Results of Incomplete Gamma and Related Functions

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+.4826856014I <- {a = 2^(1/2)+I2^(1/2), z = 2^(1/2)+I2^(1/2)}` `-.4474572306+.2704599710I <- {a = 2^(1/2)+I2^(1/2), z = 2^(1/2)-I2^(1/2)}` `23.62700226+82.69161801I <- {a = 2^(1/2)+I2^(1/2), z = -2^(1/2)-I2^(1/2)}` `3.420707652-13.57627439I <- {a = 2^(1/2)+I2^(1/2), z = -2^(1/2)+I2^(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, zexp(2PimI))= exp(2PimIa)*GAMMA(a)-GAMMA(a, z)`	`Gamma[a, 0, zExp[2PimI]]= Exp[2PimIa]*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, zexp(2PimI))= exp(2PimIa)GAMMA(a, z)+(1 - exp(2PimIa)) GAMMA(a)`	`Gamma[a, zExp[2PimI]]= Exp[2PimIa]Gamma[a, z]+(1 - Exp[2PimIa]) Gamma[a]`	Failure	Failure	Fail `-.2249049111-.4410511843e-1I <- {a = 2^(1/2)+I2^(1/2), z = 2^(1/2)+I2^(1/2), m = 1}` `-.2248750758-.4411585330e-1I <- {a = 2^(1/2)+I2^(1/2), z = 2^(1/2)+I2^(1/2), m = 2}` `-.2248750795-.4411584875e-1I <- {a = 2^(1/2)+I2^(1/2), z = 2^(1/2)+I2^(1/2), m = 3}` `-1.005323136+.3326243216I <- {a = 2^(1/2)+I2^(1/2), z = 2^(1/2)-I2^(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(- PiIa)GAMMA(a, zexp(PiI))- exp(PiIa)GAMMA(a, zexp(- PiI))= -(2PiI)/(GAMMA(1 - a))`	`Exp[- PiIa]Gamma[a, zExp[PiI]]- Exp[PiIa]Gamma[a, zExp[- PiI]]= -Divide[2PiI,Gamma[1 - a]]`	Failure	Failure	Fail `-7167.292469-174.9289096I <- {a = 2^(1/2)+I2^(1/2), z = 2^(1/2)+I2^(1/2)}` `2.16987973+12.77160007I <- {a = 2^(1/2)+I2^(1/2), z = 2^(1/2)-I2^(1/2)}` `8.705606105-17.43270949I <- {a = 2^(1/2)+I2^(1/2), z = -2^(1/2)-I2^(1/2)}` `-4.50134822-89.91653387I <- {a = 2^(1/2)+I2^(1/2), z = -2^(1/2)+I2^(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, zexp(+ PiI))= GAMMA(a)(1 - (z)^(a) exp(+ PiIa)(- z)^(-(a))(GAMMA(a)-GAMMA(a, - z))/GAMMA(a))`	`Error`	Failure	Error	Fail `20.46249972+81.80630504I <- {a = 2^(1/2)+I2^(1/2), z = 2^(1/2)+I2^(1/2)}` `-1.005323138+.3326243220I <- {a = 2^(1/2)+I2^(1/2), z = -2^(1/2)+I2^(1/2)}` `1095.761010-111.2868863I <- {a = 2^(1/2)-I2^(1/2), z = 2^(1/2)+I2^(1/2)}` `-1231.554386+1108.053849I <- {a = 2^(1/2)-I2^(1/2), z = -2^(1/2)+I2^(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, zexp(- PiI))= GAMMA(a)(1 - (z)^(a) exp(- PiIa)(- z)^(-(a))(GAMMA(a)-GAMMA(a, - z))/GAMMA(a))`	`Error`	Failure	Error	Fail `1095.761010+111.2868863I <- {a = 2^(1/2)+I2^(1/2), z = 2^(1/2)-I2^(1/2)}` `-1231.554386-1108.053849I <- {a = 2^(1/2)+I2^(1/2), z = -2^(1/2)-I2^(1/2)}` `20.46249972-81.80630504I <- {a = 2^(1/2)-I2^(1/2), z = 2^(1/2)-I2^(1/2)}` `-1.005323138-.3326243220I <- {a = 2^(1/2)-I2^(1/2), z = -2^(1/2)-I2^(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-.3535533907I <- {a = 2^(1/2)+I2^(1/2), w = 2^(1/2)+I2^(1/2), z = 2^(1/2)+I2^(1/2)}` `.6464466093+.3535533907I <- {a = 2^(1/2)+I2^(1/2), w = 2^(1/2)+I2^(1/2), z = 2^(1/2)-I2^(1/2)}` `-.6464466093-.3535533907I <- {a = 2^(1/2)+I2^(1/2), w = 2^(1/2)+I2^(1/2), z = -2^(1/2)-I2^(1/2)}` `.6464466093+.3535533907I <- {a = 2^(1/2)+I2^(1/2), w = 2^(1/2)+I2^(1/2), z = -2^(1/2)+I2^(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	${\displaystyle{\displaystyle z\frac{{\mathrm{d}}^{2}\gamma^{}}{{\mathrm{d}z}^% {2}}+(a+1+z)\frac{\mathrm{d}\gamma^{}}{\mathrm{d}z}+a\gamma^{*}=0}}$	`zdiff((+)^(-(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	${\displaystyle{\displaystyle 2\int_{0}^{z}e^{-t^{2}}\mathrm{d}t=\sqrt{\pi}% \operatorname{erf}\left(z\right)}}$	`2int(exp(- (t)^(2)), t = 0..z)=sqrt(Pi)erf(z)`	`2Integrate[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+1.106937485I <- {a = 2^(1/2)+I2^(1/2)}` `-.6493698774-1.106937485I <- {a = 2^(1/2)-I2^(1/2)}` `4.564263782+2.639434666I <- {a = -2^(1/2)-I2^(1/2)}` `4.564263782-2.639434666I <- {a = -2^(1/2)+I2^(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))=(2exp((z)^(2)))/(zsqrt(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	${\displaystyle{\displaystyle 2\int_{z}^{\infty}e^{-t^{2}}\mathrm{d}t=\sqrt{\pi% }\operatorname{erfc}\left(z\right)}}$	`2int(exp(- (t)^(2)), t = z..infinity)=sqrt(Pi)erfc(z)`	`2Integrate[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)+I2^(1/2), n = 1}` `Float(undefined)+Float(undefined)I <- {z = 2^(1/2)+I2^(1/2), n = 2}` `Float(undefined)+Float(undefined)I <- {z = 2^(1/2)+I2^(1/2), n = 3}` `Float(undefined)+Float(undefined)I <- {z = 2^(1/2)-I2^(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)^(2k - 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)^(2k - 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(Pia))int(exp(zcos(t))cos(at + zsin(t)), t = 0..Pi)`	`Gamma[a, 0, z]=Divide[(z)^(a),Sin[Pia]]Integrate[Exp[zCos[t]]Cos[at + zSin[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, 2sqrt(zt)), t = 0..infinity)`	`Gamma[a, 0, z]= (z)^(Divide[1,2]a) Integrate[Exp[- t](t)^(Divide[1,2]a - 1)* BesselJ[a, 2Sqrt[zt]], {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(- at - zexp(- t)), t = 0..infinity)`	`Gamma[a, 0, z]= (z)^(a)* Integrate[Exp[- at - zExp[- 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(- zt))/((1 + t)^(1 - a)), t = 0..infinity)`	`Gamma[a, z]= (z)^(a)* Exp[- z]Integrate[Divide[Exp[- zt],(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, 2sqrt(zt)), 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, 2Sqrt[zt]], {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(at - zexp(t)), t = 0..infinity)`	`Gamma[a, z]= (z)^(a)* Integrate[Exp[at - zExp[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))/(2sin(Pia))int((t)^(a - 1)* exp(z*t), t = - 1..