Airy and Related Functions - 9.11 Products

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
9.11.E1 d 3 w d z 3 - 4 ⁢ z ⁢ d w d z - 2 ⁢ w = 0 derivative 𝑤 𝑧 3 4 𝑧 derivative 𝑤 𝑧 2 𝑤 0 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{3}w}{{\mathrm{d}z}^{3}}-4z% \frac{\mathrm{d}w}{\mathrm{d}z}-2w=0}}
\deriv[3]{w}{z}-4z\deriv{w}{z}-2w = 0		 w = w 1 ⁢ w 2 𝑤 subscript 𝑤 1 subscript 𝑤 2 {\displaystyle{\displaystyle w=w_{1}w_{2}}} 
diff(w, [z$(3)])- 4*z*diff(w, z)- 2*w = 0

D[w, {z, 3}]- 4*z*D[w, z]- 2*w == 0
Failure Failure
Failed [70 / 70]
Result: -1.732050808-1.000000000*I
Test Values: {w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}


Result: -1.732050808-1.000000000*I
Test Values: {w = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
 Skip - No test values generated
9.11.E2 𝒲 ⁡ { Ai 2 ⁡ ( z ) , Ai ⁡ ( z ) ⁢ Bi ⁡ ( z ) , Bi 2 ⁡ ( z ) } = 2 ⁢ π - 3 Wronskian Airy-Ai 2 𝑧 Airy-Ai 𝑧 Airy-Bi 𝑧 Airy-Bi 2 𝑧 2 superscript 𝜋 3 {\displaystyle{\displaystyle\mathscr{W}\left\{{\mathrm{Ai}^{2}}\left(z\right),% \mathrm{Ai}\left(z\right)\mathrm{Bi}\left(z\right),{\mathrm{Bi}^{2}}\left(z% \right)\right\}=2\pi^{-3}}}
\Wronskian@{\AiryAi^{2}@{z},\AiryAi@{z}\AiryBi@{z},\AiryBi^{2}@{z}} = 2\pi^{-3}		 

((AiryAi(z))^(2))*diff(AiryAi(z)*AiryBi(z), z)-diff((AiryAi(z))^(2), z)*(AiryAi(z)*AiryBi(z)) = 2*(Pi)^(- 3)

Wronskian[{(AiryAi[z])^(2), AiryAi[z]*AiryBi[z]}, z] == 2*(Pi)^(- 3)
Failure Failure
Failed [7 / 7]
Result: -.6075530626e-1-.7911780259e-2*I
Test Values: {z = 1/2*3^(1/2)+1/2*I}


Result: .1529112816e-1-.8621001058e-1*I
Test Values: {z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [7 / 7]
Result: Complex[-0.060755306279053636, -0.0079117802669642]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.015291128133821968, -0.08621001051231339]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
9.11.E3 Ai 2 ⁡ ( x ) = 1 4 ⁢ π ⁢ 3 ⁢ ∫ 0 ∞ J 0 ⁡ ( 1 12 ⁢ t 3 + x ⁢ t ) ⁢ t ⁢ d t Airy-Ai 2 𝑥 1 4 𝜋 3 superscript subscript 0 Bessel-J 0 1 12 superscript 𝑡 3 𝑥 𝑡 𝑡 𝑡 {\displaystyle{\displaystyle{\mathrm{Ai}^{2}}\left(x\right)=\frac{1}{4\pi\sqrt% {3}}\int_{0}^{\infty}J_{0}\left(\tfrac{1}{12}t^{3}+xt\right)t\mathrm{d}t}}
\AiryAi^{2}@{x} = \frac{1}{4\pi\sqrt{3}}\int_{0}^{\infty}\BesselJ{0}@{\tfrac{1}{12}t^{3}+xt}t\diff{t}		 x ≥ 0 𝑥 0 {\displaystyle{\displaystyle x\geq 0}} 
(AiryAi(x))^(2) = (1)/(4*Pi*sqrt(3))*int(BesselJ(0, (1)/(12)*(t)^(3)+ x*t)*t, t = 0..infinity)

