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	Fail Complex[1.4142135623730951, 1.4142135623730951] <- {Rule[ConditionalExpression[0, Or[Unequal[Im[z], 0], Greater[Re[z], 0]]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[1.4142135623730951, -1.4142135623730951] <- {Rule[ConditionalExpression[0, Or[Unequal[Im[z], 0], Greater[Re[z], 0]]], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[-1.4142135623730951, -1.4142135623730951] <- {Rule[ConditionalExpression[0, Or[Unequal[Im[z], 0], Greater[Re[z], 0]]], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[-1.4142135623730951, 1.4142135623730951] <- {Rule[ConditionalExpression[0, Or[Unequal[Im[z], 0], Greater[Re[z], 0]]], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
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)} -.4474572306-.2704599710*I <- {a = 2^(1/2)-I*2^(1/2), z = 2^(1/2)+I*2^(1/2)} .3504429851-.4826856014*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)} 23.62700226-82.69161801*I <- {a = 2^(1/2)-I*2^(1/2), z = -2^(1/2)+I*2^(1/2)} -.4343882366+.4808114998*I <- {a = -2^(1/2)-I*2^(1/2), z = 2^(1/2)+I*2^(1/2)} .7194242296-.2247089431*I <- {a = -2^(1/2)-I*2^(1/2), z = 2^(1/2)-I*2^(1/2)} 29.01554215+20.00785694*I <- {a = -2^(1/2)-I*2^(1/2), z = -2^(1/2)-I*2^(1/2)} 8.087330677+3.05352968*I <- {a = -2^(1/2)-I*2^(1/2), z = -2^(1/2)+I*2^(1/2)} .7194242296+.2247089431*I <- {a = -2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)} -.4343882366-.4808114998*I <- {a = -2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)} 8.087330677-3.05352968*I <- {a = -2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)} 29.01554215-20.00785694*I <- {a = -2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}	-
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} -1.005227386+.3325134381*I <- {a = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2), m = 2} -1.005227403+.3325134619*I <- {a = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2), m = 3} 20.46249955+81.80630491*I <- {a = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2), m = 1} 20.45425972+81.79804426*I <- {a = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2), m = 2} 20.45426304+81.79804594*I <- {a = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2), m = 3} .1380013835-.6459749422e-1*I <- {a = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2), m = 1} .1379895862-.6458002320e-1*I <- {a = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2), m = 2} .1379895857-.6458002881e-1*I <- {a = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2), m = 3} -7470.619632+1666.547276*I <- {a = 2^(1/2)-I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 1} 40153142.99-38054433.76*I <- {a = 2^(1/2)-I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 2} -.1078988446e12+.3850474280e12*I <- {a = 2^(1/2)-I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 3} -1231.554386+1108.053850*I <- {a = 2^(1/2)-I*2^(1/2), z = 2^(1/2)-I*2^(1/2), m = 1} 3527566.842-11442035.17*I <- {a = 2^(1/2)-I*2^(1/2), z = 2^(1/2)-I*2^(1/2), m = 2} .2056934222e11+.8406857680e11*I <- {a = 2^(1/2)-I*2^(1/2), z = 2^(1/2)-I*2^(1/2), m = 3} 1095.761010-111.286886*I <- {a = 2^(1/2)-I*2^(1/2), z = -2^(1/2)-I*2^(1/2), m = 1} -6383542.479+4755777.90*I <- {a = 2^(1/2)-I*2^(1/2), z = -2^(1/2)-I*2^(1/2), m = 2} .2195519693e11-.531867945e11*I <- {a = 2^(1/2)-I*2^(1/2), z = -2^(1/2)-I*2^(1/2), m = 3} -176581.1742-583404.7743*I <- {a = 2^(1/2)-I*2^(1/2), z = -2^(1/2)+I*2^(1/2), m = 1} 3259806629.+2963403529.*I <- {a = 2^(1/2)-I*2^(1/2), z = -2^(1/2)+I*2^(1/2), m = 2} -.3121706485e14-.6288807466e13*I <- {a = 2^(1/2)-I*2^(1/2), z = -2^(1/2)+I*2^(1/2), m = 3} -886.8142859+709.6704236*I <- {a = -2^(1/2)-I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 1} 8133491.718-1111438.055*I <- {a = -2^(1/2)-I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 2} -.5457968751e11-.2328231394e11*I <- {a = -2^(1/2)-I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 3} -528.9091261+529.9238978*I <- {a = -2^(1/2)-I*2^(1/2), z = 2^(1/2)-I*2^(1/2), m = 1} 5246689.544-1324472.318*I <- {a = -2^(1/2)-I*2^(1/2), z = 2^(1/2)-I*2^(1/2), m = 2} -.3746185707e11-.1125006634e11*I <- {a = -2^(1/2)-I*2^(1/2), z = 2^(1/2)-I*2^(1/2), m = 3} -304.8330801+212.9937977*I <- {a = -2^(1/2)-I*2^(1/2), z = -2^(1/2)-I*2^(1/2), m = 1} 2680978.291-190090.9563*I <- {a = -2^(1/2)-I*2^(1/2), z = -2^(1/2)-I*2^(1/2), m = 2} -.1733669117e11-8767812652.*I <- {a = -2^(1/2)-I*2^(1/2), z = -2^(1/2)-I*2^(1/2), m = 3} 89917.05184-32090.13160*I <- {a = -2^(1/2)-I*2^(1/2), z = -2^(1/2)+I*2^(1/2), m = 1} -676770926.6-134574125.9*I <- {a = -2^(1/2)-I*2^(1/2), z = -2^(1/2)+I*2^(1/2), m = 2} .3699038153e13+.3345802717e13*I <- {a = -2^(1/2)-I*2^(1/2), z = -2^(1/2)+I*2^(1/2), m = 3} -.2516682505e-1-.1004755343*I <- {a = -2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 1} -.2517097198e-1-.1004618178*I <- {a = -2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 2} -.2517097030e-1-.1004618191*I <- {a = -2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 3} -.5489674093e-1-.1472317109*I <- {a = -2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2), m = 1} -.5490067857e-1-.1472103307*I <- {a = -2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2), m = 2} -.5490067589e-1-.1472103348*I <- {a = -2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2), m = 3} 8.397046195+10.19508799*I <- {a = -2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2), m = 1} 