8.10

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
8.10.E1	${\displaystyle{\displaystyle x^{1-a}e^{x}\Gamma\left(a,x\right)\leq 1}}$ x^{1-a}e^{x}\incGamma@{a}{x} \leq 1	${\displaystyle{\displaystyle x>0,0<a,a\leq 1}}$	(x)^(1 - a)* exp(x)*GAMMA(a, x) <= 1	(x)^(1 - a)* Exp[x]*Gamma[a, x] <= 1	Failure	Failure	Successful [Tested: 3]	Successful [Tested: 3]
8.10.E2	${\displaystyle{\displaystyle\gamma\left(a,x\right)\geq\frac{x^{a-1}}{a}(1-e^{-x})}}$ \incgamma@{a}{x} \geq \frac{x^{a-1}}{a}(1-e^{-x})	${\displaystyle{\displaystyle x>0,0<a,a\leq 1,\Re a>0}}$	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	Successful [Tested: 3]	Successful [Tested: 3]
8.10.E3	${\displaystyle{\displaystyle x^{1-a}e^{x}\Gamma\left(a,x\right)=1+\frac{a-1}{x}\vartheta}}$ x^{1-a}e^{x}\incGamma@{a}{x} = 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	Failed [180 / 180] Result: .8735840245+.8333333335*I Test Values: {a = -1.5, x = 1.5, vartheta = 1/2*3^(1/2)+1/2*I} Result: -1.403124983+1.443375674*I Test Values: {a = -1.5, x = 1.5, vartheta = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Failed [180 / 180] Result: Complex[0.8735840235649492, 0.8333333333333331] Test Values: {Rule[a, -1.5], Rule[x, 1.5], Rule[ϑ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-1.4031249827424481, 1.4433756729740643] Test Values: {Rule[a, -1.5], Rule[x, 1.5], Rule[ϑ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
8.10.E4	${\displaystyle{\displaystyle 0<\vartheta}}$ 0 < \vartheta	${\displaystyle{\displaystyle x>0,a\leq 2}}$	0 < vartheta	0 < \[CurlyTheta]	Skipped - no semantic math	Skipped - no semantic math	-	-
8.10.E5	${\displaystyle{\displaystyle A_{n}<x^{1-a}e^{x}\Gamma\left(a,x\right)}}$ A_{n} < x^{1-a}e^{x}\incGamma@{a}{x}	${\displaystyle{\displaystyle x>0,a<1}}$	A[n](<)*(x)^(1 - a)* exp(x)*GAMMA(a, x)	Subscript[A, n][<]*(x)^(1 - a)* Exp[x]*Gamma[a, x]	Failure	Failure	Failed [75 / 300] Result: 1.5 < .4302083505 Test Values: {a = -1.5, x = 1.5, A[n] = 1.5, n = 1} Result: 1.5 < .4302083505 Test Values: {a = -1.5, x = 1.5, A[n] = 1.5, n = 2} ... skip entries to safe data	Failed [195 / 300] Result: Less[Complex[0.8660254037844387, 0.49999999999999994], 0.43020835059088497] Test Values: {Rule[a, -1.5], Rule[n, 1], Rule[x, 1.5], Rule[Subscript[A, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Less[Complex[0.8660254037844387, 0.49999999999999994], 0.43020835059088497] Test Values: {Rule[a, -1.5], Rule[n, 2], Rule[x, 1.5], Rule[Subscript[A, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