(AiryAi[x])^(2) == Divide[1,4*Pi*Sqrt[3]]*Integrate[BesselJ[0, Divide[1,12]*(t)^(3)+ x*t]*t, {t, 0, Infinity}, GenerateConditions->None]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
9.11.E4 Ai 2 ⁡ ( z ) + Bi 2 ⁡ ( z ) = 1 π 3 / 2 ⁢ ∫ 0 ∞ exp ⁡ ( z ⁢ t - 1 12 ⁢ t 3 ) ⁢ t - 1 / 2 ⁢ d t Airy-Ai 2 𝑧 Airy-Bi 2 𝑧 1 superscript 𝜋 3 2 superscript subscript 0 𝑧 𝑡 1 12 superscript 𝑡 3 superscript 𝑡 1 2 𝑡 {\displaystyle{\displaystyle{\mathrm{Ai}^{2}}\left(z\right)+{\mathrm{Bi}^{2}}% \left(z\right)=\frac{1}{\pi^{3/2}}\int_{0}^{\infty}\exp\left(zt-\tfrac{1}{12}t% ^{3}\right)t^{-1/2}\mathrm{d}t}}
\AiryAi^{2}@{z}+\AiryBi^{2}@{z} = \frac{1}{\pi^{3/2}}\int_{0}^{\infty}\exp@{zt-\tfrac{1}{12}t^{3}}t^{-1/2}\diff{t}		 

(AiryAi(z))^(2)+ (AiryBi(z))^(2) = (1)/((Pi)^(3/2))*int(exp(z*t -(1)/(12)*(t)^(3))*(t)^(- 1/2), t = 0..infinity)

(AiryAi[z])^(2)+ (AiryBi[z])^(2) == Divide[1,(Pi)^(3/2)]*Integrate[Exp[z*t -Divide[1,12]*(t)^(3)]*(t)^(- 1/2), {t, 0, Infinity}, GenerateConditions->None]
Failure Successful
Failed [4 / 7]
Result: 1.205225893+.8288376548*I
Test Values: {z = 1/2*3^(1/2)+1/2*I}


Result: 3.763924327-.1437296879*I
Test Values: {z = 1.5}

... skip entries to safe data
 Successful [Tested: 7]
9.11.E12 ∫ d z Ai 2 ⁡ ( z ) = π ⁢ Bi ⁡ ( z ) Ai ⁡ ( z ) 𝑧 Airy-Ai 2 𝑧 𝜋 Airy-Bi 𝑧 Airy-Ai 𝑧 {\displaystyle{\displaystyle\int\frac{\mathrm{d}z}{{\mathrm{Ai}^{2}}\left(z% \right)}=\pi\frac{\mathrm{Bi}\left(z\right)}{\mathrm{Ai}\left(z\right)}}}
\int\frac{\diff{z}}{\AiryAi^{2}@{z}} = \pi\frac{\AiryBi@{z}}{\AiryAi@{z}}		 

int((1)/((AiryAi(z))^(2)), z) = Pi*(AiryBi(z))/(AiryAi(z))

Integrate[Divide[1,(AiryAi[z])^(2)], z, GenerateConditions->None] == Pi*Divide[AiryBi[z],AiryAi[z]]
Failure Successful Error Successful [Tested: 7]
9.11.E13 ∫ d z Ai ⁡ ( z ) ⁢ Bi ⁡ ( z ) = π ⁢ ln ⁡ ( Bi ⁡ ( z ) Ai ⁡ ( z ) ) 𝑧 Airy-Ai 𝑧 Airy-Bi 𝑧 𝜋 Airy-Bi 𝑧 Airy-Ai 𝑧 {\displaystyle{\displaystyle\int\frac{\mathrm{d}z}{\mathrm{Ai}\left(z\right)% \mathrm{Bi}\left(z\right)}=\pi\ln\left(\frac{\mathrm{Bi}\left(z\right)}{% \mathrm{Ai}\left(z\right)}\right)}}
\int\frac{\diff{z}}{\AiryAi@{z}\AiryBi@{z}} = \pi\ln@{\frac{\AiryBi@{z}}{\AiryAi@{z}}}		 

int((1)/(AiryAi(z)*AiryBi(z)), z) = Pi*ln((AiryBi(z))/(AiryAi(z)))

