
8.5.E3 ${\displaystyle{\displaystyle\Gamma\left(a,z\right)=e^{-z}U\left(1-a,1-a,z% \right)}}$ GAMMA(a, z)= exp(- z)*KummerU(1 - a, 1 - a, z) Gamma[a, z]= Exp[- z]*HypergeometricU[1 - a, 1 - a, z] Successful Successful - -
8.5.E3 ${\displaystyle{\displaystyle e^{-z}U\left(1-a,1-a,z\right)=z^{a}e^{-z}U\left(1% ,1+a,z\right)}}$ 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 - -
8.5.E4 ${\displaystyle{\displaystyle\gamma\left(a,z\right)=a^{-1}z^{\frac{1}{2}a-\frac% {1}{2}}e^{-\frac{1}{2}z}M_{\frac{1}{2}a-\frac{1}{2},\frac{1}{2}a}\left(z\right% )}}$ 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 - -
8.5.E5 ${\displaystyle{\displaystyle\Gamma\left(a,z\right)=e^{-\frac{1}{2}z}z^{\frac{1% }{2}a-\frac{1}{2}}W_{\frac{1}{2}a-\frac{1}{2},\frac{1}{2}a}\left(z\right)}}$ 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 - -
8.6.E1 ${\displaystyle{\displaystyle\gamma\left(a,z\right)=\frac{z^{a}}{\sin\left(\pi a% \right)}\int_{0}^{\pi}e^{z\cos t}\cos\left(at+z\sin t\right)\mathrm{d}t}}$ GAMMA(a)-GAMMA(a, z)=((z)^(a))/(sin(Pi*a))*int(exp(z*cos(t))*cos(a*t + z*sin(t)), t = 0..Pi) Gamma[a, 0, z]=Divide[(z)^(a),Sin[Pi*a]]*Integrate[Exp[z*Cos[t]]*Cos[a*t + z*Sin[t]], {t, 0, Pi}] Failure Failure Skip Error
8.6.E2 ${\displaystyle{\displaystyle\gamma\left(a,z\right)=z^{\frac{1}{2}a}\int_{0}^{% \infty}e^{-t}t^{\frac{1}{2}a-1}J_{a}\left(2\sqrt{zt}\right)\mathrm{d}t}}$ GAMMA(a)-GAMMA(a, z)= (z)^((1)/(2)*a)* int(exp(- t)*(t)^((1)/(2)*a - 1)* BesselJ(a, 2*sqrt(z*t)), t = 0..infinity) Gamma[a, 0, z]= (z)^(Divide[1,2]*a)* Integrate[Exp[- t]*(t)^(Divide[1,2]*a - 1)* BesselJ[a, 2*Sqrt[z*t]], {t, 0, Infinity}] Failure Failure Skip Error
8.6.E3 ${\displaystyle{\displaystyle\gamma\left(a,z\right)=z^{a}\int_{0}^{\infty}\exp% \left(-at-ze^{-t}\right)\mathrm{d}t}}$ GAMMA(a)-GAMMA(a, z)= (z)^(a)* int(exp(- a*t - z*exp(- t)), t = 0..infinity) Gamma[a, 0, z]= (z)^(a)* Integrate[Exp[- a*t - z*Exp[- t]], {t, 0, Infinity}] Failure Failure Skip Successful
8.6.E4 ${\displaystyle{\displaystyle\Gamma\left(a,z\right)=\frac{z^{a}e^{-z}}{\Gamma% \left(1-a\right)}\int_{0}^{\infty}\frac{t^{-a}e^{-t}}{z+t}\mathrm{d}t}}$ GAMMA(a, z)=((z)^(a)* exp(- z))/(GAMMA(1 - a))*int(((t)^(- a)* exp(- t))/(z + t), t = 0..infinity) Gamma[a, z]=Divide[(z)^(a)* Exp[- z],Gamma[1 - a]]*Integrate[Divide[(t)^(- a)* Exp[- t],z + t], {t, 0, Infinity}] Failure Failure Skip Successful
8.6.E5 ${\displaystyle{\displaystyle\Gamma\left(a,z\right)=z^{a}e^{-z}\int_{0}^{\infty% }\frac{e^{-zt}}{(1+t)^{1-a}}\mathrm{d}t}}$ GAMMA(a, z)= (z)^(a)* exp(- z)*int((exp(- z*t))/((1 + t)^(1 - a)), t = 0..infinity) Gamma[a, z]= (z)^(a)* Exp[- z]*Integrate[Divide[Exp[- z*t],(1 + t)^(1 - a)], {t, 0, Infinity}] Successful Failure - Error
