8.6

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
8.6.E1	${\displaystyle{\displaystyle\gamma\left(a,z\right)=\frac{z^{a}}{\sin\left(\pi a\right)}\int_{0}^{\pi}e^{z\cos t}\cos\left(at+z\sin t\right)\mathrm{d}t}}$ \incgamma@{a}{z} = \frac{z^{a}}{\sin@{\pi a}}\int_{0}^{\pi}e^{z\cos@@{t}}\cos@{at+z\sin@@{t}}\diff{t}	${\displaystyle{\displaystyle\Re a>0}}$	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}, GenerateConditions->None]	Failure	Aborted	Failed [14 / 21] Result: 1.922649672+.1964472815*I Test Values: {a = .5, z = 1/2*3^(1/2)+1/2*I, a = 3/2} Result: 2.511118576+1.941926371*I Test Values: {a = .5, z = -1/2+1/2*I*3^(1/2), a = 3/2} ... skip entries to safe data	Skipped - Because timed out
8.6.E2	${\displaystyle{\displaystyle\gamma\left(a,z\right)=z^{\frac{1}{2}a}\int_{0}^{\infty}e^{-t}t^{\frac{1}{2}a-1}J_{a}\left(2\sqrt{zt}\right)\mathrm{d}t}}$ \incgamma@{a}{z} = z^{\frac{1}{2}a}\int_{0}^{\infty}e^{-t}t^{\frac{1}{2}a-1}\BesselJ{a}@{2\sqrt{zt}}\diff{t}	${\displaystyle{\displaystyle\Re a>0}}$	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}, GenerateConditions->None]	Failure	Aborted	Successful [Tested: 21]	Skipped - Because timed out
8.6.E3	${\displaystyle{\displaystyle\gamma\left(a,z\right)=z^{a}\int_{0}^{\infty}\exp\left(-at-ze^{-t}\right)\mathrm{d}t}}$ \incgamma@{a}{z} = z^{a}\int_{0}^{\infty}\exp@{-at-ze^{-t}}\diff{t}	${\displaystyle{\displaystyle\Re a>0}}$	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}, GenerateConditions->None]	Failure	Successful	Successful [Tested: 21]	Successful [Tested: 21]
8.6.E4	${\displaystyle{\displaystyle\Gamma\left(a,z\right)=\frac{z^{a}e^{-z}}{\Gamma\left(1-a\right)}\int_{0}^{\infty}\frac{t^{-a}e^{-t}}{z+t}\mathrm{d}t}}$ \incGamma@{a}{z} = \frac{z^{a}e^{-z}}{\EulerGamma@{1-a}}\int_{0}^{\infty}\frac{t^{-a}e^{-t}}{z+t}\diff{t}	${\displaystyle{\displaystyle|\operatorname{ph}z|<\pi,\Re a<1,\Re(1-a)>0}}$	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}, GenerateConditions->None]	Failure	Successful	Failed [12 / 28] Result: Float(infinity)+Float(infinity)*I Test Values: {a = -1.5, z = 1/2*3^(1/2)+1/2*I} Result: Float(infinity)+Float(infinity)*I Test Values: {a = -1.5, z = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Successful [Tested: 28]
8.6.E5	${\displaystyle{\displaystyle\Gamma\left(a,z\right)=z^{a}e^{-z}\int_{0}^{\infty}\frac{e^{-zt}}{(1+t)^{1-a}}\mathrm{d}t}}$ \incGamma@{a}{z} = z^{a}e^{-z}\int_{0}^{\infty}\frac{e^{-zt}}{(1+t)^{1-a}}\diff{t}	${\displaystyle{\displaystyle\Re z>0}}$	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}, GenerateConditions->None]	Successful	Successful	-	Successful [Tested: 30]
8.6.E6	${\displaystyle{\displaystyle\Gamma\left(a,z\right)=\frac{2z^{\frac{1}{2}a}e^{-z}}{\Gamma\left(1-a\right)}\int_{0}^{\infty}e^{-t}t^{-\frac{1}{2}a}K_{a}\left(2\sqrt{zt}\right)\mathrm{d}t}}$ \incGamma@{a}{z} = \frac{2z^{\frac{1}{2}a}e^{-z}}{\EulerGamma@{1-a}}\int_{0}^{\infty}e^{-t}t^{-\frac{1}{2}a}\modBesselK{a}@{2\sqrt{zt}}\diff{t}	${\displaystyle{\displaystyle\Re a<1,\Re(1-a)>0}}$	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}, GenerateConditions->None]	Successful	Aborted	-	Successful [Tested: 28]
8.6.E7	${\displaystyle{\displaystyle\Gamma\left(a,z\right)=z^{a}\int_{0}^{\infty}\exp\left(at-ze^{t}\right)\mathrm{d}t}}$ \incGamma@{a}{z} = z^{a}\int_{0}^{\infty}\exp@{at-ze^{t}}\diff{t}	${\displaystyle{\displaystyle\Re z>0}}$	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}, GenerateConditions->None]	Failure	Successful	Successful [Tested: 30]	Successful [Tested: 30]
8.6.E8	${\displaystyle{\displaystyle\gamma\left(a,z\right)=\frac{-\mathrm{i}z^{a}}{2\sin\left(\pi a\right)}\int_{-1}^{(0+)}t^{a-1}e^{zt}\mathrm{d}t}}$ \incgamma@{a}{z} = \frac{-\iunit z^{a}}{2\sin@{\pi a}}\int_{-1}^{(0+)}t^{a-1}e^{zt}\diff{t}	${\displaystyle{\displaystyle z\neq 0,\Re a>0}}$	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 +)}, GenerateConditions->None]	Error	Failure	-	Error