8.396773066+10.19328107*I <- {a = -2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2), m = 2} 8.396773131+10.19328157*I <- {a = -2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2), m = 3} -.2106780493e-1-.4693492636e-1*I <- {a = -2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2), m = 1} -.2106863628e-1-.4692785756e-1*I <- {a = -2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2), m = 2} -.2106863591e-1-.4692785866e-1*I <- {a = -2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2), m = 3}	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]]]]} Complex[-1.005227396849198, 0.3325134406545761] <- {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[-1.0052274003428132, 0.33251346061799697] <- {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[20.46249974605223, 81.80630516774626] <- {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[20.454261145614897, 81.79804542060461] <- {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[20.454262710305986, 81.79804581625109] <- {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[0.1380013834196427, -0.06459749433701602] <- {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[0.13798958572122355, -0.06458002518444685] <- {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[0.13798958588146235, -0.06458002809630577] <- {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[-7470.619644952175, 1666.5472729681096] <- {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[4.015314239600219*^7, -3.80544338108609*^7] <- {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[-1.0789884878496332*^11, 3.850474305928836*^11] <- {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[-1231.5543888002394, 1108.0538497197144] <- {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[3527566.724421209, -1.1442035111005586*^7] <- {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[2.0569341689661465*^10, 8.40685777040521*^10] <- {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[1095.7610116067235, -111.28688635268304] <- {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[-6383542.3916354915, 4755777.914966784] <- {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[2.195519752693039*^10, -5.318679484476776*^10] <- {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[-176581.17394191804, -583404.7761796014] <- {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[3.2598066316560135*^9, 2.963403487390624*^9] <- {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[-3.1217065120398496*^13, -6.288807808095331*^12] <- {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[-886.8142851470194, 709.6704251919407] <- {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[8133491.655967515, -1111437.9869255058] <- {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[-5.457968815222714*^10, -2.3282313627352165*^10] <- {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[-528.9091257238363, 529.9238982674376] <- {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[5246689.508076388, -1324472.2648729375] <- {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[-3.7461857452533104*^10, -1.1250066115913736*^10] <- {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[-304.83307996021597, 212.99379800143396] <- {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[2680978.2695867238, -190090.93321141892] <- {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[-1.7336691377977974*^10, -8.767812573847723*^9] <- {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[89917.05188669058, -32090.13194435204] <- {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[-6.767709209813998*^8, -1.34574128655497*^8] <- {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[3.69903822299039*^12, 3.345802707953902*^12] <- {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[-0.025166825060855283, -0.10047553413197906] <- {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[-0.025170971837805013, -0.10046181764225634] <- {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[-0.025170970371447107, -0.1004618189763399] <- {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[-0.054896740794276436, -0.14723171098103593] <- {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[-0.05490067793941428, -0.1472103320317673] <- {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[-0.054900675953830975, -0.14721033429049477] <- {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[8.397046195215205, 10.195088034511809] <- {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[8.396773178420329, 10.193281301506214] <- {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[8.396773082538449, 10.193281535405262] <- {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[-0.021067804958581127, -0.04693492633141024] <- {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[-0.021068636465505875, -0.046927857749596076] <- {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[-0.02106863586483843, -0.04692785852979759] <- {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]]]]}