8.6.E6 ${\displaystyle{\displaystyle\Gamma\left(a,z\right)=\frac{2z^{\frac{1}{2}a}e^{-% z}}{\Gamma\left(1-a\right)}\int_{0}^{\infty}e^{-t}t^{-\frac{1}{2}a}K_{a}\left(% 2\sqrt{zt}\right)\mathrm{d}t}}$ GAMMA(a, z)=(2*(z)^((1)/(2)*a)* exp(- z))/(GAMMA(1 - a))*int(exp(- t)*(t)^(-(1)/(2)*a)* BesselK(a, 2*sqrt(z*t)), t = 0..infinity) Gamma[a, z]=Divide[2*(z)^(Divide[1,2]*a)* Exp[- z],Gamma[1 - a]]*Integrate[Exp[- t]*(t)^(-Divide[1,2]*a)* BesselK[a, 2*Sqrt[z*t]], {t, 0, Infinity}] Successful Failure - Error
8.6.E7 ${\displaystyle{\displaystyle\Gamma\left(a,z\right)=z^{a}\int_{0}^{\infty}\exp% \left(at-ze^{t}\right)\mathrm{d}t}}$ GAMMA(a, z)= (z)^(a)* int(exp(a*t - z*exp(t)), t = 0..infinity) Gamma[a, z]= (z)^(a)* Integrate[Exp[a*t - z*Exp[t]], {t, 0, Infinity}] Failure Failure Skip Error
8.6.E8 ${\displaystyle{\displaystyle\gamma\left(a,z\right)=\frac{-\mathrm{i}z^{a}}{2% \sin\left(\pi a\right)}\int_{-1}^{(0+)}t^{a-1}e^{zt}\mathrm{d}t}}$ GAMMA(a)-GAMMA(a, z)=(- I*(z)^(a))/(2*sin(Pi*a))*int((t)^(a - 1)* exp(z*t), t = - 1..(0 +)) Gamma[a, 0, z]=Divide[- I*(z)^(a),2*Sin[Pi*a]]*Integrate[(t)^(a - 1)* Exp[z*t], {t, - 1, (0 +)}] Error Failure - Error
8.6.E9 ${\displaystyle{\displaystyle\Gamma\left(-a,ze^{+\pi i}\right)=\frac{e^{z}e^{-% \pi\mathrm{i}a}}{\Gamma\left(1+a\right)}\int_{0}^{\infty}\frac{t^{a}e^{-zt}}{t% -1}\mathrm{d}t}}$ GAMMA(- a, z*exp(+ Pi*I))=(exp(z)*exp(- Pi*I*a))/(GAMMA(1 + a))*int(((t)^(a)* exp(- z*t))/(t - 1), t = 0..infinity) Gamma[- a, z*Exp[+ Pi*I]]=Divide[Exp[z]*Exp[- Pi*I*a],Gamma[1 + a]]*Integrate[Divide[(t)^(a)* Exp[- z*t],t - 1], {t, 0, Infinity}] Failure Failure Skip Error
8.6.E9 ${\displaystyle{\displaystyle\Gamma\left(-a,ze^{-\pi i}\right)=\frac{e^{z}e^{+% \pi\mathrm{i}a}}{\Gamma\left(1+a\right)}\int_{0}^{\infty}\frac{t^{a}e^{-zt}}{t% -1}\mathrm{d}t}}$ GAMMA(- a, z*exp(- Pi*I))=(exp(z)*exp(+ Pi*I*a))/(GAMMA(1 + a))*int(((t)^(a)* exp(- z*t))/(t - 1), t = 0..infinity) Gamma[- a, z*Exp[- Pi*I]]=Divide[Exp[z]*Exp[+ Pi*I*a],Gamma[1 + a]]*Integrate[Divide[(t)^(a)* Exp[- z*t],t - 1], {t, 0, Infinity}] Failure Failure Skip Error
8.6.E10 ${\displaystyle{\displaystyle\gamma\left(a,z\right)=\frac{1}{2\pi i}\int_{c-i% \infty}^{c+i\infty}\frac{\Gamma\left(s\right)}{a-s}z^{a-s}\mathrm{d}s}}$ GAMMA(a)-GAMMA(a, z)=(1)/(2*Pi*I)*int((GAMMA(s))/(a - s)*(z)^(a - s), s = c - I*infinity..c + I*infinity) Gamma[a, 0, z]=Divide[1,2*Pi*I]*Integrate[Divide[Gamma[s],a - s]*(z)^(a - s), {s, c - I*Infinity, c + I*Infinity}] Failure Failure Skip Error
