Airy and Related Functions - 9.10 Integrals

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DLMF Formula Constraints Maple Mathematica Symbolic
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9.10.E1 ∫ z ∞ Ai ⁡ ( t ) ⁢ d t = π ⁢ ( Ai ⁡ ( z ) ⁢ Gi ′ ⁡ ( z ) - Ai ′ ⁡ ( z ) ⁢ Gi ⁡ ( z ) ) superscript subscript 𝑧 Airy-Ai 𝑡 𝑡 𝜋 Airy-Ai 𝑧 diffop Scorer-Gi 1 𝑧 diffop Airy-Ai 1 𝑧 Scorer-Gi 𝑧 {\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 ∫ - ∞ z Ai ⁡ ( t ) ⁢ d t = π ⁢ ( Ai ⁡ ( z ) ⁢ Hi ′ ⁡ ( z ) - Ai ′ ⁡ ( z ) ⁢ Hi ⁡ ( z ) ) superscript subscript 𝑧 Airy-Ai 𝑡 𝑡 𝜋 Airy-Ai 𝑧 diffop Scorer-Hi 1 𝑧 diffop Airy-Ai 1 𝑧 Scorer-Hi 𝑧 {\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 ∫ - ∞ z Bi ⁡ ( t ) ⁢ d t = ∫ 0 z Bi ⁡ ( t ) ⁢ d t superscript subscript 𝑧 Airy-Bi 𝑡 𝑡 superscript subscript 0 𝑧 Airy-Bi 𝑡 𝑡 {\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 ∫ 0 z Bi ⁡ ( t ) ⁢ d t = π ⁢ ( Bi ′ ⁡ ( z ) ⁢ Gi ⁡ ( z ) - Bi ⁡ ( z ) ⁢ Gi ′ ⁡ ( z ) ) superscript subscript 0 𝑧 Airy-Bi 𝑡 𝑡 𝜋 diffop Airy-Bi 1 𝑧 Scorer-Gi 𝑧 Airy-Bi 𝑧 diffop Scorer-Gi 1 𝑧 {\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 π ⁢ ( Bi ′ ⁡ ( z ) ⁢ Gi ⁡ ( z ) - Bi ⁡ ( z ) ⁢ Gi ′ ⁡ ( z ) ) = π ⁢ ( Bi ⁡ ( z ) ⁢ Hi ′ ⁡ ( z ) - Bi ′ ⁡ ( z ) ⁢ Hi ⁡ ( z ) ) 𝜋 diffop Airy-Bi 1 𝑧 Scorer-Gi 𝑧 Airy-Bi 𝑧 diffop Scorer-Gi 1 𝑧 𝜋 Airy-Bi 𝑧 diffop Scorer-Hi 1 𝑧 diffop Airy-Bi 1 𝑧 Scorer-Hi 𝑧 {\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 ∫ 0 ∞ Ai ⁡ ( t ) ⁢ d t = 1 3 superscript subscript 0 Airy-Ai 𝑡 𝑡 1 3 {\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 ∫ - ∞ 0 Ai ⁡ ( t ) ⁢ d t = 2 3 superscript subscript 0 Airy-Ai 𝑡 𝑡 2 3 {\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 ∫ - ∞ 0 Bi ⁡ ( t ) ⁢ d t = 0 superscript subscript 0 Airy-Bi 𝑡 𝑡 0 {\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 ∫ - ∞ ∞ e p ⁢ t ⁢ Ai ⁡ ( t ) ⁢ d t = e p 3 / 3 superscript subscript superscript 𝑒 𝑝 𝑡 Airy-Ai 𝑡 𝑡 superscript 𝑒 superscript 𝑝 3 3 {\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}		 ℜ ⁡ p > 0 𝑝 0 {\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 ∫ 0 ∞ e - p ⁢ t ⁢ Ai ⁡ ( t ) ⁢ d t = e - p 3 / 3 ⁢ ( 1 3 - p ⁢ F 1 1 ⁡ ( 1 3 ; 4 3 ; 1 3 ⁢ p 3 ) 3 4 / 3 ⁢ Γ ⁡ ( 4 3 ) + p 2 ⁢ F 1 1 ⁡ ( 2 3 ; 5 3 ; 1 3 ⁢ p 3 ) 3 5 / 3 ⁢ Γ ⁡ ( 5 3 ) ) superscript subscript 0 superscript 𝑒 𝑝 𝑡 Airy-Ai 𝑡 𝑡 superscript 𝑒 superscript 𝑝 3 3 1 3 𝑝 Kummer-confluent-hypergeometric-M-as-1F1 1 3 4 3 1 3 superscript 𝑝 3 superscript 3 4 3 Euler-Gamma 4 3 superscript 𝑝 2 Kummer-confluent-hypergeometric-M-as-1F1 2 3 5 3 1 3 superscript 𝑝 3 superscript 