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)} -2.16987973+12.77160007*I <- {a = 2^(1/2)-I*2^(1/2), z = 2^(1/2)+I*2^(1/2)} 7167.292469-174.9289096*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)} -8.705606105-17.43270949*I <- {a = 2^(1/2)-I*2^(1/2), z = -2^(1/2)+I*2^(1/2)} 3.369439236+2.788984848*I <- {a = -2^(1/2)-I*2^(1/2), z = 2^(1/2)+I*2^(1/2)} -645.4110961-918.9294888*I <- {a = -2^(1/2)-I*2^(1/2), z = 2^(1/2)-I*2^(1/2)} 10.82304704+7.831702915*I <- {a = -2^(1/2)-I*2^(1/2), z = -2^(1/2)-I*2^(1/2)} 7.664340201+4.336898369*I <- {a = -2^(1/2)-I*2^(1/2), z = -2^(1/2)+I*2^(1/2)} 645.4110961-918.9294888*I <- {a = -2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)} -3.369439236+2.788984848*I <- {a = -2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)} -7.664340201+4.336898369*I <- {a = -2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)} -10.82304704+7.831702915*I <- {a = -2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}	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]]]]} 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[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[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]]]]} 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[3.369439241027149, 2.7889848429588855] <- {Rule[a, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-645.4110982406346, -918.9294880188124] <- {Rule[a, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[10.823047044974839, 7.831702902208898] <- {Rule[a, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[7.664340200530641, 4.336898364261077] <- {Rule[a, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} Complex[645.4110982406346, -918.9294880188124] <- {Rule[a, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-3.369439241027149, 2.7889848429588855] <- {Rule[a, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[-7.664340200530641, 4.336898364261077] <- {Rule[a, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[-10.823047044974839, 7.831702902208898] <- {Rule[a, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
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)} -304.8330800+212.9937978*I <- {a = -2^(1/2)-I*2^(1/2), z = 2^(1/2)+I*2^(1/2)} -528.9091261+529.9238977*I <- {a = -2^(1/2)-I*2^(1/2), z = -2^(1/2)+I*2^(1/2)} 8.397046212+10.19508802*I <- {a = -2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)} -.5489674088e-1-.1472317112*I <- {a = -2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}	-
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)} 8.397046212-10.19508802*I <- {a = -2^(1/2)-I*2^(1/2), z = 2^(1/2)-I*2^(1/2)} -.5489674088e-1+.1472317112*I <- {a = -2^(1/2)-I*2^(1/2), z = -2^(1/2)-I*2^(1/2)} -304.8330800-212.9937978*I <- {a = -2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)} -528.9091261-529.9238977*I <- {a = -2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}	-
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)} -.3535533907+.6464466093*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)} .3535533907-.6464466093*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)} -.3535533907+.6464466093*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)} .3535533907-.6464466093*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)} -.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)} .3535533907-.6464466093*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)} -.3535533907+.6464466093*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)} .3535533907-.6464466093*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)} -.3535533907+.6464466093*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)} .3535533907+.6464466093*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)} -.3535533907-.6464466093*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)} .3535533907+.6464466093*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)} -.3535533907-.6464466093*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)} -.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)} -.3535533907-.6464466093*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)} .3535533907+.6464466093*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)} -.3535533907-.6464466093*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)} .3535533907+.6464466093*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)} .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)} 1.353553391-.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)} -1.353553391+.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)} 1.353553391-.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)} -1.353553391+.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)} -.3535533907-1.353553391*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)} .3535533907+1.353553391*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)} -.3535533907-1.353553391*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)} .3535533907+1.353553391*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)} -1.353553391+.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)} 1.353553391-.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)} -1.353553391+.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)} 1.353553391-.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)} .3535533907+1.353553391*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)} -.3535533907-1.353553391*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)} .3535533907+1.353553391*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)} -.3535533907-1.353553391*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)} .3535533907-1.353553391*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)} -.3535533907+1.353553391*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)} .3535533907-1.353553391*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)} -.3535533907+1.353553391*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)} -1.353553391-.