Airy and Related Functions - 9.8 Modulus and Phase

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
9.8.E1 Ai ⁡ ( x ) = M ⁡ ( x ) ⁢ sin ⁡ θ ⁡ ( x ) Airy-Ai 𝑥 modulus-Airy-M 𝑥 phase-Airy-Theta 𝑥 {\displaystyle{\displaystyle\mathrm{Ai}\left(x\right)=M\left(x\right)\sin% \theta\left(x\right)}}
\AiryAi@{x} = \AirymodM@{x}\sin@@{\Airyphasetheta@{x}}		 

AiryAi(x) = sqrt(AiryAi(x)^2+AiryBi(x)^2)*sin(arctan(AiryAi(x)/AiryBi(x)))

AiryAi[x] == Sqrt[AiryAi[x]^2 + AiryBi[x]^2]*Sin[ArcTan[Divide[AiryAi[x], AiryBi[x]]]]
Failure Failure Successful [Tested: 3] Successful [Tested: 3]
9.8.E2 Bi ⁡ ( x ) = M ⁡ ( x ) ⁢ cos ⁡ θ ⁡ ( x ) Airy-Bi 𝑥 modulus-Airy-M 𝑥 phase-Airy-Theta 𝑥 {\displaystyle{\displaystyle\mathrm{Bi}\left(x\right)=M\left(x\right)\cos% \theta\left(x\right)}}
\AiryBi@{x} = \AirymodM@{x}\cos@@{\Airyphasetheta@{x}}		 

AiryBi(x) = sqrt(AiryAi(x)^2+AiryBi(x)^2)*cos(arctan(AiryAi(x)/AiryBi(x)))

AiryBi[x] == Sqrt[AiryAi[x]^2 + AiryBi[x]^2]*Cos[ArcTan[Divide[AiryAi[x], AiryBi[x]]]]
Failure Failure Successful [Tested: 3] Successful [Tested: 3]
9.8.E3 M ⁡ ( x ) = Ai 2 ⁡ ( x ) + Bi 2 ⁡ ( x ) modulus-Airy-M 𝑥 Airy-Ai 2 𝑥 Airy-Bi 2 𝑥 {\displaystyle{\displaystyle M\left(x\right)=\sqrt{{\mathrm{Ai}^{2}}\left(x% \right)+{\mathrm{Bi}^{2}}\left(x\right)}}}
\AirymodM@{x} = \sqrt{\AiryAi^{2}@{x}+\AiryBi^{2}@{x}}		 

sqrt(AiryAi(x)^2+AiryBi(x)^2) = sqrt((AiryAi(x))^(2)+ (AiryBi(x))^(2))

Sqrt[AiryAi[x]^2 + AiryBi[x]^2] == Sqrt[(AiryAi[x])^(2)+ (AiryBi[x])^(2)]
Successful Successful - Successful [Tested: 3]
9.8.E4 θ ⁡ ( x ) = arctan ⁡ ( Ai ⁡ ( x ) / Bi ⁡ ( x ) ) phase-Airy-Theta 𝑥 Airy-Ai 𝑥 Airy-Bi 𝑥 {\displaystyle{\displaystyle\theta\left(x\right)=\operatorname{arctan}\left(% \mathrm{Ai}\left(x\right)/\mathrm{Bi}\left(x\right)\right)}}
\Airyphasetheta@{x} = \atan@{\AiryAi@{x}/\AiryBi@{x}}		 

arctan(AiryAi(x)/AiryBi(x)) = arctan(AiryAi(x)/AiryBi(x))

ArcTan[Divide[AiryAi[x], AiryBi[x]]] == ArcTan[AiryAi[x]/AiryBi[x]]
Successful Successful - Successful [Tested: 3]
9.8.E5 Ai ′ ⁡ ( x ) = N ⁡ ( x ) ⁢ sin ⁡ ϕ ⁡ ( x ) diffop Airy-Ai 1 𝑥 modulus-Airy-N 𝑥 phase-Airy-Phi 𝑥 {\displaystyle{\displaystyle\mathrm{Ai}'\left(x\right)=N\left(x\right)\sin\phi% \left(x\right)}}
\AiryAi'@{x} = \AirymodderivN@{x}\sin@@{\Airyphasederivphi@{x}}		 

diff( AiryAi(x), x$(1) ) = sqrt(AiryAi(1, x)^2+AiryBi(1, x)^2)*sin(arctan(AiryAi(1, x)/AiryBi(1, x)))

