Asymptotic Approximations - 2.8 Differential Equations with a Parameter

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
2.8#Ex2 ξ = ∫ f 1 / 2 ⁢ ( z ) ⁢ d z 𝜉 superscript 𝑓 1 2 𝑧 𝑧 {\displaystyle{\displaystyle\xi=\int f^{1/2}(z)\mathrm{d}z}}
\xi = \int f^{1/2}(z)\diff{z}		 

xi = int((f(z))^(1/2), z)

\[Xi] == Integrate[(f[z])^(1/2), z, GenerateConditions->None]
Failure Failure Error
Failed [100 / 100]
Result: Plus[Complex[0.8660254037844387, 0.49999999999999994], Times[Complex[-0.48296291314453416, -0.12940952255126037], Power[z, 2]]]
Test Values: {Rule[f, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ξ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Plus[Complex[-0.4999999999999998, 0.8660254037844387], Times[Complex[-0.48296291314453416, -0.12940952255126037], Power[z, 2]]]
Test Values: {Rule[f, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ξ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
2.8#Ex4 2 3 ⁢ ξ 3 / 2 = ∫ z 0 z f 1 / 2 ⁢ ( t ) ⁢ d t 2 3 superscript 𝜉 3 2 superscript subscript subscript 𝑧 0 𝑧 superscript 𝑓 1 2 𝑡 𝑡 {\displaystyle{\displaystyle\tfrac{2}{3}\xi^{3/2}=\int_{z_{0}}^{z}f^{1/2}(t)% \mathrm{d}t}}
\tfrac{2}{3}\xi^{3/2} = \int_{z_{0}}^{z}f^{1/2}(t)\diff{t}		 

(2)/(3)*(xi)^(3/2) = int((f(t))^(1/2), t = z[0]..z)

Divide[2,3]*\[Xi]^(3/2) == Integrate[(f[t])^(1/2), {t, Subscript[z, 0], z}, GenerateConditions->None]
Failure Failure
Failed [300 / 300]
Result: .4714045210+.4714045209*I
Test Values: {f = 1/2*3^(1/2)+1/2*I, xi = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, z[0] = 1/2*3^(1/2)+1/2*I}


Result: .2125854754-.4945213056*I
Test Values: {f = 1/2*3^(1/2)+1/2*I, xi = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, z[0] = -1/2+1/2*I*3^(1/2)}


Result: .2125854754-.4945213056*I
Test Values: {f = 1/2*3^(1/2)+1/2*I, xi = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, z[0] = 1/2-1/2*I*3^(1/2)}


Result: .4714045210+.4714045209*I
Test Values: {f = 1/2*3^(1/2)+1/2*I, xi = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, z[0] = -1/2*3^(1/2)-1/2*I}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[0.4714045207910317, 0.4714045207910316]
Test Values: {Rule[f, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ξ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[z, 0], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.21258547568851094, -0.4945213054980366]
Test Values: {Rule[f, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ξ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[z, 0], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
2.8#Ex6 2 ⁢ ξ 1 / 2 = ∫ z 0 z f 1 / 2 ⁢ ( t ) ⁢ d t 2 superscript 𝜉 1 2 superscript subscript subscript 𝑧 0 𝑧 superscript 𝑓 1 2 𝑡 𝑡 {\displaystyle{\displaystyle 2\xi^{1/2}=\int_{z_{0}}^{z}f^{1/2}(t)\mathrm{d}t}}
2\xi^{1/2} = \int_{z_{0}}^{z}f^{1/2}(t)\diff{t}		 

2*(xi)^(1/2) = int((f(t))^(1/2), t = z[0]..z)

2*\[Xi]^(1/2) == Integrate[(f[t])^(1/2), {t, Subscript[z, 0], z}, GenerateConditions->None]
Failure Failure
Failed [300 / 300]
Result: 1.931851653+.5176380902*I
Test Values: {f = 1/2*3^(1/2)+1/2*I, xi = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, z[0] = 1/2*3^(1/2)+1/2*I}


Result: 1.673032607-.4482877363*I
Test Values: {f = 1/2*3^(1/2)+1/2*I, xi = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, z[0] = -1/2+1/2*I*3^(1/2)}


Result: 1.673032607-.4482877363*I
Test Values: {f = 1/2*3^(1/2)+1/2*I, xi = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, z[0] = 1/2-1/2*I*3^(1/2)}


