9.5

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
9.5.E1	${\displaystyle{\displaystyle\mathrm{Ai}\left(x\right)=\frac{1}{\pi}\int_{0}^{\infty}\cos\left(\tfrac{1}{3}t^{3}+xt\right)\mathrm{d}t}}$ \AiryAi@{x} = \frac{1}{\pi}\int_{0}^{\infty}\cos@{\tfrac{1}{3}t^{3}+xt}\diff{t}		AiryAi(x) = (1)/(Pi)*int(cos((1)/(3)*(t)^(3)+ x*t), t = 0..infinity)	AiryAi[x] == Divide[1,Pi]*Integrate[Cos[Divide[1,3]*(t)^(3)+ x*t], {t, 0, Infinity}, GenerateConditions->None]	Successful	Successful	-	Successful [Tested: 3]
9.5.E2	${\displaystyle{\displaystyle\mathrm{Ai}\left(-x\right)=\frac{x^{\ifrac{1}{2}}}{\pi}\int_{-1}^{\infty}\cos\left(x^{\ifrac{3}{2}}(\tfrac{1}{3}t^{3}+t^{2}-\tfrac{2}{3})\right)\mathrm{d}t}}$ \AiryAi@{-x} = \frac{x^{\ifrac{1}{2}}}{\pi}\int_{-1}^{\infty}\cos@{x^{\ifrac{3}{2}}(\tfrac{1}{3}t^{3}+t^{2}-\tfrac{2}{3})}\diff{t}	${\displaystyle{\displaystyle x>0}}$	AiryAi(- x) = ((x)^((1)/(2)))/(Pi)*int(cos((x)^((3)/(2))*((1)/(3)*(t)^(3)+ (t)^(2)-(2)/(3))), t = - 1..infinity)	AiryAi[- x] == Divide[(x)^(Divide[1,2]),Pi]*Integrate[Cos[(x)^(Divide[3,2])*(Divide[1,3]*(t)^(3)+ (t)^(2)-Divide[2,3])], {t, - 1, Infinity}, GenerateConditions->None]	Failure	Aborted	Skipped - Because timed out	Skipped - Because timed out
9.5.E3	${\displaystyle{\displaystyle\mathrm{Bi}\left(x\right)=\frac{1}{\pi}\int_{0}^{\infty}\exp\left(-{\tfrac{1}{3}}t^{3}+xt\right)\mathrm{d}t+\frac{1}{\pi}\int_{0}^{\infty}\sin\left(\tfrac{1}{3}t^{3}+xt\right)\mathrm{d}t}}$ \AiryBi@{x} = \frac{1}{\pi}\int_{0}^{\infty}\exp@{-{\tfrac{1}{3}}t^{3}+xt}\diff{t}+\frac{1}{\pi}\int_{0}^{\infty}\sin@{\tfrac{1}{3}t^{3}+xt}\diff{t}		AiryBi(x) = (1)/(Pi)*int(exp(-(1)/(3)*(t)^(3)+ x*t), t = 0..infinity)+(1)/(Pi)*int(sin((1)/(3)*(t)^(3)+ x*t), t = 0..infinity)	AiryBi[x] == Divide[1,Pi]*Integrate[Exp[-Divide[1,3]*(t)^(3)+ x*t], {t, 0, Infinity}, GenerateConditions->None]+Divide[1,Pi]*Integrate[Sin[Divide[1,3]*(t)^(3)+ x*t], {t, 0, Infinity}, GenerateConditions->None]	Failure	Failure	Failed [3 / 3] Result: 1.510759173-.1408206709*I Test Values: {x = 1.5} Result: .2865429290-.9608783696e-1*I Test Values: {x = .5} ... skip entries to safe data	Successful [Tested: 3]
9.5.E4	${\displaystyle{\displaystyle\mathrm{Ai}\left(z\right)=\frac{1}{2\pi i}\int_{\infty e^{-\pi i/3}}^{\infty e^{\pi i/3}}\exp\left(\tfrac{1}{3}t^{3}-zt\right)\mathrm{d}t}}$ \AiryAi@{z} = \frac{1}{2\pi i}\int_{\infty e^{-\pi i/3}}^{\infty e^{\pi i/3}}\exp@{\tfrac{1}{3}t^{3}-zt}\diff{t}		AiryAi(z) = (1)/(2*Pi*I)*int(exp((1)/(3)*(t)^(3)- z*t), t = infinity*exp(- Pi*I/3)..infinity*exp(Pi*I/3))	