9.10

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
9.10.E1	${\displaystyle{\displaystyle\int_{z}^{\infty}\mathrm{Ai}\left(t\right)\mathrm{d}t=\pi\left(\mathrm{Ai}\left(z\right)\mathrm{Gi}'\left(z\right)-\mathrm{Ai}'\left(z\right)\mathrm{Gi}\left(z\right)\right)}}$ \int_{z}^{\infty}\AiryAi@{t}\diff{t} = \pi\left(\AiryAi@{z}\ScorerGi'@{z}-\AiryAi'@{z}\ScorerGi@{z}\right)		int(AiryAi(t), t = z..infinity) = Pi*(AiryAi(z)*diff( AiryBi(z)*(int(AiryAi(t), t = (z) .. infinity))+AiryAi(z)*(int(AiryBi(t), t = 0 .. (z))), z$(1) )- diff( AiryAi(z), z$(1) )*AiryBi(z)*(int(AiryAi(t), t = (z) .. infinity))+AiryAi(z)*(int(AiryBi(t), t = 0 .. (z))))	Integrate[AiryAi[t], {t, z, Infinity}, GenerateConditions->None] == Pi*(AiryAi[z]*D[ScorerGi[z], {z, 1}]- D[AiryAi[z], {z, 1}]*ScorerGi[z])	Failure	Failure	Failed [7 / 7] Result: -.3430999769-.7863536809e-1*I Test Values: {z = 1/2*3^(1/2)+1/2*I} Result: .1173558541-.6113539683*I Test Values: {z = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Successful [Tested: 7]
9.10.E2	${\displaystyle{\displaystyle\int_{-\infty}^{z}\mathrm{Ai}\left(t\right)\mathrm{d}t=\pi\left(\mathrm{Ai}\left(z\right)\mathrm{Hi}'\left(z\right)-\mathrm{Ai}'\left(z\right)\mathrm{Hi}\left(z\right)\right)}}$ \int_{-\infty}^{z}\AiryAi@{t}\diff{t} = \pi\left(\AiryAi@{z}\ScorerHi'@{z}-\AiryAi'@{z}\ScorerHi@{z}\right)		int(AiryAi(t), t = - infinity..z) = Pi*(AiryAi(z)*diff( AiryBi(z)*(int(AiryAi(t), t = -infinity .. (z)))-AiryAi(z)*(int(AiryBi(t), t = -infinity .. (z))), z$(1) )- diff( AiryAi(z), z$(1) )*AiryBi(z)*(int(AiryAi(t), t = -infinity .. (z)))-AiryAi(z)*(int(AiryBi(t), t = -infinity .. (z))))	Integrate[AiryAi[t], {t, - Infinity, z}, GenerateConditions->None] == Pi*(AiryAi[z]*D[ScorerHi[z], {z, 1}]- D[AiryAi[z], {z, 1}]*ScorerHi[z])	Failure	Failure	Failed [7 / 7] Result: .3430999769+.7863536803e-1*I Test Values: {z = 1/2*3^(1/2)+1/2*I} Result: -.1173558550+.6113539681*I Test Values: {z = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Successful [Tested: 7]
9.10.E3	${\displaystyle{\displaystyle\int_{-\infty}^{z}\mathrm{Bi}\left(t\right)\mathrm{d}t=\int_{0}^{z}\mathrm{Bi}\left(t\right)\mathrm{d}t}}$ \int_{-\infty}^{z}\AiryBi@{t}\diff{t} = \int_{0}^{z}\AiryBi@{t}\diff{t}		int(AiryBi(t), t = - infinity..z) = int(AiryBi(t), t = 0..z)	Integrate[AiryBi[t], {t, - Infinity, z}, GenerateConditions->None] == Integrate[AiryBi[t], {t, 0, z}, GenerateConditions->None]	Successful	Successful	Skip - symbolical successful subtest	Successful [Tested: 7]