D[AiryAi[x], {x, 1}] == Sqrt[AiryAiPrime[x]^2 + AiryBiPrime[x]^2]*Sin[ArcTan[Divide[AiryAiPrime[x], AiryBiPrime[x]]]]
Failure Failure Successful [Tested: 3] Successful [Tested: 3]
9.8.E6 Bi ′ ⁡ ( x ) = N ⁡ ( x ) ⁢ cos ⁡ ϕ ⁡ ( x ) diffop Airy-Bi 1 𝑥 modulus-Airy-N 𝑥 phase-Airy-Phi 𝑥 {\displaystyle{\displaystyle\mathrm{Bi}'\left(x\right)=N\left(x\right)\cos\phi% \left(x\right)}}
\AiryBi'@{x} = \AirymodderivN@{x}\cos@@{\Airyphasederivphi@{x}}		 

diff( AiryBi(x), x$(1) ) = sqrt(AiryAi(1, x)^2+AiryBi(1, x)^2)*cos(arctan(AiryAi(1, x)/AiryBi(1, x)))

D[AiryBi[x], {x, 1}] == Sqrt[AiryAiPrime[x]^2 + AiryBiPrime[x]^2]*Cos[ArcTan[Divide[AiryAiPrime[x], AiryBiPrime[x]]]]
Failure Failure Successful [Tested: 3] Successful [Tested: 3]
9.8.E7 N ⁡ ( x ) = Ai ′ 2 ⁡ ( x ) + Bi ′ 2 ⁡ ( x ) modulus-Airy-N 𝑥 diffop Airy-Ai 1 2 𝑥 diffop Airy-Bi 1 2 𝑥 {\displaystyle{\displaystyle N\left(x\right)=\sqrt{{\mathrm{Ai}'^{2}}\left(x% \right)+{\mathrm{Bi}'^{2}}\left(x\right)}}}
\AirymodderivN@{x} = \sqrt{\AiryAi'^{2}@{x}+\AiryBi'^{2}@{x}}		 

sqrt(AiryAi(1, x)^2+AiryBi(1, x)^2) = sqrt((diff( AiryAi(x), x$(1) ))^(2)+ (diff( AiryBi(x), x$(1) ))^(2))

Sqrt[AiryAiPrime[x]^2 + AiryBiPrime[x]^2] == Sqrt[(D[AiryAi[x], {x, 1}])^(2)+ (D[AiryBi[x], {x, 1}])^(2)]
Successful Successful - Successful [Tested: 3]
9.8.E8 ϕ ⁡ ( x ) = arctan ⁡ ( Ai ′ ⁡ ( x ) / Bi ′ ⁡ ( x ) ) phase-Airy-Phi 𝑥 diffop Airy-Ai 1 𝑥 diffop Airy-Bi 1 𝑥 {\displaystyle{\displaystyle\phi\left(x\right)=\operatorname{arctan}\left(% \mathrm{Ai}'\left(x\right)/\mathrm{Bi}'\left(x\right)\right)}}
\Airyphasederivphi@{x} = \atan@{\AiryAi'@{x}/\AiryBi'@{x}}		 

arctan(AiryAi(1, x)/AiryBi(1, x)) = arctan(diff( AiryAi(x), x$(1) )/diff( AiryBi(x), x$(1) ))

ArcTan[Divide[AiryAiPrime[x], AiryBiPrime[x]]] == ArcTan[D[AiryAi[x], {x, 1}]/D[AiryBi[x], {x, 1}]]
Successful Successful - Successful [Tested: 3]
9.8.E9 | x | 1 / 2 ⁢ M 2 ⁡ ( x ) = 1 2 ⁢ ξ ⁢ ( J 1 / 3 2 ⁡ ( ξ ) + Y 1 / 3 2 ⁡ ( ξ ) ) superscript 𝑥 1 2 modulus-Airy-M 2 𝑥 1 2 𝜉 Bessel-J 1 3 2 𝜉 Bessel-Y-Weber 1 3 2 𝜉 {\displaystyle{\displaystyle|x|^{1/2}{M^{2}}\left(x\right)=\tfrac{1}{2}\xi% \left({J_{1/3}^{2}}\left(\xi\right)+{Y_{1/3}^{2}}\left(\xi\right)\right)}}
|x|^{1/2}\AirymodM^{2}@{x} = \tfrac{1}{2}\xi\left(\BesselJ{1/3}^{2}@{\xi}+\BesselY{1/3}^{2}@{\xi}\right)		 

(abs(x))^(1/2)* (sqrt(AiryAi(x)^2+AiryBi(x)^2))^(2) = (1)/(2)*xi*((BesselJ(1/3, xi))^(2)+ (BesselY(1/3, xi))^(2))