8.10.E5	${\displaystyle{\displaystyle x^{1-a}e^{x}\Gamma\left(a,x\right)<B_{n}}}$ x^{1-a}e^{x}\incGamma@{a}{x} < B_{n}	${\displaystyle{\displaystyle x>0,a<1}}$	(x)^(1 - a)* exp(x)*GAMMA(a, x) < B[n]	(x)^(1 - a)* Exp[x]*Gamma[a, x] < Subscript[B, n]	Failure	Failure	Failed [105 / 300] Result: .4302083505 < -1.5 Test Values: {a = -1.5, x = 1.5, B[n] = -1.5, n = 1} Result: .4302083505 < -1.5 Test Values: {a = -1.5, x = 1.5, B[n] = -1.5, n = 2} ... skip entries to safe data	Failed [225 / 300] Result: Less[0.43020835059088497, Complex[0.8660254037844387, 0.49999999999999994]] Test Values: {Rule[a, -1.5], Rule[n, 1], Rule[x, 1.5], Rule[Subscript[B, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Less[0.43020835059088497, Complex[0.8660254037844387, 0.49999999999999994]] Test Values: {Rule[a, -1.5], Rule[n, 2], Rule[x, 1.5], Rule[Subscript[B, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
8.10#Ex1	${\displaystyle{\displaystyle A_{1}=\frac{x}{x+1-a}}}$ A_{1} = \frac{x}{x+1-a}		A[1] = (x)/(x + 1 - a)	Subscript[A, 1] == Divide[x,x + 1 - a]	Skipped - no semantic math	Skipped - no semantic math	-	-
8.10#Ex2	${\displaystyle{\displaystyle B_{1}=\frac{x+1}{x+2-a}}}$ B_{1} = \frac{x+1}{x+2-a}		B[1] = (x + 1)/(x + 2 - a)	Subscript[B, 1] == Divide[x + 1,x + 2 - a]	Skipped - no semantic math	Skipped - no semantic math	-	-
8.10#Ex3	${\displaystyle{\displaystyle A_{2}=\frac{x(x+3-a)}{x^{2}+2(2-a)x+(1-a)(2-a)}}}$ A_{2} = \frac{x(x+3-a)}{x^{2}+2(2-a)x+(1-a)(2-a)}		A[2] = (x*(x + 3 - a))/((x)^(2)+ 2*(2 - a)*x +(1 - a)*(2 - a))	Subscript[A, 2] == Divide[x*(x + 3 - a),(x)^(2)+ 2*(2 - a)*x +(1 - a)*(2 - a)]	Skipped - no semantic math	Skipped - no semantic math	-	-
8.10#Ex4	${\displaystyle{\displaystyle B_{2}=\frac{x^{2}+(5-a)x+2}{x^{2}+2(3-a)x+(2-a)(3-a)}}}$ B_{2} = \frac{x^{2}+(5-a)x+2}{x^{2}+2(3-a)x+(2-a)(3-a)}		B[2] = ((x)^(2)+(5 - a)*x + 2)/((x)^(2)+ 2*(3 - a)*x +(2 - a)*(3 - a))	Subscript[B, 2] == Divide[(x)^(2)+(5 - a)*x + 2,(x)^(2)+ 2*(3 - a)*x +(2 - a)*(3 - a)]	Skipped - no semantic math	Skipped - no semantic math	-	-
8.10.E7	${\displaystyle{\displaystyle I=\int_{0}^{x}t^{a-1}e^{t}\mathrm{d}t}}$ I = \int_{0}^{x}t^{a-1}e^{t}\diff{t}	${\displaystyle{\displaystyle\Re a>0}}$	I = int((t)^(a - 1)* exp(t), t = 0..x)	I == Integrate[(t)^(a - 1)* Exp[t], {t, 0, x}, GenerateConditions->None]	Failure	Failure	Failed [90 / 90] Result: -2.374226751+.5000000000*I Test Values: {I = 1/2*3^(1/2)+1/2*I, a = 1.5, x = 1.5} Result: .5451660792+.5000000000*I Test Values: {I = 1/2*3^(1/2)+1/2*I, a = 1.5, x = .5} ... skip entries to safe data	Failed [27 / 27] Result: Complex[-2.240252154794788, -4.0113152157384396*^-17] Test Values: {Rule[Complex[0, 1], 1], Rule[a, 1.5], Rule[x, 1.5]} Result: Complex[-1.240252154794788, -4.0113152157384396*^-17] Test Values: {Rule[Complex[0, 1], 2], Rule[a, 1.5], Rule[x, 1.5]} ... skip entries to safe data
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_{0}^{x}t^{a-1}e^{t}\diff{t} = \EulerGamma@{a}x^{a}\scincgamma@{a}{-x}	${\displaystyle{\displaystyle\Re a>0}}$	int((t)^(a - 1)* exp(t), t = 0..x) = GAMMA(a)*(x)^(a)* (- x)^(-(a))*(GAMMA(a)-GAMMA(a, - x))/GAMMA(a)	Error	Failure	Missing Macro Error	Successful [Tested: 9]	Skip - symbolical successful subtest
