Airy and Related Functions - 9.5 Integral Representations

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
9.5.E1 Ai ⁡ ( x ) = 1 π ⁢ ∫ 0 ∞ cos ⁡ ( 1 3 ⁢ t 3 + x ⁢ t ) ⁢ d t Airy-Ai 𝑥 1 𝜋 superscript subscript 0 1 3 superscript 𝑡 3 𝑥 𝑡 𝑡 {\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 Ai ⁡ ( - x ) = x 1 / 2 π ⁢ ∫ - 1 ∞ cos ⁡ ( x 3 / 2 ⁢ ( 1 3 ⁢ t 3 + t 2 - 2 3 ) ) ⁢ d t Airy-Ai 𝑥 superscript 𝑥 1 2 𝜋 superscript subscript 1 superscript 𝑥 3 2 1 3 superscript 𝑡 3 superscript 𝑡 2 2 3 𝑡 {\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}		 x > 0 𝑥 0 {\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 Bi ⁡ ( x ) = 1 π ⁢ ∫ 0 ∞ exp ⁡ ( - 1 3 ⁢ t 3 + x ⁢ t ) ⁢ d t + 1 π ⁢ ∫ 0 ∞ sin ⁡ ( 1 3 ⁢ t 3 + x ⁢ t ) ⁢ d t Airy-Bi 𝑥 1 𝜋 superscript subscript 0 1 3 superscript 𝑡 3 𝑥 𝑡 𝑡 1 𝜋 superscript subscript 0 1 3 superscript 𝑡 3 𝑥 𝑡 𝑡 {\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 Ai ⁡ ( z ) = 1 2 ⁢ π ⁢ i ⁢ ∫ ∞ ⁢ e - π ⁢ i / 3 ∞ ⁢ e π ⁢ i / 3 exp ⁡ ( 1 3 ⁢ t 3 - z ⁢ t ) ⁢ d t Airy-Ai 𝑧 1 2 𝜋 𝑖 superscript subscript superscript 𝑒 𝜋 𝑖 3 superscript 𝑒 𝜋 𝑖 3 1 3 superscript 𝑡 3 𝑧 𝑡 𝑡 {\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 Bi ⁡ ( z ) = 1 2 ⁢ π ⁢ ∫ - ∞ ∞ ⁢ e π ⁢ i / 3 exp ⁡ ( 1 3 ⁢ t 3 - z ⁢ t ) ⁢ d t + 1 2 ⁢ π ⁢ ∫ - ∞ ∞ ⁢ e - π ⁢ i / 3 exp ⁡ ( 1 3 ⁢ t 3 - z ⁢ t ) ⁢ d t Airy-Bi 𝑧 1 2 𝜋 superscript subscript superscript 𝑒 𝜋 𝑖 3 1 3 superscript 𝑡 3 𝑧 𝑡 𝑡 1 2 𝜋 superscript subscript superscript 𝑒 𝜋 𝑖 3 1 3 superscript 𝑡 3 𝑧 𝑡 𝑡 {\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 Ai ⁡ ( z ) = 3 2 ⁢ π ⁢ ∫ 0 ∞ exp ⁡ ( - t 3 3 - z 3 3 ⁢ t 3 ) ⁢ d t Airy-Ai 𝑧 3 2 𝜋 superscript subscript 0 superscript 𝑡 3 3 superscript 𝑧 3 3 superscript 𝑡 3 𝑡 {\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}		 | ph ⁡ z | < 1 6 ⁢ π phase 𝑧 1 6 {\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 Ai ⁡ ( z ) = e - ζ π ⁢ ∫ 0 ∞ exp ⁡ ( - z 1 / 2 ⁢ t 2 ) ⁢ cos ⁡ ( 1 3 ⁢ t 3 ) ⁢ d t Airy-Ai 𝑧 superscript 𝑒 𝜁 𝜋 superscript subscript 0 superscript 𝑧 1 2 superscript 𝑡 2 1 3 superscript 𝑡 3 𝑡 {\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}		 | ph ⁡ z | < π phase 𝑧 𝜋 {\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 Ai ⁡ ( z ) = e - ζ ⁢ ζ - 1 / 6 π ⁢ ( 48 ) 1 / 6 ⁢ Γ ⁡ ( 5 6 ) ⁢ ∫ 0 ∞ e - t ⁢ t - 1 / 6 ⁢ ( 2 + t ζ ) - 1 / 6 ⁢ d t Airy-Ai 𝑧 superscript 𝑒 𝜁 superscript 𝜁 1 6 𝜋 superscript 48 1 6 Euler-Gamma 5 6 superscript subscript 0 superscript 𝑒 𝑡 superscript 𝑡 1 6 superscript 2 𝑡 𝜁 1 6 𝑡 {\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}		 | ph ⁡ z | < 2 3 ⁢ π phase 𝑧 2 3 𝜋 {\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
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