AiryAi[z] == Divide[1,2*Pi*I]*Integrate[Exp[Divide[1,3]*(t)^(3)- z*t], {t, Infinity*Exp[- Pi*I/3], Infinity*Exp[Pi*I/3]}, GenerateConditions->None]	Failure	Failure	Failed [7 / 7] Result: .1401376924-.8868274596e-1*I Test Values: {z = 1/2*3^(1/2)+1/2*I} Result: .5566528573-.2432725641*I Test Values: {z = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Failed [7 / 7] Result: Plus[Complex[0.14013769245288224, -0.08868274597809751], Times[Complex[0.0, 0.15915494309189535], NIntegrate[Power[E, Plus[Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]], t], Times[Rational[1, 3], Power[t, 3]]]] Test Values: {t, DirectedInfinity[Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]], DirectedInfinity[Power[E, Times[Complex[0, Rational[1, 3]], Pi]]]}]]], {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Plus[Complex[0.5566528572571797, -0.24327256400505012], Times[Complex[0.0, 0.15915494309189535], NIntegrate[Power[E, Plus[Times[-1, Power[E, Times[Complex[0, Rational[2, 3]], Pi]], t], Times[Rational[1, 3], Power[t, 3]]]] Test Values: {t, DirectedInfinity[Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]], DirectedInfinity[Power[E, Times[Complex[0, Rational[1, 3]], Pi]]]}]]], {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
9.5.E5	${\displaystyle{\displaystyle\mathrm{Bi}\left(z\right)=\frac{1}{2\pi}\int_{-\infty}^{\infty e^{\pi i/3}}\exp\left(\tfrac{1}{3}t^{3}-zt\right)\mathrm{d}t+\dfrac{1}{2\pi}\int_{-\infty}^{\infty e^{-\pi i/3}}\exp\left(\tfrac{1}{3}t^{3}-zt\right)\mathrm{d}t}}$ \AiryBi@{z} = \frac{1}{2\pi}\int_{-\infty}^{\infty e^{\pi i/3}}\exp@{\tfrac{1}{3}t^{3}-zt}\diff{t}+\dfrac{1}{2\pi}\int_{-\infty}^{\infty e^{-\pi i/3}}\exp@{\tfrac{1}{3}t^{3}-zt}\diff{t}		AiryBi(z) = (1)/(2*Pi)*int(exp((1)/(3)*(t)^(3)- z*t), t = - infinity..infinity*exp(Pi*I/3))+(1)/(2*Pi)*int(exp((1)/(3)*(t)^(3)- z*t), t = - infinity..infinity*exp(- Pi*I/3))	AiryBi[z] == Divide[1,2*Pi]*Integrate[Exp[Divide[1,3]*(t)^(3)- z*t], {t, - Infinity, Infinity*Exp[Pi*I/3]}, GenerateConditions->None]+Divide[1,2*Pi]*Integrate[Exp[Divide[1,3]*(t)^(3)- z*t], {t, - Infinity, Infinity*Exp[- Pi*I/3]}, GenerateConditions->None]	Failure	Failure	Error	Skipped - Because timed out
9.5.E6	${\displaystyle{\displaystyle\mathrm{Ai}\left(z\right)=\frac{\sqrt{3}}{2\pi}\int_{0}^{\infty}\exp\left(-\frac{t^{3}}{3}-\frac{z^{3}}{3t^{3}}\right)\mathrm{d}t}}$ \AiryAi@{z} = \frac{\sqrt{3}}{2\pi}\int_{0}^{\infty}\exp@{-\frac{t^{3}}{3}-\frac{z^{3}}{3t^{3}}}\diff{t}	${\displaystyle{\displaystyle|\operatorname{ph}z|<\tfrac{1}{6}\pi}}$	AiryAi(z) = (sqrt(3))/(2*Pi)*int(exp(-((t)^(3))/(3)-((z)^(3))/(3*(t)^(3))), t = 0..infinity)	AiryAi[z] == Divide[Sqrt[3],2*Pi]*Integrate[Exp[-Divide[(t)^(3),3]-Divide[(z)^(3),3*(t)^(3)]], {t, 0, Infinity}, GenerateConditions->None]	Successful	Successful	-	Successful [Tested: 3]