(0 +))`	`Gamma[a, 0, z]=Divide[- I(z)^(a),2Sin[Pia]]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, zexp(+ PiI))=(exp(z)exp(- PiIa))/(GAMMA(1 + a))int(((t)^(a)* exp(- z*t))/(t - 1), t = 0..infinity)`	`Gamma[- a, zExp[+ PiI]]=Divide[Exp[z]Exp[- PiIa],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, zexp(- PiI))=(exp(z)exp(+ PiIa))/(GAMMA(1 + a))int(((t)^(a)* exp(- z*t))/(t - 1), t = 0..infinity)`	`Gamma[- a, zExp[- PiI]]=Divide[Exp[z]Exp[+ PiIa],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)/(2PiI)int((GAMMA(s))/(a - s)(z)^(a - s), s = c - Iinfinity..c + Iinfinity)`	`Gamma[a, 0, z]=Divide[1,2PiI]Integrate[Divide[Gamma[s],a - s](z)^(a - s), {s, c - IInfinity, c + IInfinity}]`	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)/(2PiI)int(GAMMA(s + a)((z)^(- s))/(s), s = c - Iinfinity..c + Iinfinity)`	`Gamma[a, z]=Divide[1,2PiI]Integrate[Gamma[s + a]Divide[(z)^(- s),s], {s, c - IInfinity, c + IInfinity}]`	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)/(2PiI)int(GAMMA(s + 1 - a)(Pi(z)^(- s))/(sin(Pis)), s = c - Iinfinity..c + I*infinity)`	`Gamma[a, z]= -Divide[(z)^(a - 1)* Exp[- z],Gamma[1 - a]]Divide[1,2PiI]Integrate[Gamma[s + 1 - a]Divide[Pi(z)^(- s),Sin[Pis]], {s, c - IInfinity, 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)= aGAMMA(a)-GAMMA(a, z)- (z)^(a) exp(- z)`	`Gamma[a + 1, 0, z]= aGamma[a, 0, z]- (z)^(a) Exp[- z]`	Failure	Successful	Fail `.135004907e-1-.2375774782I <- {a = 2^(1/2)+I2^(1/2), z = 2^(1/2)+I2^(1/2)}` `.8693672828+.710002389I <- {a = 2^(1/2)+I2^(1/2), z = 2^(1/2)-I2^(1/2)}` `107.1902160-63.3824277I <- {a = 2^(1/2)+I2^(1/2), z = -2^(1/2)-I2^(1/2)}` `-.1657436948-.7422690683I <- {a = 2^(1/2)+I2^(1/2), z = -2^(1/2)+I2^(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)= aGAMMA(a, z)+ (z)^(a) exp(- z)`	`Gamma[a + 1, z]= aGamma[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	${\displaystyle{\displaystyle Q\left(a+n,z\right)=Q\left(a,z\right)+z^{a}e^{-z}% \sum_{k=0}^{n-1}\frac{z^{k}}{\Gamma\left(a+k+1\right)}}}$	`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}]`	Successful	Successful	-	-
8.8.E13	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\gamma\left(a,z% \right)=-\frac{\mathrm{d}}{\mathrm{d}z}\Gamma\left(a,z\right)}}$	`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	-	-
8.8.E13	${\displaystyle{\displaystyle-\frac{\mathrm{d}}{\mathrm{d}z}\Gamma\left(a,z% \right)=z^{a-1}e^{-z}}}$	`- diff(GAMMA(a, z), z)= (z)^(a - 1)* exp(- z)`	`- D[Gamma[a, z], z]= (z)^(a - 1)* Exp[- z]`	Successful	Successful	-	-
8.8.E15	${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}(z^{-a}% \gamma\left(a,z\right))=(-1)^{n}z^{-a-n}\gamma\left(a+n,z\right)}}$	`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	Skip	Skip
8.8.E16	${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}(z^{-a}% \Gamma\left(a,z\right))=(-1)^{n}z^{-a-n}\Gamma\left(a+n,z\right)}}$	`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	Skip	Skip
8.8.E17	${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}(e^{z}% \gamma\left(a,z\right))=(-1)^{n}{\left(1-a\right)_{n}}e^{z}\gamma\left(a-n,z% \right)}}$	`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	Skip	Successful
8.8.E18	${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}(z^{a}e^% {z}\gamma^{}\left(a,z\right))=z^{a-n}e^{z}\gamma^{}\left(a-n,z\right)}}$	`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	Error	Skip	-
8.8.E19	${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}(e^{z}% \Gamma\left(a,z\right))=(-1)^{n}{\left(1-a\right)_{n}}e^{z}\Gamma\left(a-n,z% \right)}}$	`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	Skip	Skip
8.10.E1	${\displaystyle{\displaystyle x^{1-a}e^{x}\Gamma\left(a,x\right)<=1}}$	`(x)^(1 - a)* exp(x)*GAMMA(a, x)< = 1`	`(x)^(1 - a)* Exp[x]*Gamma[a, x]< = 1`	Failure	Failure	Skip	Successful
8.10.E2	${\displaystyle{\displaystyle\gamma\left(a,x\right)>=\frac{x^{a-1}}{a}(1-e^{-x}% )}}$	`GAMMA(a)-GAMMA(a, x)> =((x)^(a - 1))/(a)*(1 - exp(- x))`	`Gamma[a, 0, x]> =Divide[(x)^(a - 1),a]*(1 - Exp[- x])`	Failure	Failure	Skip	Successful
8.10.E3	${\displaystyle{\displaystyle x^{1-a}e^{x}\Gamma\left(a,x\right)=1+\frac{a-1}{x% }\vartheta}}$	`(x)^(1 - a)* exp(x)GAMMA(a, x)= 1 +(a - 1)/(x)vartheta`	`(x)^(1 - a)* Exp[x]Gamma[a, x]= 1 +Divide[a - 1,x]\[CurlyTheta]`	Failure	Failure	Fail `1.052938223-1.733408016I <- {a = 2^(1/2)+I2^(1/2), vartheta = 2^(1/2)+I2^(1/2), x = 1}` `.6195824495-.7346525318I <- {a = 2^(1/2)+I2^(1/2), vartheta = 2^(1/2)+I2^(1/2), x = 2}` `.4531580595-.4544327802I <- {a = 2^(1/2)+I2^(1/2), vartheta = 2^(1/2)+I2^(1/2), x = 3}` `-2.947061775-.5618351419I <- {a = 2^(1/2)+I2^(1/2), vartheta = 2^(1/2)-I2^(1/2), x = 1}` ... skip entries to safe data	Fail `Complex[1.0529382235611282, -1.733408017034722] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 1], Rule[ϑ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}` `Complex[0.6195824493248067, -0.7346525326366091] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 2], Rule[ϑ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}` `Complex[0.4531580595377106, -0.4544327806624232] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 3], Rule[ϑ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}` `Complex[-2.947061776438873, -0.5618351417809119] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 1], Rule[ϑ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}` ... skip entries to safe data
8.10.E5	${\displaystyle{\displaystyle A_{n}<x^{1-a}e^{x}\Gamma\left(a,x\right)}}$	`A[n]< (x)^(1 - a)* exp(x)*GAMMA(a, x)`	`Subscript[A, n]< (x)^(1 - a)* Exp[x]*Gamma[a, x]`	Failure	Failure	Successful	Successful
8.10.E5	${\displaystyle{\displaystyle x^{1-a}e^{x}\Gamma\left(a,x\right)<B_{n}}}$	`(x)^(1 - a)* exp(x)*GAMMA(a, x)< B[n]`	`(x)^(1 - a)* Exp[x]*Gamma[a, x]< Subscript[B, n]`	Failure	Failure	Successful	Successful
8.10.E7	${\displaystyle{\displaystyle I=\int_{0}^{x}t^{a-1}e^{t}\mathrm{d}t}}$	`I = int((t)^(a - 1)* exp(t), t = 0..x)`	`I = Integrate[(t)^(a - 1)* Exp[t], {t, 0, x}]`	Failure	Failure	Skip	Fail `Complex[-2.925303491814363, 1.0] <- {Rule[a, Rational[1, 2]], Rule[x, 1]}` `Complex[-6.687685525621974, 1.0000000000000002] <- {Rule[a, Rational[1, 2]], Rule[x, 2]}` `Complex[-14.626171384019093, 1.0000000000000007] <- {Rule[a, Rational[1, 2]], Rule[x, 3]}`