Integrate[Divide[1,AiryAi[z]*AiryBi[z]], z, GenerateConditions->None] == Pi*Log[Divide[AiryBi[z],AiryAi[z]]]
Failure Failure Error
Failed [7 / 7]
Result: Plus[Complex[-5.779215712137658, -2.873897613994506], Integrate[Times[Power[AiryAi[Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], -1], Power[AiryBi[Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], -1]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Rule[GenerateConditions, None]]]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Plus[Complex[-0.1485658721378681, -3.565476804713019], Integrate[Times[Power[AiryAi[Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], -1], Power[AiryBi[Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], -1]], Power[E, Times[Complex[0, Rational[2, 3]], Pi]], Rule[GenerateConditions, None]]]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
9.11.E14 ∫ Ai ⁡ ( z ) ⁢ Bi ⁡ ( z ) ( Ai 2 ⁡ ( z ) + Bi 2 ⁡ ( z ) ) 2 ⁢ d z = π 2 ⁢ Bi 2 ⁡ ( z ) Ai 2 ⁡ ( z ) + Bi 2 ⁡ ( z ) Airy-Ai 𝑧 Airy-Bi 𝑧 superscript Airy-Ai 2 𝑧 Airy-Bi 2 𝑧 2 𝑧 𝜋 2 Airy-Bi 2 𝑧 Airy-Ai 2 𝑧 Airy-Bi 2 𝑧 {\displaystyle{\displaystyle\int\frac{\mathrm{Ai}\left(z\right)\mathrm{Bi}% \left(z\right)}{\left({\mathrm{Ai}^{2}}\left(z\right)+{\mathrm{Bi}^{2}}\left(z% \right)\right)^{2}}\mathrm{d}z=\frac{\pi}{2}\frac{{\mathrm{Bi}^{2}}\left(z% \right)}{{\mathrm{Ai}^{2}}\left(z\right)+{\mathrm{Bi}^{2}}\left(z\right)}}}
\int\frac{\AiryAi@{z}\AiryBi@{z}}{\left(\AiryAi^{2}@{z}+\AiryBi^{2}@{z}\right)^{2}}\diff{z} = \frac{\pi}{2}\frac{\AiryBi^{2}@{z}}{\AiryAi^{2}@{z}+\AiryBi^{2}@{z}}		 

int((AiryAi(z)*AiryBi(z))/(((AiryAi(z))^(2)+ (AiryBi(z))^(2))^(2)), z) = (Pi)/(2)*((AiryBi(z))^(2))/((AiryAi(z))^(2)+ (AiryBi(z))^(2))

Integrate[Divide[AiryAi[z]*AiryBi[z],((AiryAi[z])^(2)+ (AiryBi[z])^(2))^(2)], z, GenerateConditions->None] == Divide[Pi,2]*Divide[(AiryBi[z])^(2),(AiryAi[z])^(2)+ (AiryBi[z])^(2)]
Failure Failure Error
Failed [7 / 7]
Result: Plus[Complex[-1.580056541145603, -0.03880964929600676], Integrate[Times[AiryAi[Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], AiryBi[Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Power[Plus[Power[AiryAi[Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], 2], Power[AiryBi[Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], 2]], -2]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Rule[GenerateConditions, None]]]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Plus[Complex[-0.9914863532591266, -1.6654177670843742], Integrate[Times[AiryAi[Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], AiryBi[Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Power[Plus[Power[AiryAi[Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], 2], Power[AiryBi[Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], 2]], -2]], Power[E, Times[Complex[0, Rational[2, 3]], Pi]], Rule[GenerateConditions, None]]]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
9.11.E15 ∫ 0 ∞ t α - 1 ⁢ Ai 2 ⁡ ( t ) ⁢ d t = 2 ⁢ Γ ⁡ ( α ) π 1 / 2 ⁢ 12 ( 2 ⁢ α + 5 ) / 6 ⁢ Γ ⁡ ( 1 3 ⁢ α + 5 6 ) superscript subscript 0 superscript 𝑡 𝛼 1 Airy-Ai 2 𝑡 𝑡 2 Euler-Gamma 𝛼 superscript 𝜋 1 2 superscript 12 2 𝛼 5 6 Euler-Gamma 1 3 𝛼 5 6 {\displaystyle{\displaystyle\int_{0}^{\infty}t^{\alpha-1}{\mathrm{Ai}^{2}}% \left(t\right)\mathrm{d}t=\frac{2\Gamma\left(\alpha\right)}{\pi^{1/2}12^{(2% \alpha+5)/6}\Gamma\left(\frac{1}{3}\alpha+\frac{5}{6}\right)}}}
\int_{0}^{\infty}t^{\alpha-1}\AiryAi^{2}@{t}\diff{t} = \frac{2\EulerGamma@{\alpha}}{\pi^{1/2}12^{(2\alpha+5)/6}\EulerGamma@{\frac{1}{3}\alpha+\frac{5}{6}}}		 ℜ ⁡ α > 0 , ℜ ⁡ ( α ) > 0 , ℜ ⁡ ( 1 3 ⁢ α + 5 6 ) > 0 formulae-sequence 𝛼 0 formulae-sequence 𝛼 0 1 3 𝛼 5 6 0 {\displaystyle{\displaystyle\Re\alpha>0,\Re(\alpha)>0,\Re(\frac{1}{3}\alpha+% \frac{5}{6})>0}} 
int((t)^(alpha - 1)* (AiryAi(t))^(2), t = 0..infinity) = (2*GAMMA(alpha))/((Pi)^(1/2)* (12)^((2*alpha + 5)/6)* GAMMA((1)/(3)*alpha +(5)/(6)))