8.10.E8	${\displaystyle{\displaystyle\frac{(a+1)(a+2)-x}{(a+1)(a+2+x)}<ax^{-a}e^{-x}I}}$ \frac{(a+1)(a+2)-x}{(a+1)(a+2+x)} < ax^{-a}e^{-x}I	${\displaystyle{\displaystyle x>0,a\geq 0}}$	((a + 1)*(a + 2)- x)/((a + 1)*(a + 2 + x)) < a*(x)^(- a)* exp(- x)*I	Divide[(a + 1)*(a + 2)- x,(a + 1)*(a + 2 + x)] < a*(x)^(- a)* Exp[- x]*I	Skipped - no semantic math	Skipped - no semantic math	-	-
8.10#Ex5	${\displaystyle{\displaystyle c_{a}=(\Gamma\left(1+a\right))^{1/(a-1)}}}$ c_{a} = (\EulerGamma@{1+a})^{1/(a-1)}	${\displaystyle{\displaystyle\Re(1+a)>0}}$	c[a] = (GAMMA(1 + a))^(1/(a - 1))	Subscript[c, a] == (Gamma[1 + a])^(1/(a - 1))	Failure	Failure	Failed [39 / 40] Result: -.9011204630+.5000000000*I Test Values: {a = 1.5, c[a] = 1/2*3^(1/2)+1/2*I} Result: -2.267145867+.8660254040*I Test Values: {a = 1.5, c[a] = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Failed [39 / 40] Result: Complex[-0.9011204638598199, 0.49999999999999994] Test Values: {Rule[a, 1.5], Rule[Subscript[c, a], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-2.2671458676442584, 0.8660254037844387] Test Values: {Rule[a, 1.5], Rule[Subscript[c, a], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
8.10#Ex6	${\displaystyle{\displaystyle d_{a}=(\Gamma\left(1+a\right))^{-1/a}}}$ d_{a} = (\EulerGamma@{1+a})^{-1/a}	${\displaystyle{\displaystyle\Re(1+a)>0}}$	d[a] = (GAMMA(1 + a))^(- 1/a)	Subscript[d, a] == (Gamma[1 + a])^(- 1/a)	Failure	Failure	Failed [40 / 40] Result: .388914161e-1+.5000000000*I Test Values: {a = 1.5, d[a] = 1/2*3^(1/2)+1/2*I} Result: -1.327133988+.8660254040*I Test Values: {a = 1.5, d[a] = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Failed [40 / 40] Result: Complex[0.038891415918572037, 0.49999999999999994] Test Values: {Rule[a, 1.5], Rule[Subscript[d, a], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-1.3271339878658663, 0.8660254037844387] Test Values: {Rule[a, 1.5], Rule[Subscript[d, a], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
8.10.E10	${\displaystyle{\displaystyle\frac{x}{2a}\left(\left(1+\frac{2}{x}\right)^{a}-1\right)<x^{1-a}e^{x}\Gamma\left(a,x\right)}}$ \frac{x}{2a}\left(\left(1+\frac{2}{x}\right)^{a}-1\right) < x^{1-a}e^{x}\incGamma@{a}{x}	${\displaystyle{\displaystyle x\geq 0,0<a,a<1}}$	(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 [Tested: 3]	Successful [Tested: 3]
8.10.E10	${\displaystyle{\displaystyle x^{1-a}e^{x}\Gamma\left(a,x\right)\leq\frac{x}{ac_{a}}\left(\left(1+\frac{c_{a}}{x}\right)^{a}-1\right)}}$ x^{1-a}e^{x}\incGamma@{a}{x} \leq \frac{x}{ac_{a}}\left(\left(1+\frac{c_{a}}{x}\right)^{a}-1\right)	${\displaystyle{\displaystyle x\geq 0,0<a,a<1}}$	(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	Failed [3 / 30] Result: .8100694499 <= .7912878480 Test Values: {a = .5, x = 1.5, c[a] = 2} Result: .6556795425 <= .6180339885 Test Values: {a = .5, x = .5, c[a] = 2} ... skip entries to safe data	Failed [18 / 30] Result: LessEqual[0.8100694501969615, Complex[0.8808481919138387, -0.05137441674828974]] Test Values: {Rule[a, 0.5], Rule[x, 1.5], Rule[Subscript[c, a], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: LessEqual[0.8100694501969615, Complex[1.0324243733930456, -0.1801654927820326]] Test Values: {Rule[a, 0.5], Rule[x, 1.5], Rule[Subscript[c, a], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