8.10.E7	${\displaystyle{\displaystyle\int_{0}^{x}t^{a-1}e^{t}\mathrm{d}t=\Gamma\left(a% \right)x^{a}\gamma^{*}\left(a,-x\right)}}$	`int((t)^(a - 1)* exp(t), t = 0..x)= GAMMA(a)(x)^(a) (- x)^(-(a))*(GAMMA(a)-GAMMA(a, - x))/GAMMA(a)`	`Error`	Failure	Error	Skip	-
8.10#Ex5	${\displaystyle{\displaystyle c_{a}=(\Gamma\left(1+a\right))^{1/(a-1)}}}$	`c[a]=(GAMMA(1 + a))^(1/(a - 1))`	`Subscript[c, a]=(Gamma[1 + a])^(1/(a - 1))`	Failure	Failure	Fail `-.342222950+.7512982152I <- {a = 2^(1/2)+I2^(1/2), c[a] = 2^(1/2)+I2^(1/2)}` `-.342222950-2.077128909I <- {a = 2^(1/2)+I2^(1/2), c[a] = 2^(1/2)-I2^(1/2)}` `-3.170650074-2.077128909I <- {a = 2^(1/2)+I2^(1/2), c[a] = -2^(1/2)-I2^(1/2)}` `-3.170650074+.7512982152I <- {a = 2^(1/2)+I2^(1/2), c[a] = -2^(1/2)+I2^(1/2)}` ... skip entries to safe data	Successful
8.10#Ex6	${\displaystyle{\displaystyle d_{a}=(\Gamma\left(1+a\right))^{-1/a}}}$	`d[a]=(GAMMA(1 + a))^(- 1/ a)`	`Subscript[d, a]=(Gamma[1 + a])^(- 1/ a)`	Failure	Failure	Fail `.7353701374+1.747162536I <- {a = 2^(1/2)+I2^(1/2), d[a] = 2^(1/2)+I2^(1/2)}` `.7353701374-1.081264588I <- {a = 2^(1/2)+I2^(1/2), d[a] = 2^(1/2)-I2^(1/2)}` `-2.093056987-1.081264588I <- {a = 2^(1/2)+I2^(1/2), d[a] = -2^(1/2)-I2^(1/2)}` `-2.093056987+1.747162536I <- {a = 2^(1/2)+I2^(1/2), d[a] = -2^(1/2)+I2^(1/2)}` ... skip entries to safe data	Successful
8.10.E10	${\displaystyle{\displaystyle\frac{x}{2a}\left(\left(1+\frac{2}{x}\right)^{a}-1% \right)<x^{1-a}e^{x}\Gamma\left(a,x\right)}}$	`(x)/(2a)((1 +(2)/(x))^(a)- 1)< (x)^(1 - a)* exp(x)*GAMMA(a, x)`	`Divide[x,2a]((1 +Divide[2,x])^(a)- 1)< (x)^(1 - a)* Exp[x]*Gamma[a, x]`	Failure	Failure	Successful	Successful
8.10.E10	${\displaystyle{\displaystyle x^{1-a}e^{x}\Gamma\left(a,x\right)<=\frac{x}{ac_{% a}}\left(\left(1+\frac{c_{a}}{x}\right)^{a}-1\right)}}$	`(x)^(1 - a)* exp(x)GAMMA(a, x)< =(x)/(ac[a])*((1 +(c[a])/(x))^(a)- 1)`	`(x)^(1 - a)* Exp[x]Gamma[a, x]< =Divide[x,aSubscript[c, a]]*((1 +Divide[Subscript[c, a],x])^(a)- 1)`	Failure	Failure	Successful	Successful
8.10.E11	${\displaystyle{\displaystyle(1-e^{-\alpha_{a}x})^{a}<=P\left(a,x\right)}}$	`(1 - exp(- alpha[a]*x))^(a)< = (GAMMA(a)-GAMMA(a, x))/GAMMA(a)`	`(1 - Exp[- Subscript[\[Alpha], a]*x])^(a)< = GammaRegularized[a, 0, x]`	Failure	Failure	Successful	Successful
8.10.E11	${\displaystyle{\displaystyle P\left(a,x\right)<=(1-e^{-\beta_{a}x})^{a}}}$	`(GAMMA(a)-GAMMA(a, x))/GAMMA(a)< =(1 - exp(- beta[a]*x))^(a)`	`GammaRegularized[a, 0, x]< =(1 - Exp[- Subscript[\[Beta], a]*x])^(a)`	Failure	Failure	Successful	Successful
8.10.E13	${\displaystyle{\displaystyle\frac{\Gamma\left(n,n\right)}{\Gamma\left(n\right)% }<\frac{1}{2}}}$	`(GAMMA(n, n))/(GAMMA(n))<(1)/(2)`	`Divide[Gamma[n, n],Gamma[n]]<Divide[1,2]`	Failure	Failure	Successful	Successful
8.10.E13	${\displaystyle{\displaystyle\frac{1}{2}<\frac{\Gamma\left(n,n-1\right)}{\Gamma% \left(n\right)}}}$	`(1)/(2)<(GAMMA(n, n - 1))/(GAMMA(n))`	`Divide[1,2]<Divide[Gamma[n, n - 1],Gamma[n]]`	Failure	Failure	Successful	Successful
8.11.E2	${\displaystyle{\displaystyle\Gamma\left(a,z\right)=z^{a-1}e^{-z}\left(\sum_{k=% 0}^{n-1}\frac{u_{k}}{z^{k}}+R_{n}(a,z)\right)}}$	`GAMMA(a, z)= (z)^(a - 1)* exp(- z)(sum((u[k])/((z)^(k)), k = 0..n - 1)+ R[n](a , z))`	`Gamma[a, z]= (z)^(a - 1)* Exp[- z](Sum[Divide[Subscript[u, k],(z)^(k)], {k, 0, n - 1}]+ Subscript[R, n](a , z))`	Failure	Failure	Skip	Error
8.11.E4	${\displaystyle{\displaystyle\gamma\left(a,z\right)=z^{a}e^{-z}\sum_{k=0}^{% \infty}\frac{z^{k}}{{\left(a\right)_{k+1}}}}}$	`GAMMA(a)-GAMMA(a, z)= (z)^(a)* exp(- z)*sum(((z)^(k))/(pochhammer(a, k + 1)), k = 0..infinity)`	`Gamma[a, 0, z]= (z)^(a)* Exp[- z]*Sum[Divide[(z)^(k),Pochhammer[a, k + 1]], {k, 0, Infinity}]`	Successful	Successful	-	-
8.11.E15	${\displaystyle{\displaystyle S_{n}(x)=\frac{\gamma\left(n+1,nx\right)}{(nx)^{n% }e^{-nx}}}}$	`S[n](x)=(GAMMA(n + 1)-GAMMA(n + 1, nx))/((nx)^(n) exp(- n*x))`	`Subscript[S, n](x)=Divide[Gamma[n + 1, 0, nx],(nx)^(n) Exp[- n*x]]`	Failure	Failure	Fail `.6959317335+1.414213562I <- {S[n] = 2^(1/2)+I2^(1/2), n = 1, x = 1}` `.633899074+2.828427124I <- {S[n] = 2^(1/2)+I2^(1/2), n = 1, x = 2}` `-1.119204955+4.242640686I <- {S[n] = 2^(1/2)+I2^(1/2), n = 1, x = 3}` `.219685512+1.414213562I <- {S[n] = 2^(1/2)+I2^(1/2), n = 2, x = 1}` ... skip entries to safe data	Successful
8.12.E3	${\displaystyle{\displaystyle P\left(a,z\right)=\tfrac{1}{2}\operatorname{erfc}% \left(-\eta\sqrt{a/2}\right)-S(a,\eta)}}$	`(GAMMA(a)-GAMMA(a, z))/GAMMA(a)=(1)/(2)erfc(- etasqrt(a/ 2))- S*(a , eta)`	`GammaRegularized[a, 0, z]=Divide[1,2]Erfc[- \[Eta]Sqrt[a/ 2]]- S*(a , \[Eta])`	Failure	Failure	Error	Error
8.12.E4	${\displaystyle{\displaystyle Q\left(a,z\right)=\tfrac{1}{2}\operatorname{erfc}% \left(\eta\sqrt{a/2}\right)+S(a,\eta)}}$	`GAMMA(a, z)/GAMMA(a)=(1)/(2)erfc(etasqrt(a/ 2))+ S*(a , eta)`	`GammaRegularized[a, z]=Divide[1,2]Erfc[\[Eta]Sqrt[a/ 2]]+ S*(a , \[Eta])`	Failure	Failure	Error	Error
8.12.E5	${\displaystyle{\displaystyle\frac{e^{+\pi ia}}{2i\sin\left(\pi a\right)}Q\left% (-a,ze^{+\pi i}\right)=+\tfrac{1}{2}\operatorname{erfc}\left(+i\eta\sqrt{a/2}% \right)-iT(a,\eta)}}$	`(exp(+ PiIa))/(2Isin(Pia))GAMMA(- a, zexp(+ PiI))/GAMMA(- a)= +(1)/(2)erfc(+ Ietasqrt(a/ 2))- IT*(a , eta)`	`Divide[Exp[+ PiIa],2ISin[Pia]]GammaRegularized[- a, zExp[+ PiI]]= +Divide[1,2]Erfc[+ I\[Eta]Sqrt[a/ 2]]- IT*(a , \[Eta])`	Failure	Failure	Error	Error
8.12.E5	${\displaystyle{\displaystyle\frac{e^{-\pi ia}}{2i\sin\left(\pi a\right)}Q\left% (-a,ze^{-\pi i}\right)=-\tfrac{1}{2}\operatorname{erfc}\left(-i\eta\sqrt{a/2}% \right)-iT(a,\eta)}}$	`(exp(- PiIa))/(2Isin(Pia))GAMMA(- a, zexp(- PiI))/GAMMA(- a)= -(1)/(2)erfc(- Ietasqrt(a/ 2))- IT*(a , eta)`	`Divide[Exp[- PiIa],2ISin[Pia]]GammaRegularized[- a, zExp[- PiI]]= -Divide[1,2]Erfc[- I\[Eta]Sqrt[a/ 2]]- IT*(a , \[Eta])`	Failure	Failure	Error	Error