Result: 1.931851653+.5176380902*I
Test Values: {f = 1/2*3^(1/2)+1/2*I, xi = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, z[0] = -1/2*3^(1/2)-1/2*I}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[1.9318516525781366, 0.5176380902050415]
Test Values: {Rule[f, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ξ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[z, 0], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[1.6730326074756159, -0.4482877360840267]
Test Values: {Rule[f, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ξ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[z, 0], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
2.8.E8 d 2 W / d ξ 2 = ( u 2 ⁢ ξ m + ψ ⁢ ( ξ ) ) ⁢ W derivative 𝑊 𝜉 2 superscript 𝑢 2 superscript 𝜉 𝑚 𝜓 𝜉 𝑊 {\displaystyle{\displaystyle\ifrac{{\mathrm{d}}^{2}W}{{\mathrm{d}\xi}^{2}}=% \left(u^{2}\xi^{m}+\psi(\xi)\right)W}}
\ideriv[2]{W}{\xi} = \left(u^{2}\xi^{m}+\psi(\xi)\right)W		 

diff(W, [xi$(2)]) = ((u)^(2)* (xi)^(m)+ psi(xi))*W

D[W, {\[Xi], 2}] == ((u)^(2)* \[Xi]^(m)+ \[Psi][\[Xi]])*W
Failure Failure
Failed [300 / 300]
Result: .4999999999-1.866025406*I
Test Values: {W = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, u = 1/2*3^(1/2)+1/2*I, xi = 1/2*3^(1/2)+1/2*I, m = 1}


Result: .8660254042-1.500000002*I
Test Values: {W = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, u = 1/2*3^(1/2)+1/2*I, xi = 1/2*3^(1/2)+1/2*I, m = 2}


Result: 1.000000001-1.000000001*I
Test Values: {W = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, u = 1/2*3^(1/2)+1/2*I, xi = 1/2*3^(1/2)+1/2*I, m = 3}


Result: 1.866025406+.4999999999*I
Test Values: {W = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, u = 1/2*3^(1/2)+1/2*I, xi = -1/2+1/2*I*3^(1/2), m = 1}

... skip entries to safe data
Failed [296 / 300]
Result: Complex[0.4999999999999997, -1.8660254037844388]
Test Values: {Rule[m, 1], Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[W, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ξ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ψ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.8660254037844382, -1.5000000000000002]
Test Values: {Rule[m, 2], Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[W, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ξ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ψ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
2.8.E9 d 2 W d ξ 2 = ( u 2 ξ + ρ ξ 2 ) ⁢ W derivative 𝑊 𝜉 2 superscript 𝑢 2 𝜉 𝜌 superscript 𝜉 2 𝑊 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}W}{{\mathrm{d}\xi}^{2}}=% \left(\frac{u^{2}}{\xi}+\frac{\rho}{\xi^{2}}\right)W}}
\deriv[2]{W}{\xi} = \left(\frac{u^{2}}{\xi}+\frac{\rho}{\xi^{2}}\right)W		 

diff(W, [xi$(2)]) = (((u)^(2))/(xi)+(rho)/((xi)^(2)))*W

D[W, {\[Xi], 2}] == (Divide[(u)^(2),\[Xi]]+Divide[\[Rho],\[Xi]^(2)])*W
Failure Failure
Failed [300 / 300]
Result: -1.500000001-.8660254042*I
Test Values: {W = 1/2*3^(1/2)+1/2*I, rho = 1/2*3^(1/2)+1/2*I, u = 1/2*3^(1/2)+1/2*I, xi = 1/2*3^(1/2)+1/2*I}


Result: .1339745960+.5000000004*I
Test Values: {W = 1/2*3^(1/2)+1/2*I, rho = 1/2*3^(1/2)+1/2*I, u = 1/2*3^(1/2)+1/2*I, xi = -1/2+1/2*I*3^(1/2)}


Result: 1.866025404-.5000000004*I
Test Values: {W = 1/2*3^(1/2)+1/2*I, rho = 1/2*3^(1/2)+1/2*I, u = 1/2*3^(1/2)+1/2*I, xi = 1/2-1/2*I*3^(1/2)}


Result: -.4999999996+.8660254040*I
Test Values: {W = 1/2*3^(1/2)+1/2*I, rho = 1/2*3^(1/2)+1/2*I, u = 1/2*3^(1/2)+1/2*I, xi = -1/2*3^(1/2)-1/2*I}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-1.5000000000000002, -0.8660254037844386]
Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[W, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ξ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ρ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.5000000000000004, -1.8660254037844388]
Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[W, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ξ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ρ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
2.8.E10 d 2 W / d ξ 2 = ( u 2 + ψ ⁢ ( ξ ) ) ⁢ W derivative 𝑊 𝜉 2 superscript 𝑢 2 𝜓 𝜉 𝑊 {\displaystyle{\displaystyle\ifrac{{\mathrm{d}}^{2}W}{{\mathrm{d}\xi}^{2}}=(u^% {2}+\psi(\xi))W}}
\ideriv[2]{W}{\xi} = (u^{2}+\psi(\xi))W		 

diff(W, [xi$(2)]) = ((u)^(2)+ psi(xi))*W

D[W, {\[Xi], 2}] == ((u)^(2)+ \[Psi][\[Xi]])*W
Failure Failure
Failed [288 / 300]
Result: -.6467477718e-9-2.000000002*I
Test Values: {W = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, u = 1/2*3^(1/2)+1/2*I, xi = 1/2*3^(1/2)+1/2*I}