(Abs[x])^(1/2)* (Sqrt[AiryAi[x]^2 + AiryBi[x]^2])^(2) == Divide[1,2]*\[Xi]*((BesselJ[1/3, \[Xi]])^(2)+ (BesselY[1/3, \[Xi]])^(2))
Failure Failure
Failed [30 / 30]
Result: 4.021808267-.8614613375e-2*I
Test Values: {x = 1.5, xi = 1/2*3^(1/2)+1/2*I}


Result: 3.972124824-.1350954874e-1*I
Test Values: {x = 1.5, xi = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [30 / 30]
Result: Complex[4.021808267868023, -0.008614613397096321]
Test Values: {Rule[x, 1.5], Rule[ξ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[3.972124824572757, -0.01350954875717339]
Test Values: {Rule[x, 1.5], Rule[ξ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
9.8.E10 | x | - 1 / 2 ⁢ N 2 ⁡ ( x ) = 1 2 ⁢ ξ ⁢ ( J 2 / 3 2 ⁡ ( ξ ) + Y 2 / 3 2 ⁡ ( ξ ) ) superscript 𝑥 1 2 modulus-Airy-N 2 𝑥 1 2 𝜉 Bessel-J 2 3 2 𝜉 Bessel-Y-Weber 2 3 2 𝜉 {\displaystyle{\displaystyle|x|^{-1/2}{N^{2}}\left(x\right)=\tfrac{1}{2}\xi% \left({J_{2/3}^{2}}\left(\xi\right)+{Y_{2/3}^{2}}\left(\xi\right)\right)}}
|x|^{-1/2}\AirymodderivN^{2}@{x} = \tfrac{1}{2}\xi\left(\BesselJ{2/3}^{2}@{\xi}+\BesselY{2/3}^{2}@{\xi}\right)		 

(abs(x))^(- 1/2)* (sqrt(AiryAi(1, x)^2+AiryBi(1, x)^2))^(2) = (1)/(2)*xi*((BesselJ(2/3, xi))^(2)+ (BesselY(2/3, xi))^(2))

(Abs[x])^(- 1/2)* (Sqrt[AiryAiPrime[x]^2 + AiryBiPrime[x]^2])^(2) == Divide[1,2]*\[Xi]*((BesselJ[2/3, \[Xi]])^(2)+ (BesselY[2/3, \[Xi]])^(2))
Failure Failure
Failed [30 / 30]
Result: 2.579966574+.1365442595e-1*I
Test Values: {x = 1.5, xi = 1/2*3^(1/2)+1/2*I}


Result: 2.649043945+.8067203529e-2*I
Test Values: {x = 1.5, xi = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [30 / 30]
Result: Complex[2.579966572371216, 0.013654425864881942]
Test Values: {Rule[x, 1.5], Rule[ξ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[2.6490439435787625, 0.00806720349537901]
Test Values: {Rule[x, 1.5], Rule[ξ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
9.8.E11 θ ⁡ ( x ) = 2 3 ⁢ π + arctan ⁡ ( Y 1 / 3 ⁡ ( ξ ) / J 1 / 3 ⁡ ( ξ ) ) phase-Airy-Theta 𝑥 2 3 𝜋 Bessel-Y-Weber 1 3 𝜉 Bessel-J 1 3 𝜉 {\displaystyle{\displaystyle\theta\left(x\right)=\tfrac{2}{3}\pi+\operatorname% {arctan}\left(Y_{1/3}\left(\xi\right)/J_{1/3}\left(\xi\right)\right)}}
\Airyphasetheta@{x} = \tfrac{2}{3}\pi+\atan@{\BesselY{1/3}@{\xi}/\BesselJ{1/3}@{\xi}}		 

arctan(AiryAi(x)/AiryBi(x)) = (2)/(3)*Pi + arctan(BesselY(1/3, xi)/BesselJ(1/3, xi))

ArcTan[Divide[AiryAi[x], AiryBi[x]]] == Divide[2,3]*Pi + ArcTan[BesselY[1/3, \[Xi]]/BesselJ[1/3, \[Xi]]]
Failure Failure
Failed [30 / 30]
Result: -1.560189280-.5213615815*I
Test Values: {x = 1.5, xi = 1/2*3^(1/2)+1/2*I}