8.12#Ex5	${\displaystyle{\displaystyle\Gamma\left(a+1\right)\frac{e^{+\pi ia}}{2\pi i}% \Gamma\left(-a,ze^{+\pi i}\right)=-\tfrac{1}{2}\operatorname{erfc}\left(+i\eta% \sqrt{a/2}\right)+iT(a,\eta)}}$	`GAMMA(a + 1)(exp(+ PiIa))/(2PiI)GAMMA(- a, zexp(+ PiI))= -(1)/(2)erfc(+ Ietasqrt(a/ 2))+ IT*(a , eta)`	`Gamma[a + 1]Divide[Exp[+ PiIa],2PiI]Gamma[- a, zExp[+ PiI]]= -Divide[1,2]Erfc[+ I\[Eta]Sqrt[a/ 2]]+ IT*(a , \[Eta])`	Failure	Failure	Error	Error
8.12#Ex5	${\displaystyle{\displaystyle\Gamma\left(a+1\right)\frac{e^{-\pi ia}}{2\pi i}% \Gamma\left(-a,ze^{-\pi i}\right)=+\tfrac{1}{2}\operatorname{erfc}\left(-i\eta% \sqrt{a/2}\right)+iT(a,\eta)}}$	`GAMMA(a + 1)(exp(- PiIa))/(2PiI)GAMMA(- a, zexp(- PiI))= +(1)/(2)erfc(- Ietasqrt(a/ 2))+ IT*(a , eta)`	`Gamma[a + 1]Divide[Exp[- PiIa],2PiI]Gamma[- a, zExp[- PiI]]= +Divide[1,2]Erfc[- I\[Eta]Sqrt[a/ 2]]+ IT*(a , \[Eta])`	Failure	Failure	Error	Error
8.12.E6	${\displaystyle{\displaystyle z^{-a}\gamma^{*}\left(-a,-z\right)=\cos\left(\pi a% \right)-2\sin\left(\pi a\right)\left(\frac{e^{\frac{1}{2}a\eta^{2}}}{\sqrt{\pi% }}F\left(\eta\sqrt{a/2}\right)+T(a,\eta)\right)}}$	`(z)^(- a)* (- z)^(-(- a))(GAMMA(- a)-GAMMA(- a, - z))/GAMMA(- a)= cos(Pia)- 2sin(Pia)((exp((1)/(2)a(eta)^(2)))/(sqrt(Pi))dawson(etasqrt(a/ 2))+ T(a , eta))`	`Error`	Failure	Error	Error	-
8.12.E10	${\displaystyle{\displaystyle c_{k}(\eta)=\frac{1}{\eta}\frac{\mathrm{d}}{% \mathrm{d}\eta}c_{k-1}(\eta)+(-1)^{k}\frac{g_{k}}{\mu}}}$	`c[k](eta)=(1)/(eta)diff(c[k - 1](eta), eta)+(- 1)^(k)(g[k])/(mu)`	`Subscript[c, k](\[Eta])=Divide[1,\[Eta]]D[Subscript[c, k - 1](\[Eta]), \[Eta]]+(- 1)^(k)Divide[Subscript[g, k],\[Mu]]`	Failure	Failure	Skip	Skip
8.12#Ex23	${\displaystyle{\displaystyle d(+\chi)=\sqrt{\tfrac{1}{2}\pi}e^{\chi^{2}/2}% \operatorname{erfc}\left(+\chi/\sqrt{2}\right)}}$	`d(+ chi)=sqrt((1)/(2)Pi)exp((chi)^(2)/ 2)erfc(+ chi/sqrt(2))`	`d(+ \[Chi])=Sqrt[Divide[1,2]Pi]Exp[(\[Chi])^(2)/ 2]Erfc[+ \[Chi]/Sqrt[2]]`	Failure	Failure	Fail `-.3819402210+4.260963736I <- {chi = 2^(1/2)+I2^(1/2), d = 2^(1/2)+I2^(1/2)}` `3.618059777+.2609637385I <- {chi = 2^(1/2)+I2^(1/2), d = 2^(1/2)-I2^(1/2)}` `-.3819402210-3.739036260I <- {chi = 2^(1/2)+I2^(1/2), d = -2^(1/2)-I2^(1/2)}` `-4.381940219+.2609637385I <- {chi = 2^(1/2)+I2^(1/2), d = -2^(1/2)+I2^(1/2)}` ... skip entries to safe data	Fail `Complex[-0.3819402207134648, 4.260963738906431] <- {Rule[d, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[χ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}` `Complex[3.6180597792865354, -0.26096373890643143] <- {Rule[d, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[χ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}` `Complex[1.425065647600777, -6.540234379036898] <- {Rule[d, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[χ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}` `Complex[-2.574934352399223, 2.5402343790368977] <- {Rule[d, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[χ, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}` ... skip entries to safe data
8.12#Ex23	${\displaystyle{\displaystyle d(-\chi)=\sqrt{\tfrac{1}{2}\pi}e^{\chi^{2}/2}% \operatorname{erfc}\left(-\chi/\sqrt{2}\right)}}$	`d(- chi)=sqrt((1)/(2)Pi)exp((chi)^(2)/ 2)erfc(- chi/sqrt(2))`	`d(- \[Chi])=Sqrt[Divide[1,2]Pi]Exp[(\[Chi])^(2)/ 2]Erfc[- \[Chi]/Sqrt[2]]`	Failure	Failure	Fail `1.425065646-6.540234377I <- {chi = 2^(1/2)+I2^(1/2), d = 2^(1/2)+I2^(1/2)}` `-2.574934352-2.540234379I <- {chi = 2^(1/2)+I2^(1/2), d = 2^(1/2)-I2^(1/2)}` `1.425065646+1.459765619I <- {chi = 2^(1/2)+I2^(1/2), d = -2^(1/2)-I2^(1/2)}` `5.425065644-2.540234379I <- {chi = 2^(1/2)+I2^(1/2), d = -2^(1/2)+I2^(1/2)}` ... skip entries to safe data	Fail `Complex[1.425065647600777, -6.540234379036898] <- {Rule[d, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[χ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}` `Complex[-2.574934352399223, 2.5402343790368977] <- {Rule[d, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[χ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}` `Complex[-0.3819402207134648, 4.260963738906431] <- {Rule[d, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[χ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}` `Complex[3.6180597792865354, -0.26096373890643143] <- {Rule[d, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[χ, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}` ... skip entries to safe data
8.12.E21	${\displaystyle{\displaystyle Q\left(a,x\right)=q}}$	`GAMMA(a, x)/GAMMA(a)= q`	`GammaRegularized[a, x]= q`	Failure	Failure	Fail `-.6276752047-.7874152397I <- {a = 2^(1/2)+I2^(1/2), q = 2^(1/2)+I2^(1/2), x = 1}` `-1.269609688-.9406490460I <- {a = 2^(1/2)+I2^(1/2), q = 2^(1/2)+I2^(1/2), x = 2}` `-1.440152063-1.201678512I <- {a = 2^(1/2)+I2^(1/2), q = 2^(1/2)+I2^(1/2), x = 3}` `-.6276752047+2.041011884I <- {a = 2^(1/2)+I2^(1/2), q = 2^(1/2)-I2^(1/2), x = 1}` ... skip entries to safe data	Fail `Complex[-0.6276752046971461, -0.7874152400294763] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[q, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 1]}` `Complex[-1.2696096879275383, -0.9406490461902074] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[q, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 2]}` `Complex[-1.4401520638257446, -1.2016785120794473] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[q, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 3]}` `Complex[-0.6276752046971461, 2.0410118847167142] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[q, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[x, 1]}` ... skip entries to safe data
8.13.E1	${\displaystyle{\displaystyle 1+a^{-1}<x_{-}(a)}}$	`1 + (a)^(- 1)< x[-]*(a)`	`1 + (a)^(- 1)< Subscript[x, -]*(a)`	Error	Failure	-	Error
8.13.E1	${\displaystyle{\displaystyle x_{-}(a)<\ln\|a\|}}$	`x[-]*(a)< ln(abs(a))`	`Subscript[x, -]*(a)< Log[Abs[a]]`	Error	Failure	-	Error
8.14.E1	${\displaystyle{\displaystyle\int_{0}^{\infty}e^{-ax}\frac{\gamma\left(b,x% \right)}{\Gamma\left(b\right)}\mathrm{d}x=\frac{(1+a)^{-b}}{a}}}$	`int(exp(- ax)(GAMMA(b)-GAMMA(b, x))/(GAMMA(b)), x = 0..infinity)=((1 + a)^(- b))/(a)`	`Integrate[Exp[- ax]Divide[Gamma[b, 0, x],Gamma[b]], {x, 0, Infinity}]=Divide[(1 + a)^(- b),a]`	Successful	Failure	-	Error