8.6.E11 ${\displaystyle{\displaystyle\Gamma\left(a,z\right)=\frac{1}{2\pi i}\int_{c-i% \infty}^{c+i\infty}\Gamma\left(s+a\right)\frac{z^{-s}}{s}\mathrm{d}s}}$ GAMMA(a, z)=(1)/(2*Pi*I)*int(GAMMA(s + a)*((z)^(- s))/(s), s = c - I*infinity..c + I*infinity) Gamma[a, z]=Divide[1,2*Pi*I]*Integrate[Gamma[s + a]*Divide[(z)^(- s),s], {s, c - I*Infinity, c + I*Infinity}] Failure Failure Skip Error
8.6.E12 ${\displaystyle{\displaystyle\Gamma\left(a,z\right)=-\frac{z^{a-1}e^{-z}}{% \Gamma\left(1-a\right)}\*\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\Gamma% \left(s+1-a\right)\frac{\pi z^{-s}}{\sin\left(\pi s\right)}\mathrm{d}s}}$ GAMMA(a, z)= -((z)^(a - 1)* exp(- z))/(GAMMA(1 - a))*(1)/(2*Pi*I)*int(GAMMA(s + 1 - a)*(Pi*(z)^(- s))/(sin(Pi*s)), s = c - I*infinity..c + I*infinity) Gamma[a, z]= -Divide[(z)^(a - 1)* Exp[- z],Gamma[1 - a]]*Divide[1,2*Pi*I]*Integrate[Gamma[s + 1 - a]*Divide[Pi*(z)^(- s),Sin[Pi*s]], {s, c - I*Infinity, c + I*Infinity}] Failure Failure Skip Error
8.7.E1 ${\displaystyle{\displaystyle\gamma^{*}\left(a,z\right)=e^{-z}\sum_{k=0}^{% \infty}\frac{z^{k}}{\Gamma\left(a+k+1\right)}}}$ (z)^(-(a))*(GAMMA(a)-GAMMA(a, z))/GAMMA(a)= exp(- z)*sum(((z)^(k))/(GAMMA(a + k + 1)), k = 0..infinity) Error Successful Error - -
8.7.E1 ${\displaystyle{\displaystyle e^{-z}\sum_{k=0}^{\infty}\frac{z^{k}}{\Gamma\left% (a+k+1\right)}=\frac{1}{\Gamma\left(a\right)}\sum_{k=0}^{\infty}\frac{(-z)^{k}% }{k!(a+k)}}}$ exp(- z)*sum(((z)^(k))/(GAMMA(a + k + 1)), k = 0..infinity)=(1)/(GAMMA(a))*sum(((- z)^(k))/(factorial(k)*(a + k)), k = 0..infinity) Exp[- z]*Sum[Divide[(z)^(k),Gamma[a + k + 1]], {k, 0, Infinity}]=Divide[1,Gamma[a]]*Sum[Divide[(- z)^(k),(k)!*(a + k)], {k, 0, Infinity}] Successful Successful - -
8.7.E3 ${\displaystyle{\displaystyle\Gamma\left(a,z\right)=\Gamma\left(a\right)-\sum_{% k=0}^{\infty}\frac{(-1)^{k}z^{a+k}}{k!(a+k)}}}$ GAMMA(a, z)= GAMMA(a)- sum(((- 1)^(k)* (z)^(a + k))/(factorial(k)*(a + k)), k = 0..infinity) Gamma[a, z]= Gamma[a]- Sum[Divide[(- 1)^(k)* (z)^(a + k),(k)!*(a + k)], {k, 0, Infinity}] Successful Successful - -
8.7.E3 ${\displaystyle{\displaystyle\Gamma\left(a\right)-\sum_{k=0}^{\infty}\frac{(-1)% ^{k}z^{a+k}}{k!(a+k)}=\Gamma\left(a\right)\left(1-z^{a}e^{-z}\sum_{k=0}^{% \infty}\frac{z^{k}}{\Gamma\left(a+k+1\right)}\right)}}$ GAMMA(a)- sum(((- 1)^(k)* (z)^(a + k))/(factorial(k)*(a + k)), k = 0..infinity)= GAMMA(a)*(1 - (z)^(a)* exp(- z)*sum(((z)^(k))/(GAMMA(a + k + 1)), k = 0..infinity)) Gamma[a]- Sum[Divide[(- 1)^(k)* (z)^(a + k),(k)!*(a + k)], {k, 0, Infinity}]= Gamma[a]*(1 - (z)^(a)* Exp[- z]*Sum[Divide[(z)^(k),Gamma[a + k + 1]], {k, 0, Infinity}]) Successful Successful - -