Result: -3.390111334-.9722564139*I
Test Values: {x = 1.5, xi = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [30 / 30]
Result: Complex[-1.5601892780520927, -0.5213615814894055]
Test Values: {Rule[x, 1.5], Rule[ξ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-3.390111332221422, -0.9722564141048874]
Test Values: {Rule[x, 1.5], Rule[ξ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
9.8.E12 ϕ ⁡ ( x ) = 1 3 ⁢ π + arctan ⁡ ( Y 2 / 3 ⁡ ( ξ ) / J 2 / 3 ⁡ ( ξ ) ) phase-Airy-Phi 𝑥 1 3 𝜋 Bessel-Y-Weber 2 3 𝜉 Bessel-J 2 3 𝜉 {\displaystyle{\displaystyle\phi\left(x\right)=\tfrac{1}{3}\pi+\operatorname{% arctan}\left(Y_{2/3}\left(\xi\right)/J_{2/3}\left(\xi\right)\right)}}
\Airyphasederivphi@{x} = \tfrac{1}{3}\pi+\atan@{\BesselY{2/3}@{\xi}/\BesselJ{2/3}@{\xi}}		 

arctan(AiryAi(1, x)/AiryBi(1, x)) = (1)/(3)*Pi + arctan(BesselY(2/3, xi)/BesselJ(2/3, xi))

ArcTan[Divide[AiryAiPrime[x], AiryBiPrime[x]]] == Divide[1,3]*Pi + ArcTan[BesselY[2/3, \[Xi]]/BesselJ[2/3, \[Xi]]]
Failure Failure
Failed [30 / 30]
Result: -.2068569407-.4703554156*I
Test Values: {x = 1.5, xi = 1/2*3^(1/2)+1/2*I}


Result: -1.895355428-.7064271023*I
Test Values: {x = 1.5, xi = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [30 / 30]
Result: Complex[-0.20685694111550512, -0.47035541563882277]
Test Values: {Rule[x, 1.5], Rule[ξ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-1.8953554288661256, -0.7064271020951838]
Test Values: {Rule[x, 1.5], Rule[ξ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
9.8.E13 M ⁡ ( x ) ⁢ N ⁡ ( x ) ⁢ sin ⁡ ( θ ⁡ ( x ) - ϕ ⁡ ( x ) ) = π - 1 modulus-Airy-M 𝑥 modulus-Airy-N 𝑥 phase-Airy-Theta 𝑥 phase-Airy-Phi 𝑥 superscript 𝜋 1 {\displaystyle{\displaystyle M\left(x\right)N\left(x\right)\sin\left(\theta% \left(x\right)-\phi\left(x\right)\right)=\pi^{-1}}}
\AirymodM@{x}\AirymodderivN@{x}\sin@{\Airyphasetheta@{x}-\Airyphasederivphi@{x}} = \pi^{-1}		 

sqrt(AiryAi(x)^2+AiryBi(x)^2)*sqrt(AiryAi(1, x)^2+AiryBi(1, x)^2)*sin(arctan(AiryAi(x)/AiryBi(x))- arctan(AiryAi(1, x)/AiryBi(1, x))) = (Pi)^(- 1)

Sqrt[AiryAi[x]^2 + AiryBi[x]^2]*Sqrt[AiryAiPrime[x]^2 + AiryBiPrime[x]^2]*Sin[ArcTan[Divide[AiryAi[x], AiryBi[x]]]- ArcTan[Divide[AiryAiPrime[x], AiryBiPrime[x]]]] == (Pi)^(- 1)
Failure Failure Successful [Tested: 3] Successful [Tested: 3]
9.8#Ex1 M 2 ⁡ ( x ) ⁢ θ ′ ⁡ ( x ) = - π - 1 modulus-Airy-M 2 𝑥 diffop phase-Airy-Theta 1 𝑥 superscript 𝜋 1 {\displaystyle{\displaystyle{M^{2}}\left(x\right)\theta'\left(x\right)=-\pi^{-% 1}}}
\AirymodM^{2}@{x}\Airyphasetheta'@{x} = -\pi^{-1}		 

(sqrt(AiryAi(x)^2+AiryBi(x)^2))^(2)* diff( arctan(AiryAi(x)/AiryBi(x)), x$(1) ) = - (Pi)^(- 1)

(Sqrt[AiryAi[x]^2 + AiryBi[x]^2])^(2)* D[ArcTan[Divide[AiryAi[x], AiryBi[x]]], {x, 1}] == - (Pi)^(- 1)
Failure Successful Successful [Tested: 3] Successful [Tested: 3]
9.8#Ex2 N 2 ⁡ ( x ) ⁢ ϕ ′ ⁡ ( x ) = π - 1 ⁢ x modulus-Airy-N 2 𝑥 diffop phase-Airy-Phi 1 𝑥 superscript 𝜋 1 𝑥 {\displaystyle{\displaystyle{N^{2}}\left(x\right)\phi'\left(x\right)=\pi^{-1}x}}
\AirymodderivN^{2}@{x}\Airyphasederivphi'@{x} = \pi^{-1}x		 