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)} 1.353553391+.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)} -1.353553391-.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)} 1.353553391+.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)} -.3535533907+1.353553391*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)} .3535533907-1.353553391*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)} -.3535533907+1.353553391*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)} .3535533907-1.353553391*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)} 1.353553391+.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)} -1.353553391-.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)} 1.353553391+.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)} -1.353553391-.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)}	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]]]]} Complex[-0.35355339059327373, 0.6464466094067263] <- {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[0.35355339059327373, -0.6464466094067263] <- {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[-0.35355339059327373, 0.6464466094067263] <- {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[0.35355339059327373, -0.6464466094067263] <- {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]]]]} 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[0.35355339059327373, -0.6464466094067263] <- {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[-0.35355339059327373, 0.6464466094067263] <- {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[0.35355339059327373, -0.6464466094067263] <- {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[-0.35355339059327373, 0.6464466094067263] <- {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[0.35355339059327373, 0.6464466094067263] <- {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[-0.35355339059327373, -0.6464466094067263] <- {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[0.35355339059327373, 0.6464466094067263] <- {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[-0.35355339059327373, -0.6464466094067263] <- {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]]]]} 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[-0.35355339059327373, -0.6464466094067263] <- {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[0.35355339059327373, 0.6464466094067263] <- {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[-0.35355339059327373, -0.6464466094067263] <- {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[0.35355339059327373, 0.6464466094067263] <- {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]]]]} 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[1.3535533905932735, -0.3535533905932736] <- {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[-1.3535533905932735, 0.3535533905932736] <- {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[1.3535533905932735, -0.3535533905932736] <- {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[-1.3535533905932735, 0.3535533905932736] <- {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[-0.3535533905932736, -1.3535533905932735] <- {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[0.3535533905932736, 1.3535533905932735] <- {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[-0.3535533905932736, -1.3535533905932735] <- {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[0.3535533905932736, 1.3535533905932735] <- {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[-1.3535533905932735, 0.3535533905932736] <- {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[1.3535533905932735, -0.3535533905932736] <- {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[-1.3535533905932735, 0.3535533905932736] <- {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[1.3535533905932735, -0.3535533905932736] <- {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[0.3535533905932736, 1.3535533905932735] <- {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[-0.3535533905932736, -1.3535533905932735] <- {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[0.3535533905932736, 1.3535533905932735] <- {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[-0.3535533905932736, -1.3535533905932735] <- {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[0.3535533905932736, -1.3535533905932735] <- {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[-0.3535533905932736, 1.3535533905932735] <- {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[0.3535533905932736, -1.3535533905932735] <- {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[-0.3535533905932736, 1.3535533905932735] <- {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[-1.3535533905932735, -0.3535533905932736] <- {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[1.3535533905932735, 0.3535533905932736] <- {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[-1.3535533905932735, -0.3535533905932736] <- {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[1.3535533905932735, 0.3535533905932736] <- {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[-0.3535533905932736, 1.3535533905932735] <- {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[0.3535533905932736, -1.3535533905932735] <- {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[-0.3535533905932736, 1.3535533905932735] <- {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[0.3535533905932736, -1.3535533905932735] <- {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[1.3535533905932735, 0.3535533905932736] <- {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[-1.3535533905932735, -0.3535533905932736] <- {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[1.3535533905932735, 0.3535533905932736] <- {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[-1.3535533905932735, -0.3535533905932736] <- {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]]]]}