8.6.E9	${\displaystyle{\displaystyle\Gamma\left(-a,ze^{+\pi i}\right)=\frac{e^{z}e^{-\pi\mathrm{i}a}}{\Gamma\left(1+a\right)}\int_{0}^{\infty}\frac{t^{a}e^{-zt}}{t-1}\mathrm{d}t}}$ \incGamma@{-a}{ze^{+\pi i}} = \frac{e^{z}e^{-\pi\iunit a}}{\EulerGamma@{1+a}}\int_{0}^{\infty}\frac{t^{a}e^{-zt}}{t-1}\diff{t}	${\displaystyle{\displaystyle\Re z>0,\Re a>-1,\Re(1+a)>0}}$	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}, GenerateConditions->None]	Failure	Aborted	Failed [20 / 20] Result: Float(infinity)+Float(infinity)*I Test Values: {a = 1.5, z = 1/2*3^(1/2)+1/2*I} Result: Float(infinity)+Float(infinity)*I Test Values: {a = 1.5, z = 1/2-1/2*I*3^(1/2)} ... skip entries to safe data	Skipped - Because timed out
8.6.E9	${\displaystyle{\displaystyle\Gamma\left(-a,ze^{-\pi i}\right)=\frac{e^{z}e^{+\pi\mathrm{i}a}}{\Gamma\left(1+a\right)}\int_{0}^{\infty}\frac{t^{a}e^{-zt}}{t-1}\mathrm{d}t}}$ \incGamma@{-a}{ze^{-\pi i}} = \frac{e^{z}e^{+\pi\iunit a}}{\EulerGamma@{1+a}}\int_{0}^{\infty}\frac{t^{a}e^{-zt}}{t-1}\diff{t}	${\displaystyle{\displaystyle\Re z>0,\Re a>-1,\Re(1+a)>0}}$	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}, GenerateConditions->None]	Failure	Aborted	Failed [20 / 20] Result: Float(infinity)+Float(infinity)*I Test Values: {a = 1.5, z = 1/2*3^(1/2)+1/2*I} Result: Float(infinity)+Float(infinity)*I Test Values: {a = 1.5, z = 1/2-1/2*I*3^(1/2)} ... skip entries to safe data	Skipped - Because timed out
8.6.E10	${\displaystyle{\displaystyle\gamma\left(a,z\right)=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\frac{\Gamma\left(s\right)}{a-s}z^{a-s}\mathrm{d}s}}$ \incgamma@{a}{z} = \frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\frac{\EulerGamma@{s}}{a-s}z^{a-s}\diff{s}	${\displaystyle{\displaystyle|\operatorname{ph}z|<\tfrac{1}{2}\pi,\Re s>0,\Re a>0}}$	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}, GenerateConditions->None]	Failure	Aborted	Failed [90 / 90] Result: .3508882474+.1990162824*I Test Values: {a = 1.5, c = -1.5, z = 1/2*3^(1/2)+1/2*I, a = 1} Result: .2281607298-.4280186861*I Test Values: {a = 1.5, c = -1.5, z = 1/2-1/2*I*3^(1/2), a = 1} ... skip entries to safe data	Skipped - Because timed out
8.6.E11	${\displaystyle{\displaystyle\Gamma\left(a,z\right)=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\Gamma\left(s+a\right)\frac{z^{-s}}{s}\mathrm{d}s}}$ \incGamma@{a}{z} = \frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\EulerGamma@{s+a}\frac{z^{-s}}{s}\diff{s}	${\displaystyle{\displaystyle|\operatorname{ph}z|<\tfrac{1}{2}\pi,\Re(s+a)>0}}$	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}, GenerateConditions->None]	Failure	Aborted	Failed [180 / 180] Result: .1072320848e-1-.1480251451*I Test Values: {a = -1.5, c = -1.5, z = 1/2*3^(1/2)+1/2*I} Result: -.2224046553+.6479031822e-1*I Test Values: {a = -1.5, c = -1.5, z = 1/2-1/2*I*3^(1/2)} ... skip entries to safe data	Skipped - Because timed out
8.6.E12	${\displaystyle{\displaystyle\Gamma\left(a,z\right)=-\frac{z^{a-1}e^{-z}}{\Gamma\left(1-a\right)}\*\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\Gamma\left(s+1-a\right)\frac{\pi z^{-s}}{\sin\left(\pi s\right)}\mathrm{d}s}}$ \incGamma@{a}{z} = -\frac{z^{a-1}e^{-z}}{\EulerGamma@{1-a}}\*\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\EulerGamma@{s+1-a}\frac{\pi z^{-s}}{\sin@{\pi s}}\diff{s}	${\displaystyle{\displaystyle|\operatorname{ph}z|<\tfrac{3}{2}\pi,\Re(s+1-a)>0,\Re(1-a)>0}}$	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}, GenerateConditions->None]	Failure	Aborted	Failed [168 / 168] Result: .1072320848e-1-.1480251451*I Test Values: {a = -1.5, c = -1.5, z = 1/2*3^(1/2)+1/2*I, a = -1} Result: .7867555591e-1+.8824866094*I Test Values: {a = -1.5, c = -1.5, z = -1/2+1/2*I*3^(1/2), a = -1} ... skip entries to safe data	Skipped - Because timed out