(sqrt(AiryAi(1, x)^2+AiryBi(1, x)^2))^(2)* diff( arctan(AiryAi(1, x)/AiryBi(1, x)), x$(1) ) = (Pi)^(- 1)* x

(Sqrt[AiryAiPrime[x]^2 + AiryBiPrime[x]^2])^(2)* D[ArcTan[Divide[AiryAiPrime[x], AiryBiPrime[x]]], {x, 1}] == (Pi)^(- 1)* x
Failure Successful Successful [Tested: 3] Successful [Tested: 3]
9.8#Ex3 N ⁡ ( x ) ⁢ N ′ ⁡ ( x ) = x ⁢ M ⁡ ( x ) ⁢ M ′ ⁡ ( x ) modulus-Airy-N 𝑥 diffop modulus-Airy-N 1 𝑥 𝑥 modulus-Airy-M 𝑥 diffop modulus-Airy-M 1 𝑥 {\displaystyle{\displaystyle N\left(x\right)N'\left(x\right)=xM\left(x\right)M% '\left(x\right)}}
\AirymodderivN@{x}\AirymodderivN'@{x} = x\AirymodM@{x}\AirymodM'@{x}		 

sqrt(AiryAi(1, x)^2+AiryBi(1, x)^2)*diff( sqrt(AiryAi(1, x)^2+AiryBi(1, x)^2), x$(1) ) = x*sqrt(AiryAi(x)^2+AiryBi(x)^2)*diff( sqrt(AiryAi(x)^2+AiryBi(x)^2), x$(1) )

Sqrt[AiryAiPrime[x]^2 + AiryBiPrime[x]^2]*D[Sqrt[AiryAiPrime[x]^2 + AiryBiPrime[x]^2], {x, 1}] == x*Sqrt[AiryAi[x]^2 + AiryBi[x]^2]*D[Sqrt[AiryAi[x]^2 + AiryBi[x]^2], {x, 1}]
Successful Successful - Successful [Tested: 3]
9.8.E15 N 2 ⁡ ( x ) = M ′ 2 ⁡ ( x ) + M 2 ⁡ ( x ) ⁢ θ ′ 2 ⁡ ( x ) modulus-Airy-N 2 𝑥 diffop modulus-Airy-M 1 2 𝑥 modulus-Airy-M 2 𝑥 diffop phase-Airy-Theta 1 2 𝑥 {\displaystyle{\displaystyle{N^{2}}\left(x\right)={M'^{2}}\left(x\right)+{M^{2% }}\left(x\right){\theta'^{2}}\left(x\right)}}
\AirymodderivN^{2}@{x} = \AirymodM'^{2}@{x}+\AirymodM^{2}@{x}\Airyphasetheta'^{2}@{x}		 

(sqrt(AiryAi(1, x)^2+AiryBi(1, x)^2))^(2) = (diff( sqrt(AiryAi(x)^2+AiryBi(x)^2), x$(1) ))^(2)+ (sqrt(AiryAi(x)^2+AiryBi(x)^2))^(2)* (diff( arctan(AiryAi(x)/AiryBi(x)), x$(1) ))^(2)

(Sqrt[AiryAiPrime[x]^2 + AiryBiPrime[x]^2])^(2) == (D[Sqrt[AiryAi[x]^2 + AiryBi[x]^2], {x, 1}])^(2)+ (Sqrt[AiryAi[x]^2 + AiryBi[x]^2])^(2)* (D[ArcTan[Divide[AiryAi[x], AiryBi[x]]], {x, 1}])^(2)
Successful Successful - Successful [Tested: 3]
9.8.E15 M ′ 2 ⁡ ( x ) + M 2 ⁡ ( x ) ⁢ θ ′ 2 ⁡ ( x ) = M ′ 2 ⁢ ( x ) + π - 2 ⁢ M - 2 ⁡ ( x ) diffop modulus-Airy-M 1 2 𝑥 modulus-Airy-M 2 𝑥 diffop phase-Airy-Theta 1 2 𝑥 diffop modulus-Airy-M 1 2 𝑥 superscript 𝜋 2 modulus-Airy-M 2 𝑥 {\displaystyle{\displaystyle{M'^{2}}\left(x\right)+{M^{2}}\left(x\right){% \theta'^{2}}\left(x\right)={M'^{2}}(x)+\pi^{-2}{M^{-2}}\left(x\right)}}
\AirymodM'^{2}@{x}+\AirymodM^{2}@{x}\Airyphasetheta'^{2}@{x} = \AirymodM'^{2}(x)+\pi^{-2}\AirymodM^{-2}@{x}		 