9.10.E3	${\displaystyle{\displaystyle\int_{0}^{z}\mathrm{Bi}\left(t\right)\mathrm{d}t=\pi\left(\mathrm{Bi}'\left(z\right)\mathrm{Gi}\left(z\right)-\mathrm{Bi}\left(z\right)\mathrm{Gi}'\left(z\right)\right)\\ }}$ \int_{0}^{z}\AiryBi@{t}\diff{t} = \pi\left(\AiryBi'@{z}\ScorerGi@{z}-\AiryBi@{z}\ScorerGi'@{z}\right)\\		int(AiryBi(t), t = 0..z) = Pi*(diff( AiryBi(z), z$(1) )*AiryBi(z)*(int(AiryAi(t), t = (z) .. infinity))+AiryAi(z)*(int(AiryBi(t), t = 0 .. (z)))- AiryBi(z)*diff( AiryBi(z)*(int(AiryAi(t), t = (z) .. infinity))+AiryAi(z)*(int(AiryBi(t), t = 0 .. (z))), z$(1) ))	Integrate[AiryBi[t], {t, 0, z}, GenerateConditions->None] == Pi*(D[AiryBi[z], {z, 1}]*ScorerGi[z]- AiryBi[z]*D[ScorerGi[z], {z, 1}])	Failure	Failure	Failed [7 / 7] Result: -.2028158445+.1550535689*I Test Values: {z = 1/2*3^(1/2)+1/2*I} Result: .5468682154-.3940689299*I Test Values: {z = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Successful [Tested: 7]
9.10.E3	${\displaystyle{\displaystyle\pi\left(\mathrm{Bi}'\left(z\right)\mathrm{Gi}\left(z\right)-\mathrm{Bi}\left(z\right)\mathrm{Gi}'\left(z\right)\right)\\ =\pi\left(\mathrm{Bi}\left(z\right)\mathrm{Hi}'\left(z\right)-\mathrm{Bi}'\left(z\right)\mathrm{Hi}\left(z\right)\right)}}$ \pi\left(\AiryBi'@{z}\ScorerGi@{z}-\AiryBi@{z}\ScorerGi'@{z}\right)\\ = \pi\left(\AiryBi@{z}\ScorerHi'@{z}-\AiryBi'@{z}\ScorerHi@{z}\right)		Pi*(diff( AiryBi(z), z$(1) )*AiryBi(z)*(int(AiryAi(t), t = (z) .. infinity))+AiryAi(z)*(int(AiryBi(t), t = 0 .. (z)))- AiryBi(z)*diff( AiryBi(z)*(int(AiryAi(t), t = (z) .. infinity))+AiryAi(z)*(int(AiryBi(t), t = 0 .. (z))), z$(1) )) = Pi*(AiryBi(z)*diff( AiryBi(z)*(int(AiryAi(t), t = -infinity .. (z)))-AiryAi(z)*(int(AiryBi(t), t = -infinity .. (z))), z$(1) )- diff( AiryBi(z), z$(1) )*AiryBi(z)*(int(AiryAi(t), t = -infinity .. (z)))-AiryAi(z)*(int(AiryBi(t), t = -infinity .. (z))))	Pi*(D[AiryBi[z], {z, 1}]*ScorerGi[z]- AiryBi[z]*D[ScorerGi[z], {z, 1}]) == Pi*(AiryBi[z]*D[ScorerHi[z], {z, 1}]- D[AiryBi[z], {z, 1}]*ScorerHi[z])	Failure	Successful	Failed [7 / 7] Result: .843931870+.115991466*I Test Values: {z = 1/2*3^(1/2)+1/2*I} Result: -.9844521300+1.906824069*I Test Values: {z = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Successful [Tested: 7]
9.10#Ex1	${\displaystyle{\displaystyle\int_{0}^{\infty}\mathrm{Ai}\left(t\right)\mathrm{d}t=\tfrac{1}{3}}}$ \int_{0}^{\infty}\AiryAi@{t}\diff{t} = \tfrac{1}{3}		int(AiryAi(t), t = 0..infinity) = (1)/(3)	Integrate[AiryAi[t], {t, 0, Infinity}, GenerateConditions->None] == Divide[1,3]	Successful	Successful	-	Successful [Tested: 1]
