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{{DISPLAYTITLE:Airy and Related Functions - 9.11 Products}}

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! scope="col" style="position: sticky; top: 0;" | DLMF 
! scope="col" style="position: sticky; top: 0;" | Formula 
! scope="col" style="position: sticky; top: 0;" | Constraints 
! scope="col" style="position: sticky; top: 0;" | Maple
! scope="col" style="position: sticky; top: 0;" | Mathematica
! scope="col" style="position: sticky; top: 0;" | Symbolic<br>Maple
! scope="col" style="position: sticky; top: 0;" | Symbolic<br>Mathematica
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Maple
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
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| [https://dlmf.nist.gov/9.11.E1 9.11.E1] || <math qid="Q2908">\deriv[3]{w}{z}-4z\deriv{w}{z}-2w = 0</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\deriv[3]{w}{z}-4z\deriv{w}{z}-2w = 0</syntaxhighlight> || <math>w = w_{1}w_{2}</math> || <syntaxhighlight lang=mathematica>diff(w, [z$(3)])- 4*z*diff(w, z)- 2*w = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[w, {z, 3}]- 4*z*D[w, z]- 2*w == 0</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [70 / 70]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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)}</syntaxhighlight><br>... skip entries to safe data</div></div> || Skip - No test values generated
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| [https://dlmf.nist.gov/9.11.E2 9.11.E2] || <math qid="Q2909">\Wronskian@{\AiryAi^{2}@{z},\AiryAi@{z}\AiryBi@{z},\AiryBi^{2}@{z}} = 2\pi^{-3}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Wronskian@{\AiryAi^{2}@{z},\AiryAi@{z}\AiryBi@{z},\AiryBi^{2}@{z}} = 2\pi^{-3}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>((AiryAi(z))^(2))*diff(AiryAi(z)*AiryBi(z), z)-diff((AiryAi(z))^(2), z)*(AiryAi(z)*AiryBi(z)) = 2*(Pi)^(- 3)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Wronskian[{(AiryAi[z])^(2), AiryAi[z]*AiryBi[z]}, z] == 2*(Pi)^(- 3)</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -.6075530626e-1-.7911780259e-2*I
Test Values: {z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .1529112816e-1-.8621001058e-1*I
Test Values: {z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-0.060755306279053636, -0.0079117802669642]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[0.015291128133821968, -0.08621001051231339]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/9.11.E3 9.11.E3] || <math qid="Q2910">\AiryAi^{2}@{x} = \frac{1}{4\pi\sqrt{3}}\int_{0}^{\infty}\BesselJ{0}@{\tfrac{1}{12}t^{3}+xt}t\diff{t}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\AiryAi^{2}@{x} = \frac{1}{4\pi\sqrt{3}}\int_{0}^{\infty}\BesselJ{0}@{\tfrac{1}{12}t^{3}+xt}t\diff{t}</syntaxhighlight> || <math>x \geq 0</math> || <syntaxhighlight lang=mathematica>(AiryAi(x))^(2) = (1)/(4*Pi*sqrt(3))*int(BesselJ(0, (1)/(12)*(t)^(3)+ x*t)*t, t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(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]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
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| [https://dlmf.nist.gov/9.11.E4 9.11.E4] || <math qid="Q2911">\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}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(AiryAi(z))^(2)+ (AiryBi(z))^(2) = (1)/((Pi)^(3/2))*int(exp(z*t -(1)/(12)*(t)^(3))*(t)^(- 1/2), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(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]</syntaxhighlight> || Failure || Successful || <div class="toccolours mw-collapsible mw-collapsed">Failed [4 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 1.205225893+.8288376548*I
Test Values: {z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 3.763924327-.1437296879*I
Test Values: {z = 1.5}</syntaxhighlight><br>... skip entries to safe data</div></div> || Successful [Tested: 7]
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| [https://dlmf.nist.gov/9.11.E12 9.11.E12] || <math qid="Q2919">\int\frac{\diff{z}}{\AiryAi^{2}@{z}} = \pi\frac{\AiryBi@{z}}{\AiryAi@{z}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int\frac{\diff{z}}{\AiryAi^{2}@{z}} = \pi\frac{\AiryBi@{z}}{\AiryAi@{z}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>int((1)/((AiryAi(z))^(2)), z) = Pi*(AiryBi(z))/(AiryAi(z))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Divide[1,(AiryAi[z])^(2)], z, GenerateConditions->None] == Pi*Divide[AiryBi[z],AiryAi[z]]</syntaxhighlight> || Failure || Successful || Error || Successful [Tested: 7]