8.14.E2	${\displaystyle{\displaystyle\int_{0}^{\infty}e^{-ax}\Gamma\left(b,x\right)% \mathrm{d}x=\Gamma\left(b\right)\frac{1-(1+a)^{-b}}{a}}}$	`int(exp(- ax)GAMMA(b, x), x = 0..infinity)= GAMMA(b)*(1 -(1 + a)^(- b))/(a)`	`Integrate[Exp[- ax]Gamma[b, x], {x, 0, Infinity}]= Gamma[b]*Divide[1 -(1 + a)^(- b),a]`	Failure	Failure	Skip	Error
8.14.E3	${\displaystyle{\displaystyle\int_{0}^{\infty}x^{a-1}\gamma\left(b,x\right)% \mathrm{d}x=-\frac{\Gamma\left(a+b\right)}{a}}}$	`int((x)^(a - 1)* GAMMA(b)-GAMMA(b, x), x = 0..infinity)= -(GAMMA(a + b))/(a)`	`Integrate[(x)^(a - 1)* Gamma[b, 0, x], {x, 0, Infinity}]= -Divide[Gamma[a + b],a]`	Failure	Failure	Skip	Error
8.14.E4	${\displaystyle{\displaystyle\int_{0}^{\infty}x^{a-1}\Gamma\left(b,x\right)% \mathrm{d}x=\frac{\Gamma\left(a+b\right)}{a}}}$	`int((x)^(a - 1)* GAMMA(b, x), x = 0..infinity)=(GAMMA(a + b))/(a)`	`Integrate[(x)^(a - 1)* Gamma[b, x], {x, 0, Infinity}]=Divide[Gamma[a + b],a]`	Successful	Failure	-	Skip
8.14.E5	${\displaystyle{\displaystyle\int_{0}^{\infty}x^{a-1}e^{-sx}\gamma\left(b,x% \right)\mathrm{d}x=\frac{\Gamma\left(a+b\right)}{b(1+s)^{a+b}}\*F\left(1,a+b;1% +b;1/(1+s)\right)}}$	`int((x)^(a - 1)* exp(- sx)GAMMA(b)-GAMMA(b, x), x = 0..infinity)=(GAMMA(a + b))/(b(1 + s)^(a + b)) hypergeom([1, a + b], [1 + b], 1/(1 + s))`	`Integrate[(x)^(a - 1)* Exp[- sx]Gamma[b, 0, x], {x, 0, Infinity}]=Divide[Gamma[a + b],b(1 + s)^(a + b)] Hypergeometric2F1[1, a + b, 1 + b, 1/(1 + s)]`	Failure	Failure	Skip	Error
8.14.E6	${\displaystyle{\displaystyle\int_{0}^{\infty}x^{a-1}e^{-sx}\Gamma\left(b,x% \right)\mathrm{d}x=\frac{\Gamma\left(a+b\right)}{a(1+s)^{a+b}}\*F\left(1,a+b;1% +a;s/(1+s)\right)}}$	`int((x)^(a - 1)* exp(- sx)GAMMA(b, x), x = 0..infinity)=(GAMMA(a + b))/(a(1 + s)^(a + b)) hypergeom([1, a + b], [1 + a], s/(1 + s))`	`Integrate[(x)^(a - 1)* Exp[- sx]Gamma[b, x], {x, 0, Infinity}]=Divide[Gamma[a + b],a(1 + s)^(a + b)] Hypergeometric2F1[1, a + b, 1 + a, s/(1 + s)]`	Failure	Failure	Skip	Error
8.15.E1	${\displaystyle{\displaystyle\gamma\left(a,\lambda x\right)=\lambda^{a}\sum_{k=% 0}^{\infty}\gamma\left(a+k,x\right)\frac{(1-\lambda)^{k}}{k!}}}$	`GAMMA(a)-GAMMA(a, lambdax)= (lambda)^(a) sum(GAMMA(a + k)-GAMMA(a + k, x)*((1 - lambda)^(k))/(factorial(k)), k = 0..infinity)`	`Gamma[a, 0, \[Lambda]x]= (\[Lambda])^(a) Sum[Gamma[a + k, 0, x]*Divide[(1 - \[Lambda])^(k),(k)!], {k, 0, Infinity}]`	Failure	Failure	Skip	Skip
8.17.E1	${\displaystyle{\displaystyle\mathrm{B}_{x}\left(a,b\right)=\int_{0}^{x}t^{a-1}% (1-t)^{b-1}\mathrm{d}t}}$	`int(t^(a-1)(1-t)^(b-1), t = 0 .. x)= int((t)^(a - 1)(1 - t)^(b - 1), t = 0..x)`	`Beta[x, a, b]= Integrate[(t)^(a - 1)*(1 - t)^(b - 1), {t, 0, x}]`	Successful	Failure	-	Skip
8.17.E2	${\displaystyle{\displaystyle I_{x}\left(a,b\right)=\mathrm{B}_{x}\left(a,b% \right)/\mathrm{B}\left(a,b\right)}}$	`Error`	`BetaRegularized[x, a, b]= Beta[x, a, b]/ Beta[a, b]`	Error	Successful	-	-
8.17.E3	${\displaystyle{\displaystyle\mathrm{B}\left(a,b\right)=\frac{\Gamma\left(a% \right)\Gamma\left(b\right)}{\Gamma\left(a+b\right)}}}$	`Beta(a, b)=(GAMMA(a)*GAMMA(b))/(GAMMA(a + b))`	`Beta[a, b]=Divide[Gamma[a]*Gamma[b],Gamma[a + b]]`	Failure	Successful	Error	-
8.17.E4	${\displaystyle{\displaystyle I_{x}\left(a,b\right)=1-I_{1-x}\left(b,a\right)}}$	`Error`	`BetaRegularized[x, a, b]= 1 - BetaRegularized[1 - x, b, a]`	Error	Failure	-	Fail `DirectedInfinity[] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[x, 1]}` `DirectedInfinity[] <- {Rule[a, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[x, 1]}` `DirectedInfinity[] <- {Rule[a, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[x, 1]}` `DirectedInfinity[] <- {Rule[a, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[x, 1]}` ... skip entries to safe data
8.17.E5	${\displaystyle{\displaystyle I_{x}\left(m,n-m+1\right)=\sum_{j=m}^{n}\genfrac{% (}{)}{0.0pt}{}{n}{j}x^{j}(1-x)^{n-j}}}$	`Error`	`BetaRegularized[x, m, n - m + 1]= Sum[Binomial[n,j](x)^(j)(1 - x)^(n - j), {j, m, n}]`	Error	Failure	-	Successful
8.17.E6	${\displaystyle{\displaystyle I_{x}\left(a,a\right)=\tfrac{1}{2}I_{4x(1-x)}% \left(a,\tfrac{1}{2}\right)}}$	`Error`	`BetaRegularized[x, a, a]=Divide[1,2]BetaRegularized[4x*(1 - x), a, Divide[1,2]]`	Error	Failure	-	Successful
8.17.E7	${\displaystyle{\displaystyle\mathrm{B}_{x}\left(a,b\right)=\frac{x^{a}}{a}F% \left(a,1-b;a+1;x\right)}}$	`int(t^(a-1)(1-t)^(b-1), t = 0 .. x)=((x)^(a))/(a)hypergeom([a, 1 - b], [a + 1], x)`	`Beta[x, a, b]=Divide[(x)^(a),a]*Hypergeometric2F1[a, 1 - b, a + 1, x]`	Failure	Successful	Skip	-
8.17.E8	${\displaystyle{\displaystyle\mathrm{B}_{x}\left(a,b\right)=\frac{x^{a}(1-x)^{b% }}{a}F\left(a+b,1;a+1;x\right)}}$	`int(t^(a-1)(1-t)^(b-1), t = 0 .. x)=((x)^(a)(1 - x)^(b))/(a)*hypergeom([a + b, 1], [a + 1], x)`	`Beta[x, a, b]=Divide[(x)^(a)(1 - x)^(b),a]Hypergeometric2F1[a + b, 1, a + 1, x]`	Failure	Successful	Skip	-
8.17.E9	${\displaystyle{\displaystyle\mathrm{B}_{x}\left(a,b\right)=\frac{x^{a}(1-x)^{b% -1}}{a}F\left({1,1-b\atop a+1};\frac{x}{x-1}\right)}}$	`int(t^(a-1)(1-t)^(b-1), t = 0 .. x)=((x)^(a)(1 - x)^(b - 1))/(a)*hypergeom([1, 1 - b], [a + 1], (x)/(x - 1))`	`Beta[x, a, b]=Divide[(x)^(a)(1 - x)^(b - 1),a]Hypergeometric2F1[1, 1 - b, a + 1, Divide[x,x - 1]]`	Failure	Failure	Skip	Fail `Complex[-0.27132901967319506, -0.2500814455005845] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 1]}` `Complex[-0.27137899275582306, -0.250091870275464] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 2]}` `Complex[-0.27137899275582306, -0.250091870275464] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 3]}` `Complex[0.08838841600311584, 0.0] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[x, 1]}` ... skip entries to safe data