9.10#Ex2	${\displaystyle{\displaystyle\int_{-\infty}^{0}\mathrm{Ai}\left(t\right)\mathrm{d}t=\tfrac{2}{3}}}$ \int_{-\infty}^{0}\AiryAi@{t}\diff{t} = \tfrac{2}{3}		int(AiryAi(t), t = - infinity..0) = (2)/(3)	Integrate[AiryAi[t], {t, - Infinity, 0}, GenerateConditions->None] == Divide[2,3]	Failure	Successful	Skip - No test values generated	Successful [Tested: 1]
9.10.E12	${\displaystyle{\displaystyle\int_{-\infty}^{0}\mathrm{Bi}\left(t\right)\mathrm{d}t=0}}$ \int_{-\infty}^{0}\AiryBi@{t}\diff{t} = 0		int(AiryBi(t), t = - infinity..0) = 0	Integrate[AiryBi[t], {t, - Infinity, 0}, GenerateConditions->None] == 0	Successful	Successful	-	Successful [Tested: 1]
9.10.E13	${\displaystyle{\displaystyle\int_{-\infty}^{\infty}e^{pt}\mathrm{Ai}\left(t\right)\mathrm{d}t=e^{p^{3}/3}}}$ \int_{-\infty}^{\infty}e^{pt}\AiryAi@{t}\diff{t} = e^{p^{3}/3}	${\displaystyle{\displaystyle\Re p>0}}$	int(exp(p*t)*AiryAi(t), t = - infinity..infinity) = exp((p)^(3)/3)	Integrate[Exp[p*t]*AiryAi[t], {t, - Infinity, Infinity}, GenerateConditions->None] == Exp[(p)^(3)/3]	Failure	Aborted	Successful [Tested: 5]	Skipped - Because timed out
9.10.E14	${\displaystyle{\displaystyle\int_{0}^{\infty}e^{-pt}\mathrm{Ai}\left(t\right)\mathrm{d}t=e^{-p^{3}/3}\left(\frac{1}{3}-\frac{p{{}_{1}F_{1}}\left(\tfrac{1}{3};\tfrac{4}{3};\tfrac{1}{3}p^{3}\right)}{3^{4/3}\Gamma\left(\tfrac{4}{3}\right)}+\frac{p^{2}{{}_{1}F_{1}}\left(\tfrac{2}{3};\tfrac{5}{3};\tfrac{1}{3}p^{3}\right)}{3^{5/3}\Gamma\left(\tfrac{5}{3}\right)}\right)}}$ \int_{0}^{\infty}e^{-pt}\AiryAi@{t}\diff{t} = e^{-p^{3}/3}\left(\frac{1}{3}-\frac{p\genhyperF{1}{1}@{\tfrac{1}{3}}{\tfrac{4}{3}}{\tfrac{1}{3}p^{3}}}{3^{4/3}\EulerGamma@{\tfrac{4}{3}}}+\frac{p^{2}\genhyperF{1}{1}@{\tfrac{2}{3}}{\tfrac{5}{3}}{\tfrac{1}{3}p^{3}}}{3^{5/3}\EulerGamma@{\tfrac{5}{3}}}\right)		int(exp(- p*t)*AiryAi(t), t = 0..infinity) = exp(- (p)^(3)/3)*((1)/(3)-(p*hypergeom([(1)/(3)], [(4)/(3)], (1)/(3)*(p)^(3)))/((3)^(4/3)* GAMMA((4)/(3)))+((p)^(2)* hypergeom([(2)/(3)], [(5)/(3)], (1)/(3)*(p)^(3)))/((3)^(5/3)* GAMMA((5)/(3))))	Integrate[Exp[- p*t]*AiryAi[t], {t, 0, Infinity}, GenerateConditions->None] == Exp[- (p)^(3)/3]*(Divide[1,3]-Divide[p*HypergeometricPFQ[{Divide[1,3]}, {Divide[4,3]}, Divide[1,3]*(p)^(3)],(3)^(4/3)* Gamma[Divide[4,3]]]+Divide[(p)^(2)* HypergeometricPFQ[{Divide[2,3]}, {Divide[5,3]}, Divide[1,3]*(p)^(3)],(3)^(5/3)* Gamma[Divide[5,3]]])	Successful	Successful	-	Successful [Tested: 1]