(diff( sqrt(AiryAi(x)^2+AiryBi(x)^2), x$(1) ))^(2)+ (sqrt(AiryAi(x)^2+AiryBi(x)^2))^(2)* (diff( arctan(AiryAi(x)/AiryBi(x)), x$(1) ))^(2) = (subs( temp=(x), diff( sqrt(AiryAi(temp)^2+AiryBi(temp)^2), temp$(1) ) ))^(2)+ (Pi)^(- 2)* (sqrt(AiryAi(x)^2+AiryBi(x)^2))^(- 2)

(D[Sqrt[AiryAi[x]^2 + AiryBi[x]^2], {x, 1}])^(2)+ (Sqrt[AiryAi[x]^2 + AiryBi[x]^2])^(2)* (D[ArcTan[Divide[AiryAi[x], AiryBi[x]]], {x, 1}])^(2) == ((D[Sqrt[AiryAi[temp]^2 + AiryBi[temp]^2], {temp, 1}]/.temp-> (x)))^(2)+ (Pi)^(- 2)* (Sqrt[AiryAi[x]^2 + AiryBi[x]^2])^(- 2)
Translation Error Translation Error - -
9.8.E16 x 2 ⁢ M 2 ⁡ ( x ) = N ′ 2 ⁡ ( x ) + N 2 ⁡ ( x ) ⁢ ϕ ′ 2 ⁡ ( x ) superscript 𝑥 2 modulus-Airy-M 2 𝑥 diffop modulus-Airy-N 1 2 𝑥 modulus-Airy-N 2 𝑥 diffop phase-Airy-Phi 1 2 𝑥 {\displaystyle{\displaystyle x^{2}{M^{2}}\left(x\right)={N'^{2}}\left(x\right)% +{N^{2}}\left(x\right){\phi'^{2}}\left(x\right)}}
x^{2}\AirymodM^{2}@{x} = \AirymodderivN'^{2}@{x}+\AirymodderivN^{2}@{x}\Airyphasederivphi'^{2}@{x}		 

(x)^(2)* (sqrt(AiryAi(x)^2+AiryBi(x)^2))^(2) = (diff( sqrt(AiryAi(1, x)^2+AiryBi(1, x)^2), x$(1) ))^(2)+ (sqrt(AiryAi(1, x)^2+AiryBi(1, x)^2))^(2)* (diff( arctan(AiryAi(1, x)/AiryBi(1, x)), x$(1) ))^(2)

(x)^(2)* (Sqrt[AiryAi[x]^2 + AiryBi[x]^2])^(2) == (D[Sqrt[AiryAiPrime[x]^2 + AiryBiPrime[x]^2], {x, 1}])^(2)+ (Sqrt[AiryAiPrime[x]^2 + AiryBiPrime[x]^2])^(2)* (D[ArcTan[Divide[AiryAiPrime[x], AiryBiPrime[x]]], {x, 1}])^(2)
Successful Successful Skip - symbolical successful subtest Successful [Tested: 3]
9.8.E16 N ′ 2 ⁡ ( x ) + N 2 ⁡ ( x ) ⁢ ϕ ′ 2 ⁡ ( x ) = N ′ 2 ⁡ ( x ) + π - 2 ⁢ x 2 ⁢ N - 2 ⁡ ( x ) diffop modulus-Airy-N 1 2 𝑥 modulus-Airy-N 2 𝑥 diffop phase-Airy-Phi 1 2 𝑥 diffop modulus-Airy-N 1 2 𝑥 superscript 𝜋 2 superscript 𝑥 2 modulus-Airy-N 2 𝑥 {\displaystyle{\displaystyle{N'^{2}}\left(x\right)+{N^{2}}\left(x\right){\phi'% ^{2}}\left(x\right)={N'^{2}}\left(x\right)+\pi^{-2}x^{2}{N^{-2}}\left(x\right)}}
\AirymodderivN'^{2}@{x}+\AirymodderivN^{2}@{x}\Airyphasederivphi'^{2}@{x} = \AirymodderivN'^{2}@{x}+\pi^{-2}x^{2}\AirymodderivN^{-2}@{x}		 

(diff( sqrt(AiryAi(1, x)^2+AiryBi(1, x)^2), x$(1) ))^(2)+ (sqrt(AiryAi(1, x)^2+AiryBi(1, x)^2))^(2)* (diff( arctan(AiryAi(1, x)/AiryBi(1, x)), x$(1) ))^(2) = (diff( sqrt(AiryAi(1, x)^2+AiryBi(1, x)^2), x$(1) ))^(2)+ (Pi)^(- 2)* (x)^(2)* (sqrt(AiryAi(1, x)^2+AiryBi(1, x)^2))^(- 2)

