9.11

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
9.11.E1	${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{3}w}{{\mathrm{d}z}^{3}}-4z\frac{\mathrm{d}w}{\mathrm{d}z}-2w=0}}$ \deriv[3]{w}{z}-4z\deriv{w}{z}-2w = 0	${\displaystyle{\displaystyle w=w_{1}w_{2}}}$	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	${\displaystyle{\displaystyle\mathscr{W}\left\{{\mathrm{Ai}^{2}}\left(z\right),\mathrm{Ai}\left(z\right)\mathrm{Bi}\left(z\right),{\mathrm{Bi}^{2}}\left(z\right)\right\}=2\pi^{-3}}}$ \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	${\displaystyle{\displaystyle{\mathrm{Ai}^{2}}\left(x\right)=\frac{1}{4\pi\sqrt{3}}\int_{0}^{\infty}J_{0}\left(\tfrac{1}{12}t^{3}+xt\right)t\mathrm{d}t}}$ \AiryAi^{2}@{x} = \frac{1}{4\pi\sqrt{3}}\int_{0}^{\infty}\BesselJ{0}@{\tfrac{1}{12}t^{3}+xt}t\diff{t}	${\displaystyle{\displaystyle x\geq 0}}$	(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	${\displaystyle{\displaystyle{\mathrm{Ai}^{2}}\left(z\right)+{\mathrm{Bi}^{2}}\left(z\right)=\frac{1}{\pi^{3/2}}\int_{0}^{\infty}\exp\left(zt-\tfrac{1}{12}t^{3}\right)t^{-1/2}\mathrm{d}t}}$ \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	${\displaystyle{\displaystyle\int\frac{\mathrm{d}z}{{\mathrm{Ai}^{2}}\left(z\right)}=\pi\frac{\mathrm{Bi}\left(z\right)}{\mathrm{Ai}\left(z\right)}}}$ \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	${\displaystyle{\displaystyle\int\frac{\mathrm{d}z}{\mathrm{Ai}\left(z\right)\mathrm{Bi}\left(z\right)}=\pi\ln\left(\frac{\mathrm{Bi}\left(z\right)}{\mathrm{Ai}\left(z\right)}\right)}}$ \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	${\displaystyle{\displaystyle\int\frac{\mathrm{Ai}\left(z\right)\mathrm{Bi}\left(z\right)}{\left({\mathrm{Ai}^{2}}\left(z\right)+{\mathrm{Bi}^{2}}\left(z\right)\right)^{2}}\mathrm{d}z=\frac{\pi}{2}\frac{{\mathrm{Bi}^{2}}\left(z\right)}{{\mathrm{Ai}^{2}}\left(z\right)+{\mathrm{Bi}^{2}}\left(z\right)}}}$ \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	${\displaystyle{\displaystyle\int_{0}^{\infty}t^{\alpha-1}{\mathrm{Ai}^{2}}\left(t\right)\mathrm{d}t=\frac{2\Gamma\left(\alpha\right)}{\pi^{1/2}12^{(2\alpha+5)/6}\Gamma\left(\frac{1}{3}\alpha+\frac{5}{6}\right)}}}$ \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}}}	${\displaystyle{\displaystyle\Re\alpha>0,\Re(\alpha)>0,\Re(\frac{1}{3}\alpha+\frac{5}{6})>0}}$	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	${\displaystyle{\displaystyle\int_{-\infty}^{\infty}{\mathrm{Ai}^{3}}\left(t\right)\mathrm{d}t=\frac{{\Gamma^{2}}\left(\frac{1}{3}\right)}{4\pi^{2}}}}$ \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	${\displaystyle{\displaystyle\int_{-\infty}^{\infty}{\mathrm{Ai}^{2}}\left(t\right)\mathrm{Bi}\left(t\right)\mathrm{d}t=\frac{{\Gamma^{2}}\left(\frac{1}{3}\right)}{4\sqrt{3}\pi^{2}}}}$ \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	${\displaystyle{\displaystyle\int_{0}^{\infty}{\mathrm{Ai}^{4}}\left(t\right)\mathrm{d}t=\frac{\ln 3}{24\pi^{2}}}}$ \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	${\displaystyle{\displaystyle\int_{0}^{\infty}\frac{\mathrm{d}t}{{\mathrm{Ai}^{2}}\left(t\right)+{\mathrm{Bi}^{2}}\left(t\right)}=\int_{0}^{\infty}\frac{t\mathrm{d}t}{{\mathrm{Ai}'^{2}}\left(t\right)+{\mathrm{Bi}'^{2}}\left(t\right)}}}$ \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	${\displaystyle{\displaystyle\int_{0}^{\infty}\frac{t\mathrm{d}t}{{\mathrm{Ai}'^{2}}\left(t\right)+{\mathrm{Bi}'^{2}}\left(t\right)}=\frac{\pi^{2}}{6}}}$ \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]