9.10.E15	${\displaystyle{\displaystyle\int_{0}^{\infty}e^{-pt}\mathrm{Ai}\left(-t\right)\mathrm{d}t={\frac{1}{3}e^{p^{3}/3}\left(\frac{\Gamma\left(\tfrac{1}{3},\tfrac{1}{3}p^{3}\right)}{\Gamma\left(\tfrac{1}{3}\right)}+\frac{\Gamma\left(\tfrac{2}{3},\tfrac{1}{3}p^{3}\right)}{\Gamma\left(\tfrac{2}{3}\right)}\right)}}}$ \int_{0}^{\infty}e^{-pt}\AiryAi@{-t}\diff{t} = {\frac{1}{3}e^{p^{3}/3}\left(\frac{\incGamma@{\tfrac{1}{3}}{\tfrac{1}{3}p^{3}}}{\EulerGamma@{\tfrac{1}{3}}}+\frac{\incGamma@{\tfrac{2}{3}}{\tfrac{1}{3}p^{3}}}{\EulerGamma@{\tfrac{2}{3}}}\right)}	${\displaystyle{\displaystyle\Re p>0}}$	int(exp(- p*t)*AiryAi(- t), t = 0..infinity) = (1)/(3)*exp((p)^(3)/3)*((GAMMA((1)/(3), (1)/(3)*(p)^(3)))/(GAMMA((1)/(3)))+(GAMMA((2)/(3), (1)/(3)*(p)^(3)))/(GAMMA((2)/(3))))	Integrate[Exp[- p*t]*AiryAi[- t], {t, 0, Infinity}, GenerateConditions->None] == Divide[1,3]*Exp[(p)^(3)/3]*(Divide[Gamma[Divide[1,3], Divide[1,3]*(p)^(3)],Gamma[Divide[1,3]]]+Divide[Gamma[Divide[2,3], Divide[1,3]*(p)^(3)],Gamma[Divide[2,3]]])	Failure	Aborted	Failed [1 / 5] Result: -.1e-9+.6037469539*I Test Values: {p = 1/2-1/2*I*3^(1/2)}	Skipped - Because timed out
9.10.E16	${\displaystyle{\displaystyle\int_{0}^{\infty}e^{-pt}\mathrm{Bi}\left(-t\right)\mathrm{d}t={\frac{1}{\sqrt{3}}e^{p^{3}/3}\left(\frac{\Gamma\left(\tfrac{2}{3},\tfrac{1}{3}p^{3}\right)}{\Gamma\left(\tfrac{2}{3}\right)}-\frac{\Gamma\left(\tfrac{1}{3},\tfrac{1}{3}p^{3}\right)}{\Gamma\left(\tfrac{1}{3}\right)}\right)}}}$ \int_{0}^{\infty}e^{-pt}\AiryBi@{-t}\diff{t} = {\frac{1}{\sqrt{3}}e^{p^{3}/3}\left(\frac{\incGamma@{\tfrac{2}{3}}{\tfrac{1}{3}p^{3}}}{\EulerGamma@{\tfrac{2}{3}}}-\frac{\incGamma@{\tfrac{1}{3}}{\tfrac{1}{3}p^{3}}}{\EulerGamma@{\tfrac{1}{3}}}\right)}	${\displaystyle{\displaystyle\Re p>0}}$	int(exp(- p*t)*AiryBi(- t), t = 0..infinity) = (1)/(sqrt(3))*exp((p)^(3)/3)*((GAMMA((2)/(3), (1)/(3)*(p)^(3)))/(GAMMA((2)/(3)))-(GAMMA((1)/(3), (1)/(3)*(p)^(3)))/(GAMMA((1)/(3))))	Integrate[Exp[- p*t]*AiryBi[- t], {t, 0, Infinity}, GenerateConditions->None] == Divide[1,Sqrt[3]]*Exp[(p)^(3)/3]*(Divide[Gamma[Divide[2,3], Divide[1,3]*(p)^(3)],Gamma[Divide[2,3]]]-Divide[Gamma[Divide[1,3], Divide[1,3]*(p)^(3)],Gamma[Divide[1,3]]])	Failure	Aborted	Failed [1 / 5] Result: -.5e-9-.1692833917*I Test Values: {p = 1/2-1/2*I*3^(1/2)}	Skipped - Because timed out
9.10.E18	${\displaystyle{\displaystyle\mathrm{Ai}\left(z\right)=\frac{3z^{5/4}e^{-(2/3)z^{3/2}}}{4\pi}\*\int_{0}^{\infty}\frac{t^{-3/4}e^{-(2/3)t^{3/2}}\mathrm{Ai}\left(t\right)}{z^{3/2}+t^{3/2}}\mathrm{d}t}}$ \AiryAi@{z} = \frac{3z^{5/4}e^{-(2/3)z^{3/2}}}{4\pi}\*\int_{0}^{\infty}\frac{t^{-3/4}e^{-(2/3)t^{3/2}}\AiryAi@{t}}{z^{3/2}+t^{3/2}}\diff{t}	${\displaystyle{\displaystyle|\operatorname{ph}z|<\tfrac{2}{3}\pi}}$	AiryAi(z) = (3*(z)^(5/4)* exp(-(2/3)*(z)^(3/2)))/(4*Pi)* int(((t)^(- 3/4)* exp(-(2/3)*(t)^(3/2))*AiryAi(t))/((z)^(3/2)+ (t)^(3/2)), t = 0..infinity)	AiryAi[z] == Divide[3*(z)^(5/4)* Exp[-(2/3)*(z)^(3/2)],4*Pi]* Integrate[Divide[(t)^(- 3/4)* Exp[-(2/3)*(t)^(3/2)]*AiryAi[t],(z)^(3/2)+ (t)^(3/2)], {t, 0, Infinity}, GenerateConditions->None]	Failure	Aborted	Successful [Tested: 5]	Skipped - Because timed out