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	${\displaystyle{\displaystyle\gamma^{*}\left(\tfrac{1}{2},-z^{2}\right)=\frac{2e^{z^{2}}}{z\sqrt{\pi}}F\left(z\right)}}$	(- (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	-	Fail Complex[1.4142135623730951, 1.4142135623730951] <- {Rule[ConditionalExpression[0, And[Greater[Re[z], 0], Equal[Im[z], 0]]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[1.4142135623730951, -1.4142135623730951] <- {Rule[ConditionalExpression[0, And[Greater[Re[z], 0], Equal[Im[z], 0]]], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[-1.4142135623730951, -1.4142135623730951] <- {Rule[ConditionalExpression[0, And[Greater[Re[z], 0], Equal[Im[z], 0]]], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[-1.4142135623730951, 1.4142135623730951] <- {Rule[ConditionalExpression[0, And[Greater[Re[z], 0], Equal[Im[z], 0]]], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
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	Fail Complex[1.4142135623730951, 1.4142135623730951] <- {Rule[ConditionalExpression[Plus[ExpIntegralEi[Times[-1, z]], Gamma[0, z]], And[Greater[Re[z], 0], Equal[Im[z], 0]]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[1.4142135623730951, -1.4142135623730951] <- {Rule[ConditionalExpression[Plus[ExpIntegralEi[Times[-1, z]], Gamma[0, z]], And[Greater[Re[z], 0], Equal[Im[z], 0]]], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[-1.4142135623730951, -1.4142135623730951] <- {Rule[ConditionalExpression[Plus[ExpIntegralEi[Times[-1, z]], Gamma[0, z]], And[Greater[Re[z], 0], Equal[Im[z], 0]]], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[-1.4142135623730951, 1.4142135623730951] <- {Rule[ConditionalExpression[Plus[ExpIntegralEi[Times[-1, z]], Gamma[0, z]], And[Greater[Re[z], 0], Equal[Im[z], 0]]], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
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} 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} 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} 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}	-
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]]]]} Complex[-7.400077458243353, 11.126893574158686] <- {Rule[n, 2], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} Complex[11.806291479358972, 12.531631677265604] <- {Rule[n, 3], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
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]]]]} 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[-2.220446049250313*^-16, -3.1415926535897936] <- {Rule[n, 1], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[5.551115123125783*^-17, 1.5707963267948966] <- {Rule[n, 2], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[-8.326672684688674*^-17, -0.5235987755982988] <- {Rule[n, 3], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[-2.220446049250313*^-16, 3.1415926535897936] <- {Rule[n, 1], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} Complex[5.551115123125783*^-17, -1.5707963267948966] <- {Rule[n, 2], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} Complex[-8.326672684688674*^-17, 0.5235987755982988] <- {Rule[n, 3], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
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	Fail Complex[1.4142135623730951, 1.4142135623730951] <- {Rule[a, Rational[1, 2]], Rule[ConditionalExpression[Plus[Times[-1, Gamma[a]], Gamma[a, z], Gamma[a, 0, z]], Greater[Re[a], 0]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[1.4142135623730951, -1.4142135623730951] <- {Rule[a, Rational[1, 2]], Rule[ConditionalExpression[Plus[Times[-1, Gamma[a]], Gamma[a, z], Gamma[a, 0, z]], Greater[Re[a], 0]], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[-1.4142135623730951, -1.4142135623730951] <- {Rule[a, Rational[1, 2]], Rule[ConditionalExpression[Plus[Times[-1, Gamma[a]], Gamma[a, z], Gamma[a, 0, z]], Greater[Re[a], 0]], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[-1.4142135623730951, 1.4142135623730951] <- {Rule[a, Rational[1, 2]], Rule[ConditionalExpression[Plus[Times[-1, Gamma[a]], Gamma[a, z], Gamma[a, 0, z]], Greater[Re[a], 0]], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
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	Skip
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)} .8693672828-.710002389*I <- {a = 2^(1/2)-I*2^(1/2), z = 2^(1/2)+I*2^(1/2)} .135004907e-1+.2375774782*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)} 107.1902160+63.3824277*I <- {a = 2^(1/2)-I*2^(1/2), z = -2^(1/2)+I*2^(1/2)} -.1520611888+.9119148087e-1*I <- {a = -2^(1/2)-I*2^(1/2), z = 2^(1/2)+I*2^(1/2)} -.1417494577e-1+.2037473416e-1*I <- {a = -2^(1/2)-I*2^(1/2), z = 2^(1/2)-I*2^(1/2)} .7143741874e-1-.1030661023*I <- {a = -2^(1/2)-I*2^(1/2), z = -2^(1/2)-I*2^(1/2)} 34.87574808-12.92049251*I <- {a = -2^(1/2)-I*2^(1/2), z = -2^(1/2)+I*2^(1/2)} -.1417494577e-1-.2037473416e-1*I <- {a = -2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)} -.1520611888-.9119148087e-1*I <- {a = -2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)} 34.87574808+12.92049251*I <- {a = -2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)} .7143741874e-1+.1030661023*I <- {a = -2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}	-