(D[Sqrt[AiryAiPrime[x]^2 + AiryBiPrime[x]^2], {x, 1}])^(2)+ (Sqrt[AiryAiPrime[x]^2 + AiryBiPrime[x]^2])^(2)* (D[ArcTan[Divide[AiryAiPrime[x], AiryBiPrime[x]]], {x, 1}])^(2) == (D[Sqrt[AiryAiPrime[x]^2 + AiryBiPrime[x]^2], {x, 1}])^(2)+ (Pi)^(- 2)* (x)^(2)* (Sqrt[AiryAiPrime[x]^2 + AiryBiPrime[x]^2])^(- 2)
Failure Successful Successful [Tested: 3] Successful [Tested: 3]
9.8.E17 tan ⁡ ( θ ⁡ ( x ) - ϕ ⁡ ( x ) ) = 1 / ( π ⁢ M ⁡ ( x ) ⁢ M ′ ⁡ ( x ) ) phase-Airy-Theta 𝑥 phase-Airy-Phi 𝑥 1 𝜋 modulus-Airy-M 𝑥 diffop modulus-Airy-M 1 𝑥 {\displaystyle{\displaystyle\tan\left(\theta\left(x\right)-\phi\left(x\right)% \right)=1/(\pi M\left(x\right)M'\left(x\right))}}
\tan@{\Airyphasetheta@{x}-\Airyphasederivphi@{x}} = 1/(\pi\AirymodM@{x}\AirymodM'@{x})		 

tan(arctan(AiryAi(x)/AiryBi(x))- arctan(AiryAi(1, x)/AiryBi(1, x))) = 1/(Pi*sqrt(AiryAi(x)^2+AiryBi(x)^2)*diff( sqrt(AiryAi(x)^2+AiryBi(x)^2), x$(1) ))

Tan[ArcTan[Divide[AiryAi[x], AiryBi[x]]]- ArcTan[Divide[AiryAiPrime[x], AiryBiPrime[x]]]] == 1/(Pi*Sqrt[AiryAi[x]^2 + AiryBi[x]^2]*D[Sqrt[AiryAi[x]^2 + AiryBi[x]^2], {x, 1}])
Failure Successful Successful [Tested: 3] Successful [Tested: 3]
9.8.E17 1 / ( π ⁢ M ⁡ ( x ) ⁢ M ′ ⁡ ( x ) ) = - M ⁡ ( x ) ⁢ θ ′ ⁡ ( x ) / M ′ ⁡ ( x ) 1 𝜋 modulus-Airy-M 𝑥 diffop modulus-Airy-M 1 𝑥 modulus-Airy-M 𝑥 diffop phase-Airy-Theta 1 𝑥 diffop modulus-Airy-M 1 𝑥 {\displaystyle{\displaystyle 1/(\pi M\left(x\right)M'\left(x\right))=-M\left(x% \right)\theta'\left(x\right)/M'\left(x\right)}}
1/(\pi\AirymodM@{x}\AirymodM'@{x}) = -\AirymodM@{x}\Airyphasetheta'@{x}/\AirymodM'@{x}		 

1/(Pi*sqrt(AiryAi(x)^2+AiryBi(x)^2)*diff( sqrt(AiryAi(x)^2+AiryBi(x)^2), x$(1) )) = - sqrt(AiryAi(x)^2+AiryBi(x)^2)*diff( arctan(AiryAi(x)/AiryBi(x)), x$(1) )/diff( sqrt(AiryAi(x)^2+AiryBi(x)^2), x$(1) )

1/(Pi*Sqrt[AiryAi[x]^2 + AiryBi[x]^2]*D[Sqrt[AiryAi[x]^2 + AiryBi[x]^2], {x, 1}]) == - Sqrt[AiryAi[x]^2 + AiryBi[x]^2]*D[ArcTan[Divide[AiryAi[x], AiryBi[x]]], {x, 1}]/D[Sqrt[AiryAi[x]^2 + AiryBi[x]^2], {x, 1}]
Failure Successful Successful [Tested: 3] Successful [Tested: 3]
9.8#Ex4 M ′′ ⁡ ( x ) = x ⁢ M ⁡ ( x ) + π - 2 ⁢ M - 3 ⁡ ( x ) diffop modulus-Airy-M 2 𝑥 𝑥 modulus-Airy-M 𝑥 superscript 𝜋 2 modulus-Airy-M 3 𝑥 {\displaystyle{\displaystyle M''\left(x\right)=xM\left(x\right)+\pi^{-2}{M^{-3% }}\left(x\right)}}
\AirymodM''@{x} = x\AirymodM@{x}+\pi^{-2}\AirymodM^{-3}@{x}		 

diff( sqrt(AiryAi(x)^2+AiryBi(x)^2), x$(2) ) = x*sqrt(AiryAi(x)^2+AiryBi(x)^2)+ (Pi)^(- 2)* (sqrt(AiryAi(x)^2+AiryBi(x)^2))^(- 3)