9.10#Ex3	${\displaystyle{\displaystyle\mathrm{Ai}\left(z\right)=\frac{z^{5/4}e^{-(2/3)z^{3/2}}}{2^{7/2}\pi}\*\int_{0}^{\infty}\frac{t^{-1/2}e^{-(2/3)t^{3/2}}\mathrm{Ai}\left(t\right)}{z^{3/2}+t^{3/2}}\mathrm{d}t}}$ \AiryAi@{z} = \frac{z^{5/4}e^{-(2/3)z^{3/2}}}{2^{7/2}\pi}\*\int_{0}^{\infty}\frac{t^{-1/2}e^{-(2/3)t^{3/2}}\AiryAi@{t}}{z^{3/2}+t^{3/2}}\diff{t}		AiryAi(z) = ((z)^(5/4)* exp(-(2/3)*(z)^(3/2)))/((2)^(7/2)* Pi)* int(((t)^(- 1/2)* exp(-(2/3)*(t)^(3/2))*AiryAi(t))/((z)^(3/2)+ (t)^(3/2)), t = 0..infinity)	AiryAi[z] == Divide[(z)^(5/4)* Exp[-(2/3)*(z)^(3/2)],(2)^(7/2)* Pi]* Integrate[Divide[(t)^(- 1/2)* Exp[-(2/3)*(t)^(3/2)]*AiryAi[t],(z)^(3/2)+ (t)^(3/2)], {t, 0, Infinity}, GenerateConditions->None]	Failure	Aborted	Error	Skipped - Because timed out
9.10.E20	${\displaystyle{\displaystyle\int_{0}^{x}\!\!\int_{0}^{v}\mathrm{Ai}\left(t\right)\mathrm{d}t\mathrm{d}v=x\int_{0}^{x}\mathrm{Ai}\left(t\right)\mathrm{d}t-\mathrm{Ai}'\left(x\right)+\mathrm{Ai}'\left(0\right)}}$ \int_{0}^{x}\!\!\int_{0}^{v}\AiryAi@{t}\diff{t}\diff{v} = x\int_{0}^{x}\AiryAi@{t}\diff{t}-\AiryAi'@{x}+\AiryAi'@{0}		int(int(AiryAi(t), t = 0..v), v = 0..x) = x*int(AiryAi(t), t = 0..x)- diff( AiryAi(x), x$(1) )+ subs( temp=0, diff( AiryAi(temp), temp$(1) ) )	Integrate[Integrate[AiryAi[t], {t, 0, v}, GenerateConditions->None], {v, 0, x}, GenerateConditions->None] == x*Integrate[AiryAi[t], {t, 0, x}, GenerateConditions->None]- D[AiryAi[x], {x, 1}]+ (D[AiryAi[temp], {temp, 1}]/.temp-> 0)	Failure	Failure	Successful [Tested: 3]	Successful [Tested: 3]
9.10.E21	${\displaystyle{\displaystyle\int_{0}^{x}\!\!\int_{0}^{v}\mathrm{Bi}\left(t\right)\mathrm{d}t\mathrm{d}v=x\int_{0}^{x}\mathrm{Bi}\left(t\right)\mathrm{d}t-\mathrm{Bi}'\left(x\right)+\mathrm{Bi}'\left(0\right)}}$ \int_{0}^{x}\!\!\int_{0}^{v}\AiryBi@{t}\diff{t}\diff{v} = x\int_{0}^{x}\AiryBi@{t}\diff{t}-\AiryBi'@{x}+\AiryBi'@{0}		int(int(AiryBi(t), t = 0..v), v = 0..x) = x*int(AiryBi(t), t = 0..x)- diff( AiryBi(x), x$(1) )+ subs( temp=0, diff( AiryBi(temp), temp$(1) ) )	Integrate[Integrate[AiryBi[t], {t, 0, v}, GenerateConditions->None], {v, 0, x}, GenerateConditions->None] == x*Integrate[AiryBi[t], {t, 0, x}, GenerateConditions->None]- D[AiryBi[x], {x, 1}]+ (D[AiryBi[temp], {temp, 1}]/.temp-> 0)	Failure	Failure	Successful [Tested: 3]	Successful [Tested: 3]