Integrate[(t)^(\[Alpha]- 1)* (AiryAi[t])^(2), {t, 0, Infinity}, GenerateConditions->None] == Divide[2*Gamma[\[Alpha]],(Pi)^(1/2)* (12)^((2*\[Alpha]+ 5)/6)* Gamma[Divide[1,3]*\[Alpha]+Divide[5,6]]]
Failure Failure Successful [Tested: 3]
Failed [1 / 3]
Result: DirectedInfinity[]
Test Values: {Rule[α, 0.5]}

9.11.E16 ∫ - ∞ ∞ Ai 3 ⁡ ( t ) ⁢ d t = Γ 2 ⁡ ( 1 3 ) 4 ⁢ π 2 superscript subscript Airy-Ai 3 𝑡 𝑡 Euler-Gamma 2 1 3 4 superscript 𝜋 2 {\displaystyle{\displaystyle\int_{-\infty}^{\infty}{\mathrm{Ai}^{3}}\left(t% \right)\mathrm{d}t=\frac{{\Gamma^{2}}\left(\frac{1}{3}\right)}{4\pi^{2}}}}
\int_{-\infty}^{\infty}\AiryAi^{3}@{t}\diff{t} = \frac{\EulerGamma^{2}@{\frac{1}{3}}}{4\pi^{2}}		 

int((AiryAi(t))^(3), t = - infinity..infinity) = ((GAMMA((1)/(3)))^(2))/(4*(Pi)^(2))

Integrate[(AiryAi[t])^(3), {t, - Infinity, Infinity}, GenerateConditions->None] == Divide[(Gamma[Divide[1,3]])^(2),4*(Pi)^(2)]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
9.11.E17 ∫ - ∞ ∞ Ai 2 ⁡ ( t ) ⁢ Bi ⁡ ( t ) ⁢ d t = Γ 2 ⁡ ( 1 3 ) 4 ⁢ 3 ⁢ π 2 superscript subscript Airy-Ai 2 𝑡 Airy-Bi 𝑡 𝑡 Euler-Gamma 2 1 3 4 3 superscript 𝜋 2 {\displaystyle{\displaystyle\int_{-\infty}^{\infty}{\mathrm{Ai}^{2}}\left(t% \right)\mathrm{Bi}\left(t\right)\mathrm{d}t=\frac{{\Gamma^{2}}\left(\frac{1}{3% }\right)}{4\sqrt{3}\pi^{2}}}}
\int_{-\infty}^{\infty}\AiryAi^{2}@{t}\AiryBi@{t}\diff{t} = \frac{\EulerGamma^{2}@{\frac{1}{3}}}{4\sqrt{3}\pi^{2}}		 

int((AiryAi(t))^(2)* AiryBi(t), t = - infinity..infinity) = ((GAMMA((1)/(3)))^(2))/(4*sqrt(3)*(Pi)^(2))