9.5.E7	${\displaystyle{\displaystyle\mathrm{Ai}\left(z\right)=\frac{e^{-\zeta}}{\pi}\int_{0}^{\infty}\exp\left(-z^{\ifrac{1}{2}}t^{2}\right)\cos\left(\tfrac{1}{3}t^{3}\right)\mathrm{d}t}}$ \AiryAi@{z} = \frac{e^{-\zeta}}{\pi}\int_{0}^{\infty}\exp@{-z^{\ifrac{1}{2}}t^{2}}\cos@{\tfrac{1}{3}t^{3}}\diff{t}	${\displaystyle{\displaystyle|\operatorname{ph}z|<\pi}}$	AiryAi(z) = (exp(-(2)/(3)*(z)^((3)/(2))))/(Pi)*int(exp(- (z)^((1)/(2))* (t)^(2))*cos((1)/(3)*(t)^(3)), t = 0..infinity)	AiryAi[z] == Divide[Exp[-Divide[2,3]*(z)^(Divide[3,2])],Pi]*Integrate[Exp[- (z)^(Divide[1,2])* (t)^(2)]*Cos[Divide[1,3]*(t)^(3)], {t, 0, Infinity}, GenerateConditions->None]	Failure	Aborted	Failed [1 / 7] Result: .2560433475+.3687851240*I Test Values: {z = -1/2*3^(1/2)-1/2*I}	Failed [1 / 7] Result: Complex[1.0282029471418963, 0.1796919597060948] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}
9.5.E8	${\displaystyle{\displaystyle\mathrm{Ai}\left(z\right)=\frac{e^{-\zeta}\zeta^{\ifrac{-1}{6}}}{\sqrt{\pi}(48)^{\ifrac{1}{6}}\Gamma\left(\frac{5}{6}\right)}\int_{0}^{\infty}e^{-t}t^{-\ifrac{1}{6}}\left(2+\frac{t}{\zeta}\right)^{-\ifrac{1}{6}}\mathrm{d}t}}$ \AiryAi@{z} = \frac{e^{-\zeta}\zeta^{\ifrac{-1}{6}}}{\sqrt{\pi}(48)^{\ifrac{1}{6}}\EulerGamma@{\frac{5}{6}}}\int_{0}^{\infty}e^{-t}t^{-\ifrac{1}{6}}\left(2+\frac{t}{\zeta}\right)^{-\ifrac{1}{6}}\diff{t}	${\displaystyle{\displaystyle|\operatorname{ph}z|<\frac{2}{3}\pi}}$	AiryAi(z) = (exp(-(2)/(3)*(z)^((3)/(2)))*(2)/(3)*((z)^((3)/(2)))^((- 1)/(6)))/(sqrt(Pi)*(48)^((1)/(6))* GAMMA((5)/(6)))*int(exp(- t)*(t)^(-(1)/(6))*(2 +(t)/((2)/(3)*(z)^((3)/(2))))^(-(1)/(6)), t = 0..infinity)	AiryAi[z] == Divide[Exp[-Divide[2,3]*(z)^(Divide[3,2])]*Divide[2,3]*((z)^(Divide[3,2]))^(Divide[- 1,6]),Sqrt[Pi]*(48)^(Divide[1,6])* Gamma[Divide[5,6]]]*Integrate[Exp[- t]*(t)^(-Divide[1,6])*(2 +Divide[t,Divide[2,3]*(z)^(Divide[3,2])])^(-Divide[1,6]), {t, 0, Infinity}, GenerateConditions->None]	Failure	Aborted	Failed [5 / 5] Result: .5281740434e-1-.3342421534e-1*I Test Values: {z = 1/2*3^(1/2)+1/2*I} Result: .674352291e-1+.776049915e-1*I Test Values: {z = 1/2-1/2*I*3^(1/2)} ... skip entries to safe data	Failed [5 / 5] Result: Complex[0.0528174043849943, -0.03342421567182417] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.06743522883170047, 0.07760499149873934] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]} ... skip entries to safe data