8.17.E10	${\displaystyle{\displaystyle I_{x}\left(a,b\right)=\frac{x^{a}(1-x)^{b}}{2\pi i% }\int_{c-i\infty}^{c+i\infty}s^{-a}(1-s)^{-b}\frac{\mathrm{d}s}{s-x}}}$	`Error`	`BetaRegularized[x, a, b]=Divide[(x)^(a)(1 - x)^(b),2PiI]Integrate[(s)^(- a)(1 - s)^(- b)Divide[1,s - x], {s, c - IInfinity, c + IInfinity}]`	Error	Failure	-	Error
8.17.E13	${\displaystyle{\displaystyle(a+b)I_{x}\left(a,b\right)=aI_{x}\left(a+1,b\right% )+bI_{x}\left(a,b+1\right)}}$	`Error`	`(a + b)* BetaRegularized[x, a, b]= aBetaRegularized[x, a + 1, b]+ bBetaRegularized[x, a, b + 1]`	Error	Successful	-	-
8.17.E14	${\displaystyle{\displaystyle(a+bx)I_{x}\left(a,b\right)=xbI_{x}\left(a-1,b+1% \right)+aI_{x}\left(a+1,b\right)}}$	`Error`	`(a + bx) BetaRegularized[x, a, b]= xbBetaRegularized[x, a - 1, b + 1]+ a*BetaRegularized[x, a + 1, b]`	Error	Successful	-	-
8.17.E16	${\displaystyle{\displaystyle aI_{x}\left(a+1,b\right)=(a+cx)I_{x}\left(a,b% \right)-cxI_{x}\left(a-1,b\right)}}$	`Error`	`aBetaRegularized[x, a + 1, b]=(a + cx)* BetaRegularized[x, a, b]- cxBetaRegularized[x, a - 1, b]`	Error	Failure	-	Skip
8.17.E24	${\displaystyle{\displaystyle I_{x}\left(m,n\right)=(1-x)^{n}\sum_{j=m}^{\infty% }\genfrac{(}{)}{0.0pt}{}{n+j-1}{j}x^{j}}}$	`Error`	`BetaRegularized[x, m, n]=(1 - x)^(n)* Sum[Binomial[n + j - 1,j]*(x)^(j), {j, m, Infinity}]`	Error	Failure	-	Successful
8.18.E2	${\displaystyle{\displaystyle\xi=-\ln x}}$	`xi = - ln(x)`	`\[Xi]= - Log[x]`	Failure	Failure	Fail `1.414213562+1.414213562I <- {xi = 2^(1/2)+I2^(1/2), x = 1}` `2.107360743+1.414213562I <- {xi = 2^(1/2)+I2^(1/2), x = 2}` `2.512825851+1.414213562I <- {xi = 2^(1/2)+I2^(1/2), x = 3}` `1.414213562-1.414213562I <- {xi = 2^(1/2)-I2^(1/2), x = 1}` ... skip entries to safe data	Fail `Complex[1.4142135623730951, 1.4142135623730951] <- {Rule[x, 1], Rule[ξ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}` `Complex[2.1073607429330403, 1.4142135623730951] <- {Rule[x, 2], Rule[ξ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}` `Complex[2.512825851041205, 1.4142135623730951] <- {Rule[x, 3], Rule[ξ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}` `Complex[1.4142135623730951, -1.4142135623730951] <- {Rule[x, 1], Rule[ξ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}` ... skip entries to safe data
8.18#Ex1	${\displaystyle{\displaystyle F_{0}=a^{-b}Q\left(b,a\xi\right)}}$	`F[0]= (a)^(- b)* GAMMA(b, a*xi)/GAMMA(b)`	`Subscript[F, 0]= (a)^(- b)* GammaRegularized[b, a*\[Xi]]`	Failure	Failure	Fail `2.106630597+.9392389431I <- {a = 2^(1/2)+I2^(1/2), b = 2^(1/2)+I2^(1/2), xi = 2^(1/2)+I2^(1/2), F[0] = 2^(1/2)+I2^(1/2)}` `2.106630597-1.889188181I <- {a = 2^(1/2)+I2^(1/2), b = 2^(1/2)+I2^(1/2), xi = 2^(1/2)+I2^(1/2), F[0] = 2^(1/2)-I2^(1/2)}` `-.7217965272-1.889188181I <- {a = 2^(1/2)+I2^(1/2), b = 2^(1/2)+I2^(1/2), xi = 2^(1/2)+I2^(1/2), F[0] = -2^(1/2)-I2^(1/2)}` `-.7217965272+.9392389431I <- {a = 2^(1/2)+I2^(1/2), b = 2^(1/2)+I2^(1/2), xi = 2^(1/2)+I2^(1/2), F[0] = -2^(1/2)+I2^(1/2)}` ... skip entries to safe data	Skip
8.18#Ex2	${\displaystyle{\displaystyle F_{1}=\frac{b-a\xi}{a}F_{0}+\frac{\xi^{b}e^{-a\xi% }}{a\Gamma\left(b\right)}}}$	`F[1]=(b - axi)/(a)F[0]+((xi)^(b)* exp(- axi))/(aGAMMA(b))`	`Subscript[F, 1]=Divide[b - a\[Xi],a]Subscript[F, 0]+Divide[(\[Xi])^(b)* Exp[- a\[Xi]],aGamma[b]]`	Failure	Failure	Skip	Skip
8.18.E10	${\displaystyle{\displaystyle-\tfrac{1}{2}\eta^{2}=x_{0}\ln\left(\frac{x}{x_{0}% }\right)+(1-x_{0})\ln\left(\frac{1-x}{1-x_{0}}\right)}}$	`-(1)/(2)(eta)^(2)= x[0]ln((x)/(x[0]))+(1 - x[0])* ln((1 - x)/(1 - x[0]))`	`-Divide[1,2](\[Eta])^(2)= Subscript[x, 0]Log[Divide[x,Subscript[x, 0]]]+(1 - Subscript[x, 0])* Log[Divide[1 - x,1 - Subscript[x, 0]]]`	Failure	Failure	Fail `Float(undefined)-Float(infinity)I <- {eta = 2^(1/2)+I2^(1/2), x[0] = 2^(1/2)+I2^(1/2), x = 1}` `.547175857-1.970232147I <- {eta = 2^(1/2)+I2^(1/2), x[0] = 2^(1/2)+I2^(1/2), x = 2}` `.260872566-1.563388258I <- {eta = 2^(1/2)+I2^(1/2), x[0] = 2^(1/2)+I2^(1/2), x = 3}` `Float(undefined)+Float(infinity)I <- {eta = 2^(1/2)+I2^(1/2), x[0] = 2^(1/2)-I2^(1/2), x = 1}` ... skip entries to safe data	Successful
8.18.E15	${\displaystyle{\displaystyle\mu\ln\zeta-\zeta=\ln x+\mu\ln\left(1-x\right)+(1+% \mu)\ln\left(1+\mu\right)-\mu}}$	`muln(zeta)- zeta = ln(x)+ muln(1 - x)+(1 + mu)* ln(1 + mu)- mu`	`\[Mu]Log[\[zeta]]- \[zeta]= Log[x]+ \[Mu]Log[1 - x]+(1 + \[Mu])* Log[1 + \[Mu]]- \[Mu]`	Failure	Failure	Fail `Float(infinity)+Float(infinity)I <- {mu = 2^(1/2)+I2^(1/2), zeta = 2^(1/2)+I2^(1/2), x = 1}` `1.884730882-5.086259752I <- {mu = 2^(1/2)+I2^(1/2), zeta = 2^(1/2)+I2^(1/2), x = 2}` `.499007630-6.066517895I <- {mu = 2^(1/2)+I2^(1/2), zeta = 2^(1/2)+I2^(1/2), x = 3}` `Float(infinity)+Float(infinity)I <- {mu = 2^(1/2)+I2^(1/2), zeta = 2^(1/2)-I2^(1/2), x = 1}` ... skip entries to safe data	Error
8.18.E18	${\displaystyle{\displaystyle I_{x}\left(a,b\right)=p}}$	`Error`	`BetaRegularized[x, a, b]= p`	Error	Failure	-	Successful
8.19.E1	${\displaystyle{\displaystyle E_{p}\left(z\right)=z^{p-1}\Gamma\left(1-p,z% \right)}}$	`Ei(p, z)= (z)^(p - 1)* GAMMA(1 - p, z)`	`ExpIntegralE[p, z]= (z)^(p - 1)* Gamma[1 - p, z]`	Successful	Successful	-	-
8.19.E2	${\displaystyle{\displaystyle E_{p}\left(z\right)=z^{p-1}\int_{z}^{\infty}\frac% {e^{-t}}{t^{p}}\mathrm{d}t}}$	`Ei(p, z)= (z)^(p - 1)* int((exp(- t))/((t)^(p)), t = z..infinity)`	`ExpIntegralE[p, z]= (z)^(p - 1)* Integrate[Divide[Exp[- t],(t)^(p)], {t, z, Infinity}]`	Successful	Failure	-	Skip
8.19.E3	${\displaystyle{\displaystyle E_{p}\left(z\right)=\int_{1}^{\infty}\frac{e^{-zt% }}{t^{p}}\mathrm{d}t}}$	`Ei(p, z)= int((exp(- z*t))/((t)^(p)), t = 1..infinity)`	`ExpIntegralE[p, z]= Integrate[Divide[Exp[- z*t],(t)^(p)], {t, 1, Infinity}]`	Successful	Failure	-	Error
8.19.E4	${\displaystyle{\displaystyle E_{p}\left(z\right)=\frac{z^{p-1}e^{-z}}{\Gamma% \left(p\right)}\int_{0}^{\infty}\frac{t^{p-1}e^{-zt}}{1+t}\mathrm{d}t}}$	`Ei(p, z)=((z)^(p - 1)* exp(- z))/(GAMMA(p))int(((t)^(p - 1) exp(- z*t))/(1 + t), t = 0..infinity)`	`ExpIntegralE[p, z]=Divide[(z)^(p - 1)* Exp[- z],Gamma[p]]Integrate[Divide[(t)^(p - 1) Exp[- z*t],1 + t], {t, 0, Infinity}]`	Successful	Failure	-	Error