Integrate[(AiryAi[t])^(2)* AiryBi[t], {t, - Infinity, Infinity}, GenerateConditions->None] == Divide[(Gamma[Divide[1,3]])^(2),4*Sqrt[3]*(Pi)^(2)]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
9.11.E18 ∫ 0 ∞ Ai 4 ⁡ ( t ) ⁢ d t = ln ⁡ 3 24 ⁢ π 2 superscript subscript 0 Airy-Ai 4 𝑡 𝑡 3 24 superscript 𝜋 2 {\displaystyle{\displaystyle\int_{0}^{\infty}{\mathrm{Ai}^{4}}\left(t\right)% \mathrm{d}t=\frac{\ln 3}{24\pi^{2}}}}
\int_{0}^{\infty}\AiryAi^{4}@{t}\diff{t} = \frac{\ln@@{3}}{24\pi^{2}}		 

int((AiryAi(t))^(4), t = 0..infinity) = (ln(3))/(24*(Pi)^(2))

Integrate[(AiryAi[t])^(4), {t, 0, Infinity}, GenerateConditions->None] == Divide[Log[3],24*(Pi)^(2)]
Failure Failure Successful [Tested: 0] Successful [Tested: 1]
9.11.E19 ∫ 0 ∞ d t Ai 2 ⁡ ( t ) + Bi 2 ⁡ ( t ) = ∫ 0 ∞ t ⁢ d t Ai ′ 2 ⁡ ( t ) + Bi ′ 2 ⁡ ( t ) superscript subscript 0 𝑡 Airy-Ai 2 𝑡 Airy-Bi 2 𝑡 superscript subscript 0 𝑡 𝑡 diffop Airy-Ai 1 2 𝑡 diffop Airy-Bi 1 2 𝑡 {\displaystyle{\displaystyle\int_{0}^{\infty}\frac{\mathrm{d}t}{{\mathrm{Ai}^{% 2}}\left(t\right)+{\mathrm{Bi}^{2}}\left(t\right)}=\int_{0}^{\infty}\frac{t% \mathrm{d}t}{{\mathrm{Ai}'^{2}}\left(t\right)+{\mathrm{Bi}'^{2}}\left(t\right)% }}}
\int_{0}^{\infty}\frac{\diff{t}}{\AiryAi^{2}@{t}+\AiryBi^{2}@{t}} = \int_{0}^{\infty}\frac{t\diff{t}}{\AiryAi'^{2}@{t}+\AiryBi'^{2}@{t}}		 

int((1)/((AiryAi(t))^(2)+ (AiryBi(t))^(2)), t = 0..infinity) = int((t)/((diff( AiryAi(t), t$(1) ))^(2)+ (diff( AiryBi(t), t$(1) ))^(2)), t = 0..infinity)

Integrate[Divide[1,(AiryAi[t])^(2)+ (AiryBi[t])^(2)], {t, 0, Infinity}, GenerateConditions->None] == Integrate[Divide[t,(D[AiryAi[t], {t, 1}])^(2)+ (D[AiryBi[t], {t, 1}])^(2)], {t, 0, Infinity}, GenerateConditions->None]
Failure Aborted Successful [Tested: 0] Successful [Tested: 1]
9.11.E19 ∫ 0 ∞ t ⁢ d t Ai ′ 2 ⁡ ( t ) + Bi ′ 2 ⁡ ( t ) = π 2 6 superscript subscript 0 𝑡 𝑡 diffop Airy-Ai 1 2 𝑡 diffop Airy-Bi 1 2 𝑡 superscript 𝜋 2 6 {\displaystyle{\displaystyle\int_{0}^{\infty}\frac{t\mathrm{d}t}{{\mathrm{Ai}'% ^{2}}\left(t\right)+{\mathrm{Bi}'^{2}}\left(t\right)}=\frac{\pi^{2}}{6}}}
\int_{0}^{\infty}\frac{t\diff{t}}{\AiryAi'^{2}@{t}+\AiryBi'^{2}@{t}} = \frac{\pi^{2}}{6}		 

int((t)/((diff( AiryAi(t), t$(1) ))^(2)+ (diff( AiryBi(t), t$(1) ))^(2)), t = 0..infinity) = ((Pi)^(2))/(6)

Integrate[Divide[t,(D[AiryAi[t], {t, 1}])^(2)+ (D[AiryBi[t], {t, 1}])^(2)], {t, 0, Infinity}, GenerateConditions->None] == Divide[(Pi)^(2),6]
Failure Failure Successful [Tested: 0] Successful [Tested: 1]
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