8.8.E1 ${\displaystyle{\displaystyle\gamma\left(a+1,z\right)=a\gamma\left(a,z\right)-z% ^{a}e^{-z}}}$ 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
Fail
.135004907e-1-.2375774782*I <- {a = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}
.8693672828+.710002389*I <- {a = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}
107.1902160-63.3824277*I <- {a = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
-.1657436948-.7422690683*I <- {a = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
-
8.8.E2 ${\displaystyle{\displaystyle\Gamma\left(a+1,z\right)=a\Gamma\left(a,z\right)+z% ^{a}e^{-z}}}$ 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 -
8.8.E4 ${\displaystyle{\displaystyle z\gamma^{*}\left(a+1,z\right)=\gamma^{*}\left(a,z% \right)-\frac{e^{-z}}{\Gamma\left(a+1\right)}}}$ 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 Error Successful -
8.8.E5 ${\displaystyle{\displaystyle P\left(a+1,z\right)=P\left(a,z\right)-\frac{z^{a}% e^{-z}}{\Gamma\left(a+1\right)}}}$ (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 -
8.8.E6 ${\displaystyle{\displaystyle Q\left(a+1,z\right)=Q\left(a,z\right)+\frac{z^{a}% e^{-z}}{\Gamma\left(a+1\right)}}}$ 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 -
8.8.E7 ${\displaystyle{\displaystyle\gamma\left(a+n,z\right)={\left(a\right)_{n}}% \gamma\left(a,z\right)-z^{a}e^{-z}\sum_{k=0}^{n-1}\frac{\Gamma\left(a+n\right)% }{\Gamma\left(a+k+1\right)}z^{k}}}$ 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}] Failure Successful Skip -
8.8.E8 ${\displaystyle{\displaystyle\gamma\left(a,z\right)=\frac{\Gamma\left(a\right)}% {\Gamma\left(a-n\right)}\gamma\left(a-n,z\right)-z^{a-1}e^{-z}\sum_{k=0}^{n-1}% \frac{\Gamma\left(a\right)}{\Gamma\left(a-k\right)}z^{-k}}}$ 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}] Failure Successful Skip -
8.8.E9 ${\displaystyle{\displaystyle\Gamma\left(a+n,z\right)={\left(a\right)_{n}}% \Gamma\left(a,z\right)+z^{a}e^{-z}\sum_{k=0}^{n-1}\frac{\Gamma\left(a+n\right)% }{\Gamma\left(a+k+1\right)}z^{k}}}$ 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}] Successful Successful - -
8.8.E10 ${\displaystyle{\displaystyle\Gamma\left(a,z\right)=\frac{\Gamma\left(a\right)}% {\Gamma\left(a-n\right)}\Gamma\left(a-n,z\right)+z^{a-1}e^{-z}\sum_{k=0}^{n-1}% \frac{\Gamma\left(a\right)}{\Gamma\left(a-k\right)}z^{-k}}}$ 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}] Failure Successful Skip -
8.8.E11 ${\displaystyle{\displaystyle P\left(a+n,z\right)=P\left(a,z\right)-z^{a}e^{-z}% \sum_{k=0}^{n-1}\frac{z^{k}}{\Gamma\left(a+k+1\right)}}}$ (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}] Successful Successful - -
8.8.E12