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| [https://dlmf.nist.gov/9.11.E13 9.11.E13] || <math qid="Q2920">\int\frac{\diff{z}}{\AiryAi@{z}\AiryBi@{z}} = \pi\ln@{\frac{\AiryBi@{z}}{\AiryAi@{z}}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int\frac{\diff{z}}{\AiryAi@{z}\AiryBi@{z}} = \pi\ln@{\frac{\AiryBi@{z}}{\AiryAi@{z}}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>int((1)/(AiryAi(z)*AiryBi(z)), z) = Pi*ln((AiryBi(z))/(AiryAi(z)))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Divide[1,AiryAi[z]*AiryBi[z]], z, GenerateConditions->None] == Pi*Log[Divide[AiryBi[z],AiryAi[z]]]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/9.11.E14 9.11.E14] || <math qid="Q2921">\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}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>int((AiryAi(z)*AiryBi(z))/(((AiryAi(z))^(2)+ (AiryBi(z))^(2))^(2)), z) = (Pi)/(2)*((AiryBi(z))^(2))/((AiryAi(z))^(2)+ (AiryBi(z))^(2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>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)]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/9.11.E15 9.11.E15] || <math qid="Q2922">\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}}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}}}</syntaxhighlight> || <math>\realpart@@{\alpha} > 0, \realpart@@{(\alpha)} > 0, \realpart@@{(\frac{1}{3}\alpha+\frac{5}{6})} > 0</math> || <syntaxhighlight lang=mathematica>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)))</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]]]</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || <div class="toccolours mw-collapsible mw-collapsed">Failed [1 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: DirectedInfinity[]
Test Values: {Rule[α, 0.5]}</syntaxhighlight><br></div></div>
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| [https://dlmf.nist.gov/9.11.E16 9.11.E16] || <math qid="Q2923">\int_{-\infty}^{\infty}\AiryAi^{3}@{t}\diff{t} = \frac{\EulerGamma^{2}@{\frac{1}{3}}}{4\pi^{2}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{-\infty}^{\infty}\AiryAi^{3}@{t}\diff{t} = \frac{\EulerGamma^{2}@{\frac{1}{3}}}{4\pi^{2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>int((AiryAi(t))^(3), t = - infinity..infinity) = ((GAMMA((1)/(3)))^(2))/(4*(Pi)^(2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[(AiryAi[t])^(3), {t, - Infinity, Infinity}, GenerateConditions->None] == Divide[(Gamma[Divide[1,3]])^(2),4*(Pi)^(2)]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
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| [https://dlmf.nist.gov/9.11.E17 9.11.E17] || <math qid="Q2924">\int_{-\infty}^{\infty}\AiryAi^{2}@{t}\AiryBi@{t}\diff{t} = \frac{\EulerGamma^{2}@{\frac{1}{3}}}{4\sqrt{3}\pi^{2}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{-\infty}^{\infty}\AiryAi^{2}@{t}\AiryBi@{t}\diff{t} = \frac{\EulerGamma^{2}@{\frac{1}{3}}}{4\sqrt{3}\pi^{2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>int((AiryAi(t))^(2)* AiryBi(t), t = - infinity..infinity) = ((GAMMA((1)/(3)))^(2))/(4*sqrt(3)*(Pi)^(2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[(AiryAi[t])^(2)* AiryBi[t], {t, - Infinity, Infinity}, GenerateConditions->None] == Divide[(Gamma[Divide[1,3]])^(2),4*Sqrt[3]*(Pi)^(2)]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
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| [https://dlmf.nist.gov/9.11.E18 9.11.E18] || <math qid="Q2925">\int_{0}^{\infty}\AiryAi^{4}@{t}\diff{t} = \frac{\ln@@{3}}{24\pi^{2}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\infty}\AiryAi^{4}@{t}\diff{t} = \frac{\ln@@{3}}{24\pi^{2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>int((AiryAi(t))^(4), t = 0..infinity) = (ln(3))/(24*(Pi)^(2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[(AiryAi[t])^(4), {t, 0, Infinity}, GenerateConditions->None] == Divide[Log[3],24*(Pi)^(2)]</syntaxhighlight> || Failure || Failure || Successful [Tested: 0] || Successful [Tested: 1]
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| [https://dlmf.nist.gov/9.11.E19 9.11.E19] || <math qid="Q2926">\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}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>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)</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Failure || Aborted || Successful [Tested: 0] || Successful [Tested: 1]
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| [https://dlmf.nist.gov/9.11.E19 9.11.E19] || <math qid="Q2926">\int_{0}^{\infty}\frac{t\diff{t}}{\AiryAi'^{2}@{t}+\AiryBi'^{2}@{t}} = \frac{\pi^{2}}{6}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\infty}\frac{t\diff{t}}{\AiryAi'^{2}@{t}+\AiryBi'^{2}@{t}} = \frac{\pi^{2}}{6}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>int((t)/((diff( AiryAi(t), t$(1) ))^(2)+ (diff( AiryBi(t), t$(1) ))^(2)), t = 0..infinity) = ((Pi)^(2))/(6)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Divide[t,(D[AiryAi[t], {t, 1}])^(2)+ (D[AiryBi[t], {t, 1}])^(2)], {t, 0, Infinity}, GenerateConditions->None] == Divide[(Pi)^(2),6]</syntaxhighlight> || Failure || Failure || Successful [Tested: 0] || Successful [Tested: 1]
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