3 5 3 Euler-Gamma 5 3 {\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 ∫ 0 ∞ e - p ⁢ t ⁢ Ai ⁡ ( - t ) ⁢ d t = 1 3 ⁢ e p 3 / 3 ⁢ ( Γ ⁡ ( 1 3 , 1 3 ⁢ p 3 ) Γ ⁡ ( 1 3 ) + Γ ⁡ ( 2 3 , 1 3 ⁢ p 3 ) Γ ⁡ ( 2 3 ) ) superscript subscript 0 superscript 𝑒 𝑝 𝑡 Airy-Ai 𝑡 𝑡 1 3 superscript 𝑒 superscript 𝑝 3 3 incomplete-Gamma 1 3 1 3 superscript 𝑝 3 Euler-Gamma 1 3 incomplete-Gamma 2 3 1 3 superscript 𝑝 3 Euler-Gamma 2 3 {\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)}		 ℜ ⁡ p > 0 𝑝 0 {\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 ∫ 0 ∞ e - p ⁢ t ⁢ Bi ⁡ ( - t ) ⁢ d t = 1 3 ⁢ e p 3 / 3 ⁢ ( Γ ⁡ ( 2 3 , 1 3 ⁢ p 3 ) Γ ⁡ ( 2 3 ) - Γ ⁡ ( 1 3 , 1 3 ⁢ p 3 ) Γ ⁡ ( 1 3 ) ) superscript subscript 0 superscript 𝑒 𝑝 𝑡 Airy-Bi 𝑡 𝑡 1 3 superscript 𝑒 superscript 𝑝 3 3 incomplete-Gamma 2 3 1 3 superscript 𝑝 3 Euler-Gamma 2 3 incomplete-Gamma 1 3 1 3 superscript 𝑝 3 Euler-Gamma 1 3 {\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)}		 ℜ ⁡ p > 0 𝑝 0 {\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 Ai ⁡ ( z ) = 3 ⁢ z 5 / 4 ⁢ e - ( 2 / 3 ) ⁢ z 3 / 2 4 ⁢ π ⁢ ∫ 0 ∞ t - 3 / 4 ⁢ e - ( 2 / 3 ) ⁢ t 3 / 2 ⁢ Ai ⁡ ( t ) z 3 / 2 + t 3 / 2 ⁢ d t Airy-Ai 𝑧 3 superscript 𝑧 5 4 superscript 𝑒 2 3 superscript 𝑧 3 2 4 𝜋 superscript subscript 0 superscript 𝑡 3 4 superscript 𝑒 2 3 superscript 𝑡 3 2 Airy-Ai 𝑡 superscript 𝑧 3 2 superscript 𝑡 3 2 𝑡 {\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}		 | ph ⁡ z | < 2 3 ⁢ π phase 𝑧 2 3 𝜋 {\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 Ai ⁡ ( z ) = z 5 / 4 ⁢ e - ( 2 / 3 ) ⁢ z 3 / 2 2 7 / 2 ⁢ π ⁢ ∫ 0 ∞ t - 1 / 2 ⁢ e - ( 2 / 3 ) ⁢ t 3 / 2 ⁢ Ai ⁡ ( t ) z 3 / 2 + t 3 / 2 ⁢ d t Airy-Ai 𝑧 superscript 𝑧 5 4 superscript 𝑒 2 3 superscript 𝑧 3 2 superscript 2 7 2 𝜋 superscript subscript 0 superscript 𝑡 1 2 superscript 𝑒 2 3 superscript 𝑡 3 2 Airy-Ai 𝑡 superscript 𝑧 3 2 superscript 𝑡 3 2 𝑡 {\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 ∫ 0 x ∫ 0 v Ai ⁡ ( t ) ⁢ d t ⁢ d v = x ⁢ ∫ 0 x Ai ⁡ ( t ) ⁢ d t - Ai ′ ⁡ ( x ) + Ai ′ ⁡ ( 0 ) superscript subscript 0 𝑥 superscript subscript 0 𝑣 Airy-Ai 𝑡 𝑡 𝑣 𝑥 superscript subscript 0 𝑥 Airy-Ai 𝑡 𝑡 diffop Airy-Ai 1 𝑥 diffop Airy-Ai 1 0 {\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 ∫ 0 x ∫ 0 v Bi ⁡ ( t ) ⁢ d t ⁢ d v = x ⁢ ∫ 0 x Bi ⁡ ( t ) ⁢ d t - Bi ′ ⁡ ( x ) + Bi ′ ⁡ ( 0 ) superscript subscript 0 𝑥 superscript subscript 0 𝑣 Airy-Bi 𝑡 𝑡 𝑣 𝑥 superscript subscript 0 𝑥 Airy-Bi 𝑡 𝑡 diffop Airy-Bi 1 𝑥 diffop Airy-Bi 1 0 {\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]
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