8.19.E5	${\displaystyle{\displaystyle E_{0}\left(z\right)=z^{-1}e^{-z}}}$	`Ei(0, z)= (z)^(- 1)* exp(- z)`	`ExpIntegralE[0, z]= (z)^(- 1)* Exp[- z]`	Successful	Failure	-	Successful
8.19.E6	${\displaystyle{\displaystyle E_{p}\left(0\right)=\frac{1}{p-1}}}$	`Ei(p, 0)=(1)/(p - 1)`	`ExpIntegralE[p, 0]=Divide[1,p - 1]`	Successful	Successful	-	-
8.19.E7	${\displaystyle{\displaystyle E_{n}\left(z\right)=\frac{(-z)^{n-1}}{(n-1)!}E_{1% }\left(z\right)+\frac{e^{-z}}{(n-1)!}\sum_{k=0}^{n-2}(n-k-2)!(-z)^{k}}}$	`Ei(n, z)=((- z)^(n - 1))/(factorial(n - 1))Ei(z)+(exp(- z))/(factorial(n - 1))sum(factorial(n - k - 2)*(- z)^(k), k = 0..n - 2)`	`ExpIntegralE[n, z]=Divide[(- z)^(n - 1),(n - 1)!]-ExpIntegralEi[-(z)]+Divide[Exp[- z],(n - 1)!]Sum[(n - k - 2)!*(- z)^(k), {k, 0, n - 2}]`	Failure	Failure	Skip	Fail `Complex[0.0, -3.141592653589793] <- {Rule[n, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}` `Complex[-4.442882938158367, 4.442882938158366] <- {Rule[n, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}` `Complex[6.283185307179586, 0.0] <- {Rule[n, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}` `Complex[0.0, 3.141592653589793] <- {Rule[n, 1], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}` ... skip entries to safe data
8.19.E9	${\displaystyle{\displaystyle E_{n}\left(z\right)=\frac{(-1)^{n}z^{n-1}}{(n-1)!% }\ln z+\frac{e^{-z}}{(n-1)!}\sum_{k=1}^{n-1}(-z)^{k-1}\Gamma\left(n-k\right)+% \frac{e^{-z}(-z)^{n-1}}{(n-1)!}\sum_{k=0}^{\infty}\frac{z^{k}}{k!}\psi\left(k+% 1\right)}}$	`Ei(n, z)=((- 1)^(n)* (z)^(n - 1))/(factorial(n - 1))ln(z)+(exp(- z))/(factorial(n - 1))sum((- z)^(k - 1)* GAMMA(n - k), k = 1..n - 1)+(exp(- z)(- z)^(n - 1))/(factorial(n - 1))sum(((z)^(k))/(factorial(k))*Psi(k + 1), k = 0..infinity)`	`ExpIntegralE[n, z]=Divide[(- 1)^(n)* (z)^(n - 1),(n - 1)!]Log[z]+Divide[Exp[- z],(n - 1)!]Sum[(- z)^(k - 1)* Gamma[n - k], {k, 1, n - 1}]+Divide[Exp[- z](- z)^(n - 1),(n - 1)!]Sum[Divide[(z)^(k),(k)!]*PolyGamma[k + 1], {k, 0, Infinity}]`	Error	Failure	-	Successful
8.19.E10	${\displaystyle{\displaystyle E_{p}\left(z\right)=z^{p-1}\Gamma\left(1-p\right)% -\sum_{k=0}^{\infty}\frac{(-z)^{k}}{k!(1-p+k)}}}$	`Ei(p, z)= (z)^(p - 1)* GAMMA(1 - p)- sum(((- z)^(k))/(factorial(k)*(1 - p + k)), k = 0..infinity)`	`ExpIntegralE[p, z]= (z)^(p - 1)* Gamma[1 - p]- Sum[Divide[(- z)^(k),(k)!*(1 - p + k)], {k, 0, Infinity}]`	Successful	Successful	-	-
8.19.E11	${\displaystyle{\displaystyle E_{p}\left(z\right)=\Gamma\left(1-p\right)\left(z% ^{p-1}-e^{-z}\sum_{k=0}^{\infty}\frac{z^{k}}{\Gamma\left(2-p+k\right)}\right)}}$	`Ei(p, z)= GAMMA(1 - p)((z)^(p - 1)- exp(- z)sum(((z)^(k))/(GAMMA(2 - p + k)), k = 0..infinity))`	`ExpIntegralE[p, z]= Gamma[1 - p]((z)^(p - 1)- Exp[- z]Sum[Divide[(z)^(k),Gamma[2 - p + k]], {k, 0, Infinity}])`	Successful	Successful	-	-
8.19.E12	${\displaystyle{\displaystyle pE_{p+1}\left(z\right)+zE_{p}\left(z\right)=e^{-z% }}}$	`pEi(p + 1, z)+ zEi(p, z)= exp(- z)`	`pExpIntegralE[p + 1, z]+ zExpIntegralE[p, z]= Exp[- z]`	Successful	Successful	-	-
8.19.E13	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}E_{p}\left(z\right)=% -E_{p-1}\left(z\right)}}$	`diff(Ei(p, z), z)= - Ei(p - 1, z)`	`D[ExpIntegralE[p, z], z]= - ExpIntegralE[p - 1, z]`	Successful	Successful	-	-
8.19.E14	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}(e^{z}E_{p}\left(z% \right))=e^{z}E_{p}\left(z\right)\left(1+\frac{p-1}{z}\right)-\frac{1}{z}}}$	`diff(exp(z)Ei(p, z), z)= exp(z)Ei(p, z)*(1 +(p - 1)/(z))-(1)/(z)`	`D[Exp[z]ExpIntegralE[p, z], z]= Exp[z]ExpIntegralE[p, z]*(1 +Divide[p - 1,z])-Divide[1,z]`	Successful	Successful	-	-
8.19.E15	${\displaystyle{\displaystyle\frac{{\partial}^{j}E_{p}\left(z\right)}{{\partial p% }^{j}}=(-1)^{j}\int_{1}^{\infty}(\ln t)^{j}t^{-p}e^{-zt}\mathrm{d}t}}$	`diff(Ei(p, z), [p$(j)])=(- 1)^(j)* int((ln(t))^(j)* (t)^(- p)* exp(- z*t), t = 1..infinity)`	`D[ExpIntegralE[p, z], {p, j}]=(- 1)^(j)* Integrate[(Log[t])^(j)* (t)^(- p)* Exp[- z*t], {t, 1, Infinity}]`	Failure	Failure	Skip	Error
8.19.E16	${\displaystyle{\displaystyle E_{p}\left(z\right)=z^{p-1}e^{-z}U\left(p,p,z% \right)}}$	`Ei(p, z)= (z)^(p - 1)* exp(- z)*KummerU(p, p, z)`	`ExpIntegralE[p, z]= (z)^(p - 1)* Exp[- z]*HypergeometricU[p, p, z]`	Successful	Successful	-	-
8.19.E19	${\displaystyle{\displaystyle\frac{n-1}{n}E_{n}\left(x\right)<E_{n+1}\left(x% \right)}}$	`(n - 1)/(n)*Ei(n, x)< Ei(n + 1, x)`	`Divide[n - 1,n]*ExpIntegralE[n, x]< ExpIntegralE[n + 1, x]`	Failure	Failure	Successful	Successful
8.19.E19	${\displaystyle{\displaystyle E_{n+1}\left(x\right)<E_{n}\left(x\right)}}$	`Ei(n + 1, x)< Ei(n, x)`	`ExpIntegralE[n + 1, x]< ExpIntegralE[n, x]`	Failure	Failure	Successful	Successful
8.19.E20	${\displaystyle{\displaystyle\left(E_{n}\left(x\right)\right)^{2}<E_{n-1}\left(% x\right)E_{n+1}\left(x\right)}}$	`(Ei(n, x))^(2)< Ei(n - 1, x)*Ei(n + 1, x)`	`(ExpIntegralE[n, x])^(2)< ExpIntegralE[n - 1, x]*ExpIntegralE[n + 1, x]`	Failure	Failure	Successful	Successful
8.19.E21	${\displaystyle{\displaystyle\frac{1}{x+n}<e^{x}E_{n}\left(x\right)}}$	`(1)/(x + n)< exp(x)*Ei(n, x)`	`Divide[1,x + n]< Exp[x]*ExpIntegralE[n, x]`	Failure	Failure	Successful	Successful
8.19.E21	${\displaystyle{\displaystyle e^{x}E_{n}\left(x\right)<=\frac{1}{x+n-1}}}$	`exp(x)*Ei(n, x)< =(1)/(x + n - 1)`	`Exp[x]*ExpIntegralE[n, x]< =Divide[1,x + n - 1]`	Failure	Failure	Successful	Successful
8.19.E22	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}\frac{E_{n}\left(x% \right)}{E_{n-1}\left(x\right)}>0}}$	`diff((Ei(n, x))/(Ei(n - 1, x)), x)> 0`	`D[Divide[ExpIntegralE[n, x],ExpIntegralE[n - 1, x]], x]> 0`	Failure	Failure	Successful	Successful
8.19.E23	${\displaystyle{\displaystyle\int_{z}^{\infty}E_{p-1}\left(t\right)\mathrm{d}t=% E_{p}\left(z\right)}}$	`int(Ei(p - 1, t), t = z..infinity)= Ei(p, z)`	`Integrate[ExpIntegralE[p - 1, t], {t, z, Infinity}]= ExpIntegralE[p, z]`	Failure	Failure	Skip	Successful