8.10.E11	${\displaystyle{\displaystyle(1-e^{-\alpha_{a}x})^{a}\leq P\left(a,x\right)}}$ (1-e^{-\alpha_{a}x})^{a} \leq \normincGammaP@{a}{x}	${\displaystyle{\displaystyle x\geq 0,a>0}}$	(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	Failed [78 / 270] Result: .8461432717 <= .6083748237 Test Values: {a = 1.5, alpha = 1.5, x = 1.5, alpha[a] = 1.5} Result: .9262567903 <= .6083748237 Test Values: {a = 1.5, alpha = 1.5, x = 1.5, alpha[a] = 2} ... skip entries to safe data	Failed [240 / 270] Result: LessEqual[Complex[0.7016331692747775, 0.2500919864059583], 0.608374823728911] Test Values: {Rule[a, 1.5], Rule[x, 1.5], Rule[α, 1.5], Rule[Subscript[α, a], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: LessEqual[Complex[-1.369574242346028, 2.679891945423719], 0.608374823728911] Test Values: {Rule[a, 1.5], Rule[x, 1.5], Rule[α, 1.5], Rule[Subscript[α, a], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
8.10.E11	${\displaystyle{\displaystyle P\left(a,x\right)\leq(1-e^{-\beta_{a}x})^{a}}}$ \normincGammaP@{a}{x} \leq (1-e^{-\beta_{a}x})^{a}	${\displaystyle{\displaystyle x\geq 0,a>0}}$	(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	Failed [30 / 270] Result: .6083748237 <= .3832643966 Test Values: {a = 1.5, beta = 1.5, x = 1.5, beta[a] = .5} Result: .1987480431 <= .1040340193 Test Values: {a = 1.5, beta = 1.5, x = .5, beta[a] = .5} ... skip entries to safe data	Failed [192 / 270] Result: LessEqual[0.608374823728911, Complex[0.7016331692747775, 0.2500919864059583]] Test Values: {Rule[a, 1.5], Rule[x, 1.5], Rule[β, 1.5], Rule[Subscript[β, a], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: LessEqual[0.608374823728911, Complex[-1.369574242346028, 2.679891945423719]] Test Values: {Rule[a, 1.5], Rule[x, 1.5], Rule[β, 1.5], Rule[Subscript[β, a], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
8.10.E13	${\displaystyle{\displaystyle\frac{\Gamma\left(n,n\right)}{\Gamma\left(n\right)}<\frac{1}{2}}}$ \frac{\incGamma@{n}{n}}{\EulerGamma@{n}} < \frac{1}{2}	${\displaystyle{\displaystyle\Re n>0}}$	(GAMMA(n, n))/(GAMMA(n)) < (1)/(2)	Divide[Gamma[n, n],Gamma[n]] < Divide[1,2]	Failure	Failure	Successful [Tested: 3]	Successful [Tested: 1]
8.10.E13	${\displaystyle{\displaystyle\frac{1}{2}<\frac{\Gamma\left(n,n-1\right)}{\Gamma\left(n\right)}}}$ \frac{1}{2} < \frac{\incGamma@{n}{n-1}}{\EulerGamma@{n}}	${\displaystyle{\displaystyle\Re n>0}}$	(1)/(2) < (GAMMA(n, n - 1))/(GAMMA(n))	Divide[1,2] < Divide[Gamma[n, n - 1],Gamma[n]]	Failure	Failure	Successful [Tested: 3]	Successful [Tested: 1]