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	${\displaystyle{\displaystyle Q\left(a+n,z\right)=Q\left(a,z\right)+z^{a}e^{-z}\sum_{k=0}^{n-1}\frac{z^{k}}{\Gamma\left(a+k+1\right)}}}$	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	Skip
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 1.052938223-1.733408016*I <- {a = 2^(1/2)+I*2^(1/2), vartheta = 2^(1/2)+I*2^(1/2), x = 1} .6195824495-.7346525318*I <- {a = 2^(1/2)+I*2^(1/2), vartheta = 2^(1/2)+I*2^(1/2), x = 2} .4531580595-.4544327802*I <- {a = 2^(1/2)+I*2^(1/2), vartheta = 2^(1/2)+I*2^(1/2), x = 3} -2.947061775-.5618351419*I <- {a = 2^(1/2)+I*2^(1/2), vartheta = 2^(1/2)-I*2^(1/2), x = 1} -1.380417550-.1488660948*I <- {a = 2^(1/2)+I*2^(1/2), vartheta = 2^(1/2)-I*2^(1/2), x = 2} -.8801752730-.639084889e-1*I <- {a = 2^(1/2)+I*2^(1/2), vartheta = 2^(1/2)-I*2^(1/2), x = 3} -1.775488901+3.438164856*I <- {a = 2^(1/2)+I*2^(1/2), vartheta = -2^(1/2)-I*2^(1/2), x = 1} -.7946311125+1.851133904*I <- {a = 2^(1/2)+I*2^(1/2), vartheta = -2^(1/2)-I*2^(1/2), x = 2} -.4896509817+1.269424844*I <- {a = 2^(1/2)+I*2^(1/2), vartheta = -2^(1/2)-I*2^(1/2), x = 3} 2.224511097+2.266591982*I <- {a = 2^(1/2)+I*2^(1/2), vartheta = -2^(1/2)+I*2^(1/2), x = 1} 1.205368886+1.265347467*I <- {a = 2^(1/2)+I*2^(1/2), vartheta = -2^(1/2)+I*2^(1/2), x = 2} .8436823508+.8789005523*I <- {a = 2^(1/2)+I*2^(1/2), vartheta = -2^(1/2)+I*2^(1/2), x = 3} -2.947061775+.5618351419*I <- {a = 2^(1/2)-I*2^(1/2), vartheta = 2^(1/2)+I*2^(1/2), x = 1} -1.380417550+.1488660948*I <- {a = 2^(1/2)-I*2^(1/2), vartheta = 2^(1/2)+I*2^(1/2), x = 2} -.8801752730+.639084889e-1*I <- {a = 2^(1/2)-I*2^(1/2), vartheta = 2^(1/2)+I*2^(1/2), x = 3} 1.052938223+1.733408016*I <- {a = 2^(1/2)-I*2^(1/2), vartheta = 2^(1/2)-I*2^(1/2), x = 1} .6195824495+.7346525318*I <- {a = 2^(1/2)-I*2^(1/2), vartheta = 2^(1/2)-I*2^(1/2), x = 2} .4531580595+.4544327802*I <- {a = 2^(1/2)-I*2^(1/2), vartheta = 2^(1/2)-I*2^(1/2), x = 3} 2.224511097-2.266591982*I <- {a = 2^(1/2)-I*2^(1/2), vartheta = -2^(1/2)-I*2^(1/2), x = 1} 1.205368886-1.265347467*I <- {a = 2^(1/2)-I*2^(1/2), vartheta = -2^(1/2)-I*2^(1/2), x = 2} .8436823508-.8789005523*I <- {a = 2^(1/2)-I*2^(1/2), vartheta = -2^(1/2)-I*2^(1/2), x = 3} -1.775488901-3.438164856*I <- {a = 2^(1/2)-I*2^(1/2), vartheta = -2^(1/2)+I*2^(1/2), x = 1} -.7946311125-1.851133904*I <- {a = 2^(1/2)-I*2^(1/2), vartheta = -2^(1/2)+I*2^(1/2), x = 2} -.4896509817-1.269424844*I <- {a = 2^(1/2)-I*2^(1/2), vartheta = -2^(1/2)+I*2^(1/2), x = 3} .7137479990+5.279470749*I <- {a = -2^(1/2)-I*2^(1/2), vartheta = 2^(1/2)+I*2^(1/2), x = 1} .1689182623+2.559481797*I <- {a = -2^(1/2)-I*2^(1/2), vartheta = 2^(1/2)+I*2^(1/2), x = 2} .333014351e-1+1.662585688*I <- {a = -2^(1/2)-I*2^(1/2), vartheta = 2^(1/2)+I*2^(1/2), x = 3} 4.713747997-1.548956373*I <- {a = -2^(1/2)-I*2^(1/2), vartheta = 2^(1/2)-I*2^(1/2), x = 1} 2.168918261-.8547317639*I <- {a = -2^(1/2)-I*2^(1/2), vartheta = 2^(1/2)-I*2^(1/2), x = 2} 1.366634768-.6135566855*I <- {a = -2^(1/2)-I*2^(1/2), vartheta = 2^(1/2)-I*2^(1/2), x = 3} -2.114679125-5.548956371*I <- {a = -2^(1/2)-I*2^(1/2), vartheta = -2^(1/2)-I*2^(1/2), x = 1} -1.245295300-2.854731763*I <- {a = -2^(1/2)-I*2^(1/2), vartheta = -2^(1/2)-I*2^(1/2), x = 2} -.9095076061-1.946890018*I <- {a = -2^(1/2)-I*2^(1/2), vartheta = -2^(1/2)-I*2^(1/2), x = 3} -6.114679123+1.279470751*I <- {a = -2^(1/2)-I*2^(1/2), vartheta = -2^(1/2)+I*2^(1/2), x = 1} -3.245295299+.5594817981*I <- {a = -2^(1/2)-I*2^(1/2), vartheta = -2^(1/2)+I*2^(1/2), x = 2} -2.242840938+.3292523557*I <- {a = -2^(1/2)-I*2^(1/2), vartheta = -2^(1/2)+I*2^(1/2), x = 3} 4.713747997+1.548956373*I <- {a = -2^(1/2)+I*2^(1/2), vartheta = 2^(1/2)+I*2^(1/2), x = 1} 2.168918261+.8547317639*I <- {a = -2^(1/2)+I*2^(1/2), vartheta = 2^(1/2)+I*2^(1/2), x = 2} 1.366634768+.6135566855*I <- {a = -2^(1/2)+I*2^(1/2), vartheta = 2^(1/2)+I*2^(1/2), x = 3} .7137479990-5.279470749*I <- {a = -2^(1/2)+I*2^(1/2), vartheta = 2^(1/2)-I*2^(1/2), x = 1} .1689182623-2.559481797*I <- {a = -2^(1/2)+I*2^(1/2), vartheta = 2^(1/2)-I*2^(1/2), x = 2} .333014351e-1-1.662585688*I <- {a = -2^(1/2)+I*2^(1/2), vartheta = 2^(1/2)-I*2^(1/2), x = 3} -6.114679123-1.279470751*I <- {a = -2^(1/2)+I*2^(1/2), vartheta = -2^(1/2)-I*2^(1/2), x = 1} -3.245295299-.5594817981*I <- {a = -2^(1/2)+I*2^(1/2), vartheta = -2^(1/2)-I*2^(1/2), x = 2} -2.242840938-.3292523557*I <- {a = -2^(1/2)+I*2^(1/2), vartheta = -2^(1/2)-I*2^(1/2), x = 3} -2.114679125+5.548956371*I <- {a = -2^(1/2)+I*2^(1/2), vartheta = -2^(1/2)+I*2^(1/2), x = 1} -1.245295300+2.854731763*I <- {a = -2^(1/2)+I*2^(1/2), vartheta = -2^(1/2)+I*2^(1/2), x = 2} -.9095076061+1.946890018*I <- {a = -2^(1/2)+I*2^(1/2), vartheta = -2^(1/2)+I*2^(1/2), x = 3}	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]]]]} Complex[-1.3804175506751934, -0.14886609500970405] <- {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[-0.8801752737956228, -0.06390848891115308] <- {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[-1.7754889011850625, 3.438164858219089] <- {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[-0.7946311130482883, 