8.19.E24	${\displaystyle{\displaystyle\int_{0}^{\infty}e^{-at}E_{n}\left(t\right)\mathrm% {d}t=\frac{(-1)^{n-1}}{a^{n}}\left(\ln\left(1+a\right)+\sum_{k=1}^{n-1}\frac{(% -1)^{k}a^{k}}{k}\right)}}$	`int(exp(- at)Ei(n, t), t = 0..infinity)=((- 1)^(n - 1))/((a)^(n))(ln(1 + a)+ sum(((- 1)^(k) (a)^(k))/(k), k = 1..n - 1))`	`Integrate[Exp[- at]ExpIntegralE[n, t], {t, 0, Infinity}]=Divide[(- 1)^(n - 1),(a)^(n)](Log[1 + a]+ Sum[Divide[(- 1)^(k) (a)^(k),k], {k, 1, n - 1}])`	Failure	Failure	Skip	Successful
8.19.E25	${\displaystyle{\displaystyle\int_{0}^{\infty}e^{-at}t^{b-1}E_{p}\left(t\right)% \mathrm{d}t=\frac{\Gamma\left(b\right)(1+a)^{-b}}{p+b-1}\*F\left(1,b;p+b;a/(1+% a)\right)}}$	`int(exp(- at)(t)^(b - 1)* Ei(p, t), t = 0..infinity)=(GAMMA(b)(1 + a)^(- b))/(p + b - 1) hypergeom([1, b], [p + b], a/(1 + a))`	`Integrate[Exp[- at](t)^(b - 1)* ExpIntegralE[p, t], {t, 0, Infinity}]=Divide[Gamma[b](1 + a)^(- b),p + b - 1] Hypergeometric2F1[1, b, p + b, a/(1 + a)]`	Failure	Failure	Skip	Error
8.19.E26	${\displaystyle{\displaystyle\int_{0}^{\infty}E_{p}\left(t\right)E_{q}\left(t% \right)\mathrm{d}t=\frac{L(p)+L(q)}{p+q-1}}}$	`int(Ei(p, t)Ei(q, t), t = 0..infinity)=(L(p)+ L*(q))/(p + q - 1)`	`Integrate[ExpIntegralE[p, t]ExpIntegralE[q, t], {t, 0, Infinity}]=Divide[L(p)+ L*(q),p + q - 1]`	Failure	Failure	Skip	Fail `Complex[-1.9874359859908697, -2.1876726427121085] <- {Rule[L, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[p, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[q, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}` `Complex[-1.9874359859908697, -2.1876726427121085] <- {Rule[L, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[p, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[q, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}` `Complex[-1.9874359859908697, 2.1876726427121085] <- {Rule[L, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[p, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[q, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}` `Complex[-1.9874359859908697, 2.1876726427121085] <- {Rule[L, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[p, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[q, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}` ... skip entries to safe data
8.19.E27	${\displaystyle{\displaystyle L(p)=\int_{0}^{\infty}e^{-t}E_{p}\left(t\right)% \mathrm{d}t}}$	`L(p)= int(exp(- t)Ei(p, t), t = 0..infinity)`	`L(p)= Integrate[Exp[- t]ExpIntegralE[p, t], {t, 0, Infinity}]`	Failure	Failure	Skip	Skip
8.19.E27	${\displaystyle{\displaystyle\int_{0}^{\infty}e^{-t}E_{p}\left(t\right)\mathrm{% d}t=\frac{1}{2p}F\left(1,1;1+p;\tfrac{1}{2}\right)}}$	`int(exp(- t)Ei(p, t), t = 0..infinity)=(1)/(2p)*hypergeom([1, 1], [1 + p], (1)/(2))`	`Integrate[Exp[- t]ExpIntegralE[p, t], {t, 0, Infinity}]=Divide[1,2p]*Hypergeometric2F1[1, 1, 1 + p, Divide[1,2]]`	Failure	Failure	Skip	Skip
8.20.E1	${\displaystyle{\displaystyle E_{p}\left(z\right)=\frac{e^{-z}}{z}\left(\sum_{k% =0}^{n-1}(-1)^{k}\frac{{\left(p\right)_{k}}}{z^{k}}+(-1)^{n}\frac{{\left(p% \right)_{n}}e^{z}}{z^{n-1}}E_{n+p}\left(z\right)\right)}}$	`Ei(p, z)=(exp(- z))/(z)(sum((- 1)^(k)(pochhammer(p, k))/((z)^(k)), k = 0..n - 1)+(- 1)^(n)(pochhammer(p, n)exp(z))/((z)^(n - 1))*Ei(n + p, z))`	`ExpIntegralE[p, z]=Divide[Exp[- z],z](Sum[(- 1)^(k)Divide[Pochhammer[p, k],(z)^(k)], {k, 0, n - 1}]+(- 1)^(n)Divide[Pochhammer[p, n]Exp[z],(z)^(n - 1)]*ExpIntegralE[n + p, z])`	Failure	Successful	Skip	-
8.20.E4	${\displaystyle{\displaystyle A_{k+1}(\lambda)=(1-2k\lambda)A_{k}(\lambda)+% \lambda(\lambda+1)\frac{\mathrm{d}A_{k}(\lambda)}{\mathrm{d}\lambda}}}$	`A[k + 1](lambda)=(1 - 2klambda) A[k](lambda)+ lambda(lambda + 1)* diff(A[k]*(lambda), lambda)`	`Subscript[A, k + 1](\[Lambda])=(1 - 2k\[Lambda]) Subscript[A, k](\[Lambda])+ \[Lambda](\[Lambda]+ 1)* D[Subscript[A, k]*(\[Lambda]), \[Lambda]]`	Failure	Failure	Fail `-28.28427122+24.28427123I <- {lambda = 2^(1/2)+I2^(1/2), A[k] = 2^(1/2)+I2^(1/2), A[k+1] = 2^(1/2)+I2^(1/2), k = 3}` `-24.28427122+20.28427123I <- {lambda = 2^(1/2)+I2^(1/2), A[k] = 2^(1/2)+I2^(1/2), A[k+1] = 2^(1/2)-I2^(1/2), k = 3}` `-28.28427122+16.28427123I <- {lambda = 2^(1/2)+I2^(1/2), A[k] = 2^(1/2)+I2^(1/2), A[k+1] = -2^(1/2)-I2^(1/2), k = 3}` `-32.28427122+20.28427123I <- {lambda = 2^(1/2)+I2^(1/2), A[k] = 2^(1/2)+I2^(1/2), A[k+1] = -2^(1/2)+I2^(1/2), k = 3}` ... skip entries to safe data	Successful
8.21.E3	${\displaystyle{\displaystyle\int_{0}^{\infty}t^{a-1}e^{+\mathrm{i}t}\mathrm{d}% t=e^{+\frac{1}{2}\pi\mathrm{i}a}\Gamma\left(a\right)}}$	`int((t)^(a - 1)* exp(+ It), t = 0..infinity)= exp(+(1)/(2)PiIa)*GAMMA(a)`	`Integrate[(t)^(a - 1)* Exp[+ It], {t, 0, Infinity}]= Exp[+Divide[1,2]PiIa]*Gamma[a]`	Successful	Failure	-	Successful
8.21.E3	${\displaystyle{\displaystyle\int_{0}^{\infty}t^{a-1}e^{-\mathrm{i}t}\mathrm{d}% t=e^{-\frac{1}{2}\pi\mathrm{i}a}\Gamma\left(a\right)}}$	`int((t)^(a - 1)* exp(- It), t = 0..infinity)= exp(-(1)/(2)PiIa)*GAMMA(a)`	`Integrate[(t)^(a - 1)* Exp[- It], {t, 0, Infinity}]= Exp[-Divide[1,2]PiIa]*Gamma[a]`	Successful	Failure	-	Successful
8.22.E1	${\displaystyle{\displaystyle\frac{\Gamma\left(p\right)}{2\pi}z^{1-p}E_{p}\left% (z\right)=\frac{\Gamma\left(p\right)}{2\pi}\Gamma\left(1-p,z\right)}}$	`(GAMMA(p))/(2Pi)(z)^(1 - p)* Ei(p, z)=(GAMMA(p))/(2Pi)GAMMA(1 - p, z)`	`Divide[Gamma[p],2Pi](z)^(1 - p)* ExpIntegralE[p, z]=Divide[Gamma[p],2Pi]Gamma[1 - p, z]`	Successful	Successful	-	-
8.22.E3	${\displaystyle{\displaystyle\zeta_{x}(s)=\sum_{k=1}^{\infty}k^{-s}P\left(s,kx% \right)}}$	`zeta[x](s)= sum((k)^(- s) (GAMMA(s)-GAMMA(s, k*x))/GAMMA(s), k = 1..infinity)`	`Subscript[\[zeta], x](s)= Sum[(k)^(- s) GammaRegularized[s, 0, k*x], {k, 1, Infinity}]`	Failure	Failure	Skip	Error

Results of Incomplete Gamma and Related Functions

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