D[Sqrt[AiryAi[x]^2 + AiryBi[x]^2], {x, 2}] == x*Sqrt[AiryAi[x]^2 + AiryBi[x]^2]+ (Pi)^(- 2)* (Sqrt[AiryAi[x]^2 + AiryBi[x]^2])^(- 3)
Failure Successful Successful [Tested: 3] Successful [Tested: 3]
9.8#Ex5 M 2 ′′′ ⁡ ( x ) - 4 ⁢ x ⁢ M 2 ′ ⁡ ( x ) - 2 ⁢ M 2 ⁡ ( x ) = 0 diffop modulus-Airy-M 2 3 𝑥 4 𝑥 diffop modulus-Airy-M 2 1 𝑥 2 modulus-Airy-M 2 𝑥 0 {\displaystyle{\displaystyle{M^{2}}'''\left(x\right)-4x{M^{2}}'\left(x\right)-% 2{M^{2}}\left(x\right)=0}}
\AirymodM^{2}'''@{x}-4x\AirymodM^{2}'@{x}-2\AirymodM^{2}@{x} = 0		 

(diff( sqrt(AiryAi(x)^2+AiryBi(x)^2), x$(3) ))^(2)- 4*x*(diff( sqrt(AiryAi(x)^2+AiryBi(x)^2), x$(1) ))^(2)- 2*(sqrt(AiryAi(x)^2+AiryBi(x)^2))^(2) = 0

(D[Sqrt[AiryAi[x]^2 + AiryBi[x]^2], {x, 3}])^(2)- 4*x*(D[Sqrt[AiryAi[x]^2 + AiryBi[x]^2], {x, 1}])^(2)- 2*(Sqrt[AiryAi[x]^2 + AiryBi[x]^2])^(2) == 0
Failure Aborted
Failed [3 / 3]
Result: -6.622186001
Test Values: {x = 1.5}


Result: -1.215136643
Test Values: {x = .5}

... skip entries to safe data
Failed [3 / 3]
Result: -6.622186137209987
Test Values: {Rule[x, 1.5]}


Result: -1.2151366442842328
Test Values: {Rule[x, 0.5]}

... skip entries to safe data
9.8.E19 θ ′ 2 ⁡ ( x ) + 1 2 ⁢ ( θ ′′′ ⁡ ( x ) / θ ′ ⁡ ( x ) ) - 3 4 ⁢ ( θ ′′ ⁡ ( x ) / θ ′ ⁡ ( x ) ) 2 = - x diffop phase-Airy-Theta 1 2 𝑥 1 2 diffop phase-Airy-Theta 3 𝑥 diffop phase-Airy-Theta 1 𝑥 3 4 superscript diffop phase-Airy-Theta 2 𝑥 diffop phase-Airy-Theta 1 𝑥 2 𝑥 {\displaystyle{\displaystyle{\theta'^{2}}\left(x\right)+\tfrac{1}{2}(\theta'''% \left(x\right)/\theta'\left(x\right))-\tfrac{3}{4}(\theta''\left(x\right)/% \theta'\left(x\right))^{2}=-x}}
\Airyphasetheta'^{2}@{x}+\tfrac{1}{2}(\Airyphasetheta'''@{x}/\Airyphasetheta'@{x})-\tfrac{3}{4}(\Airyphasetheta''@{x}/\Airyphasetheta'@{x})^{2} = -x		 

(diff( arctan(AiryAi(x)/AiryBi(x)), x$(1) ))^(2)+(1)/(2)*(diff( arctan(AiryAi(x)/AiryBi(x)), x$(3) )/diff( arctan(AiryAi(x)/AiryBi(x)), x$(1) ))-(3)/(4)*(diff( arctan(AiryAi(x)/AiryBi(x)), x$(2) )/diff( arctan(AiryAi(x)/AiryBi(x)), x$(1) ))^(2) = - x

(D[ArcTan[Divide[AiryAi[x], AiryBi[x]]], {x, 1}])^(2)+Divide[1,2]*(D[ArcTan[Divide[AiryAi[x], AiryBi[x]]], {x, 3}]/D[ArcTan[Divide[AiryAi[x], AiryBi[x]]], {x, 1}])-Divide[3,4]*(D[ArcTan[Divide[AiryAi[x], AiryBi[x]]], {x, 2}]/D[ArcTan[Divide[AiryAi[x], AiryBi[x]]], {x, 1}])^(2) == - x
Successful Successful - Successful [Tested: 3]
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