1.851133904990296] <- {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[-0.48965098204435287, 1.2694248444221805] <- {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[2.224511098814938, 2.266591982965279] <- {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[1.2053688869517116, 1.2653474673633909] <- {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[0.8436823512889807, 0.8789005526709103] <- {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]]]]} Complex[-1.3804175506751934, 0.14886609500970405] <- {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[-0.8801752737956228, 0.06390848891115308] <- {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[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[2.224511098814938, -2.266591982965279] <- {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[1.2053688869517116, -1.2653474673633909] <- {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[0.8436823512889807, -0.8789005526709103] <- {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[-1.7754889011850625, -3.438164858219089] <- {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[-0.7946311130482883, -1.851133904990296] <- {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[-0.48965098204435287, -1.2694248444221805] <- {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[0.7137479994437111, 5.2794707516485415] <- {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[0.16891826235482466, 2.5594817982620857] <- {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[0.03330143512454342, 1.6625856892027477] <- {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[4.713747999443712, -1.5489563730976488] <- {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[2.1689182623548247, -0.8547317641110086] <- {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[1.3666347684578768, -0.6135566857126489] <- {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[-2.114679125302478, -5.54895637309765] <- {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[-1.2452953000182698, -2.8547317641110084] <- {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[-0.9095076064575197, -1.9468900190459824] <- {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[-6.114679125302479, 1.2794707516485402] <- {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[-3.2452953000182694, 0.5594817982620859] <- {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[-2.2428409397908533, 0.32925235586941426] <- {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[4.713747999443712, 1.5489563730976488] <- {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[2.1689182623548247, 0.8547317641110086] <- {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[1.3666347684578768, 0.6135566857126489] <- {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[0.7137479994437111, -5.2794707516485415] <- {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[0.16891826235482466, -2.5594817982620857] <- {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[0.03330143512454342, -1.6625856892027477] <- {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[-6.114679125302479, -1.2794707516485402] <- {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[-3.2452953000182694, -0.5594817982620859] <- {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[-2.2428409397908533, -0.32925235586941426] <- {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[-2.114679125302478, 5.54895637309765] <- {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[-1.2452953000182698, 2.8547317641110084] <- {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[-0.9095076064575197, 1.9468900190459824] <- {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]]]]}
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	Skip
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+.7512982152*I <- {a = 2^(1/2)+I*2^(1/2), c[a] = 2^(1/2)+I*2^(1/2)} -.342222950-2.077128909*I <- {a = 2^(1/2)+I*2^(1/2), c[a] = 2^(1/2)-I*2^(1/2)} -3.170650074-2.077128909*I <- {a = 2^(1/2)+I*2^(1/2), c[a] = -2^(1/2)-I*2^(1/2)} -3.170650074+.7512982152*I <- {a = 2^(1/2)+I*2^(1/2), c[a] = -2^(1/2)+I*2^(1/2)} -.342222950+2.077128909*I <- {a = 2^(1/2)-I*2^(1/2), c[a] = 2^(1/2)+I*2^(1/2)} -.342222950-.7512982152*I <- {a = 2^(1/2)-I*2^(1/2), c[a] = 2^(1/2)-I*2^(1/2)} -3.170650074-.7512982152*I <- {a = 2^(1/2)-I*2^(1/2), c[a] = -2^(1/2)-I*2^(1/2)} -3.170650074+2.077128909*I <- {a = 2^(1/2)-I*2^(1/2), c[a] = -2^(1/2)+I*2^(1/2)} .9491134946+2.345189180*I <- {a = -2^(1/2)-I*2^(1/2), c[a] = 2^(1/2)+I*2^(1/2)} .9491134946-.4832379435*I <- {a = -2^(1/2)-I*2^(1/2), c[a] = 2^(1/2)-I*2^(1/2)} -1.879313629-.4832379435*I <- {a = -2^(1/2)-I*2^(1/2), c[a] = -2^(1/2)-I*2^(1/2)} -1.879313629+2.345189180*I <- {a = -2^(1/2)-I*2^(1/2), c[a] = -2^(1/2)+I*2^(1/2)} .9491134946+.4832379435*I <- {a = -2^(1/2)+I*2^(1/2), c[a] = 2^(1/2)+I*2^(1/2)} .9491134946-2.345189180*I <- {a = -2^(1/2)+I*2^(1/2), c[a] = 2^(1/2)-I*2^(1/2)} -1.879313629-2.345189180*I <- {a = -2^(1/2)+I*2^(1/2), c[a] = -2^(1/2)-I*2^(1/2)} -1.879313629+.4832379435*I <- {a = -2^(1/2)+I*2^(1/2), c[a] = -2^(1/2)+I*2^(1/2)}	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+1.747162536*I <- {a = 2^(1/2)+I*2^(1/2), d[a] = 2^(1/2)+I*2^(1/2)} .7353701374-1.081264588*I <- {a = 2^(1/2)+I*2^(1/2), d[a] = 2^(1/2)-I*2^(1/2)} -2.093056987-1.081264588*I <- {a = 2^(1/2)+I*2^(1/2), d[a] = -2^(1/2)-I*2^(1/2)} -2.093056987+1.747162536*I <- {a = 2^(1/2)+I*2^(1/2), d[a] = -2^(1/2)+I*2^(1/2)} .7353701374+1.081264588*I <- {a = 2^(1/2)-I*2^(1/2), d[a] = 2^(1/2)+I*2^(1/2)} .7353701374-1.747162536*I <- {a = 2^(1/2)-I*2^(1/2), d[a] = 2^(1/2)-I*2^(1/2)} -2.093056987-1.747162536*I <- {a = 2^(1/2)-I*2^(1/2), d[a] = -2^(1/2)-I*2^(1/2)} -2.093056987+1.081264588*I <- {a = 2^(1/2)-I*2^(1/2), d[a] = -2^(1/2)+I*2^(1/2)} 1.333446246+.15004730e-1*I <- {a = -2^(1/2)-I*2^(1/2), d[a] = 2^(1/2)+I*2^(1/2)} 1.333446246-2.813422394*I <- {a = -2^(1/2)-I*2^(1/2), d[a] = 2^(1/2)-I*2^(1/2)} -1.494980878-2.813422394*I <- {a = -2^(1/2)-I*2^(1/2), d[a] = -2^(1/2)-I*2^(1/2)} -1.494980878+.15004730e-1*I <- {a = -2^(1/2)-I*2^(1/2), d[a] = -2^(1/2)+I*2^(1/2)} 1.333446246+2.813422394*I <- {a = -2^(1/2)+I*2^(1/2), d[a] = 2^(1/2)+I*2^(1/2)} 1.333446246-.15004730e-1*I <- {a = -2^(1/2)+I*2^(1/2), d[a] = 2^(1/2)-I*2^(1/2)} -1.494980878-.15004730e-1*I <- {a = -2^(1/2)+I*2^(1/2), d[a] = -2^(1/2)-I*2^(1/2)} -1.494980878+2.813422394*I <- {a = -2^(1/2)+I*2^(1/2), d[a] = -2^(1/2)+I*2^(1/2)}	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)/(2*a)*((1 +(2)/(x))^(a)- 1)< (x)^(1 - a)* exp(x)*GAMMA(a, x)	Divide[x,2*a]*((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)/(a*c[a])*((1 +(c[a])/(x))^(a)- 1)	(x)^(1 - a)* Exp[x]*Gamma[a, x]< =Divide[x,a*Subscript[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, n*x))/((n*x)^(n)* exp(- n*x))	Subscript[S, n]*(x)=Divide[Gamma[n + 1, 0, n*x],(n*x)^(n)* Exp[- n*x]]	Failure	Failure	Fail .6959317335+1.414213562*I <- {S[n] = 2^(1/2)+I*2^(1/2), n = 1, x = 1} .633899074+2.828427124*I <- {S[n] = 2^(1/2)+I*2^(1/2), n = 1, x = 2} -1.119204955+4.242640686*I <- {S[n] = 2^(1/2)+I*2^(1/2), n = 1, x = 3} .219685512+1.414213562*I <- {S[n] = 2^(1/2)+I*2^(1/2), n = 2, x = 1} -2.371341630+2.828427124*I <- {S[n] = 2^(1/2)+I*2^(1/2), n = 2, x = 2} -16.78118116+4.242640686*I <- {S[n] = 2^(1/2)+I*2^(1/2), n = 2, x = 3} -.160350198+1.414213562*I <- {S[n] = 2^(1/2)+I*2^(1/2), n = 3, x = 1} -6.683483804+2.828427124*I <- {S[n] = 2^(1/2)+I*2^(1/2), n = 3, x = 2} -61.03377023+4.242640686*I <- {S[n] = 2^(1/2)+I*2^(1/2), n = 3, x = 3} .6959317335-1.414213562*I <- {S[n] = 2^(1/2)-I*2^(1/2), n = 1, x = 1} .633899074-2.828427124*I <- {S[n] = 2^(1/2)-I*2^(1/2), n = 1, x = 2} -1.119204955-4.242640686*I <- {S[n] = 2^(1/2)-I*2^(1/2), n = 1, x = 3} .219685512-1.414213562*I <- {S[n] = 2^(1/2)-I*2^(1/2), n = 2, x = 1} -2.371341630-2.828427124*I <- {S[n] = 2^(1/2)-I*2^(1/2), n = 2, x = 2} -16.78118116-4.242640686*I <- {S[n] = 2^(1/2)-I*2^(1/2), n = 2, x = 3} -.160350198-1.414213562*I <- {S[n] = 2^(1/2)-I*2^(1/2), n = 3, x = 1} -6.683483804-2.828427124*I <- {S[n] = 2^(1/2)-I*2^(1/2), n = 3, x = 2} -61.03377023-4.242640686*I <- {S[n] = 2^(1/2)-I*2^(1/2), n = 3, x = 3} -2.132495390-1.414213562*I <- {S[n] = -2^(1/2)-I*2^(1/2), n = 1, x = 1} -5.022955174-2.828427124*I <- {S[n] = -2^(1/2)-I*2^(1/2), n = 1, x = 2} -9.604486327-4.242640686*I <- {S[n] = -2^(1/2)-I*2^(1/2), n = 1, x = 3} -2.608741612-1.414213562*I <- {S[n] = -2^(1/2)-I*2^(1/2), n = 2, x = 1} -8.028195878-2.828427124*I <- {S[n] = -2^(1/2)-I*2^(1/2), n = 2, x = 2} -25.26646254-4.242640686*I <- {S[n] = -2^(1/2)-I*2^(1/2), n = 2, x = 3} -2.988777322-1.414213562*I <- {S[n] = -2^(1/2)-I*2^(1/2), n = 3, x = 1} -12.34033805-2.828427124*I <- {S[n] = -2^(1/2)-I*2^(1/2), n = 3, x = 2} -69.51905161-4.242640686*I <- {S[n] = -2^(1/2)-I*2^(1/2), n = 3, x = 3} -2.132495390+1.414213562*I <- {S[n] = -2^(1/2)+I*2^(1/2), n = 1, x = 1} -5.022955174+2.828427124*I <- {S[n] = -2^(1/2)+I*2^(1/2), n = 1, x = 2} -9.604486327+4.242640686*I <- {S[n] = -2^(1/2)+I*2^(1/2), n = 1, x = 3} -2.608741612+1.414213562*I <- {S[n] = -2^(1/2)+I*2^(1/2), n = 2, x = 1} -8.028195878+2.828427124*I <- {S[n] = -2^(1/2)+I*2^(1/2), n = 2, x = 2} -25.26646254+4.242640686*I <- {S[n] = -2^(1/2)+I*2^(1/2), n = 2, x = 3} -2.988777322+1.414213562*I <- {S[n] = -2^(1/2)+I*2^(1/2), n = 3, x = 1} -12.34033805+2.828427124*I <- {S[n] = -2^(1/2)+I*2^(1/2), n = 3, x = 2} -69.51905161+4.242640686*I <- {S[n] = -2^(1/2)+I*2^(1/2), n = 3, x = 3}	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(- eta*sqrt(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(eta*sqrt(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(+ Pi*I*a))/(2*I*sin(Pi*a))*GAMMA(- a, z*exp(+ Pi*I))/GAMMA(- a)= +(1)/(2)*erfc(+ I*eta*sqrt(a/ 2))- I*T*(a , eta)	Divide[Exp[+ Pi*I*a],2*I*Sin[Pi*a]]*GammaRegularized[- a, z*Exp[+ Pi*I]]= +Divide[1,2]*Erfc[+ I*\[Eta]*Sqrt[a/ 2]]- I*T*(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(- Pi*I*a))/(2*I*sin(Pi*a))*GAMMA(- a, z*exp(- Pi*I))/GAMMA(- a)= -(1)/(2)*erfc(- I*eta*sqrt(a/ 2))- I*T*(a , eta)	Divide[Exp[- Pi*I*a],2*I*Sin[Pi*a]]*GammaRegularized[- a, z*Exp[- Pi*I]]= -Divide[1,2]*Erfc[- I*\[Eta]*Sqrt[a/ 2]]- I*T*(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(+ Pi*I*a))/(2*Pi*I)*GAMMA(- a, z*exp(+ Pi*I))= -(1)/(2)*erfc(+ I*eta*sqrt(a/ 2))+ I*T*(a , eta)	Gamma[a + 1]*Divide[Exp[+ Pi*I*a],2*Pi*I]*Gamma[- a, z*Exp[+ Pi*I]]= -Divide[1,2]*Erfc[+ I*\[Eta]*Sqrt[a/ 2]]+ I*T*(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(- Pi*I*a))/(2*Pi*I)*GAMMA(- a, z*exp(- Pi*I))= +(1)/(2)*erfc(- I*eta*sqrt(a/ 2))+ I*T*(a , eta)	Gamma[a + 1]*Divide[Exp[- Pi*I*a],2*Pi*I]*Gamma[- a, z*Exp[- Pi*I]]= +Divide[1,2]*Erfc[- I*\[Eta]*Sqrt[a/ 2]]+ I*T*(a , \[Eta])	Failure	Failure	Error	Error
8.12.E6	$z^{-a}{\gamma }^{*}(-a,-z)=\cos \left(\pi a\right)-2\sin \left(\pi a\right)\left(\frac{e^{\frac{1}{2}a{\eta }^2}}{\sqrt{\pi }}F\left(\eta \sqrt{a/2}\right)+T(a,\eta )\right)$