Result: 1.000000000-1.000000001*I
Test Values: {W = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, u = 1/2*3^(1/2)+1/2*I, xi = -1/2+1/2*I*3^(1/2)}


Result: -1.000000001-1.000000000*I
Test Values: {W = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, u = 1/2*3^(1/2)+1/2*I, xi = 1/2-1/2*I*3^(1/2)}


Result: .7500000002+.2990381054*I
Test Values: {W = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, u = 1/2*3^(1/2)+1/2*I, xi = -1.5}

... skip entries to safe data
Failed [288 / 300]
Result: Complex[0.0, -2.0]
Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[W, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ξ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ψ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.9999999999999998, -1.0000000000000002]
Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[W, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ξ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ψ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
2.8.E14 d 2 W / d ξ 2 = ( u 2 ⁢ ξ + ψ ⁢ ( ξ ) ) ⁢ W derivative 𝑊 𝜉 2 superscript 𝑢 2 𝜉 𝜓 𝜉 𝑊 {\displaystyle{\displaystyle\ifrac{{\mathrm{d}}^{2}W}{{\mathrm{d}\xi}^{2}}=(u^% {2}\xi+\psi(\xi))W}}
\ideriv[2]{W}{\xi} = (u^{2}\xi+\psi(\xi))W		 

diff(W, [xi$(2)]) = ((u)^(2)* xi + psi(xi))*W

D[W, {\[Xi], 2}] == ((u)^(2)* \[Xi]+ \[Psi][\[Xi]])*W
Failure Failure
Failed [300 / 300]
Result: .4999999999-1.866025406*I
Test Values: {W = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, u = 1/2*3^(1/2)+1/2*I, xi = 1/2*3^(1/2)+1/2*I}


Result: 1.866025406+.4999999999*I
Test Values: {W = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, u = 1/2*3^(1/2)+1/2*I, xi = -1/2+1/2*I*3^(1/2)}


Result: -1.866025406-.4999999999*I
Test Values: {W = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, u = 1/2*3^(1/2)+1/2*I, xi = 1/2-1/2*I*3^(1/2)}


Result: -.4999999999+1.866025406*I
Test Values: {W = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, u = 1/2*3^(1/2)+1/2*I, xi = -1/2*3^(1/2)-1/2*I}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[0.4999999999999997, -1.8660254037844388]
Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[W, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ξ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ψ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[1.5, -0.8660254037844387]
Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[W, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ξ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ψ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
2.8.E19 Ai ⁡ ( x ) = Bi ⁡ ( x ) Airy-Ai 𝑥 Airy-Bi 𝑥 {\displaystyle{\displaystyle\mathrm{Ai}\left(x\right)=\mathrm{Bi}\left(x\right% )}}
\AiryAi@{x} = \AiryBi@{x}		 

AiryAi(x) = AiryBi(x)

AiryAi[x] == AiryBi[x]
Failure Failure
Failed [3 / 3]
Result: -1.807192007
Test Values: {x = 1.5}


Result: -.6225834366
Test Values: {x = .5}


Result: -3.263170870
Test Values: {x = 2}

Failed [3 / 3]
Result: -1.8071920067397889
Test Values: {Rule[x, 1.5]}


Result: -0.622583436622322
Test Values: {Rule[x, 0.5]}

... skip entries to safe data
2.8.E24 d 2 W d ξ 2 = ( u 2 4 ⁢ ξ + ν 2 - 1 4 ⁢ ξ 2 + ψ ⁢ ( ξ ) ξ ) ⁢ W derivative 𝑊 𝜉 2 superscript 𝑢 2 4 𝜉 superscript 𝜈 2 1 4 superscript 𝜉 2 𝜓 𝜉 𝜉 𝑊 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}W}{{\mathrm{d}\xi}^{2}}=% \left(\frac{u^{2}}{4\xi}+\frac{\nu^{2}-1}{4\xi^{2}}+\frac{\psi(\xi)}{\xi}% \right)W}}
\deriv[2]{W}{\xi} = \left(\frac{u^{2}}{4\xi}+\frac{\nu^{2}-1}{4\xi^{2}}+\frac{\psi(\xi)}{\xi}\right)W		 

diff(W, [xi$(2)]) = (((u)^(2))/(4*xi)+((nu)^(2)- 1)/(4*(xi)^(2))+(psi(xi))/(xi))*W

D[W, {\[Xi], 2}] == (Divide[(u)^(2),4*\[Xi]]+Divide[\[Nu]^(2)- 1,4*\[Xi]^(2)]+Divide[\[Psi][\[Xi]],\[Xi]])*W
Failure Failure
Failed [300 / 300]
Result: -.6250000006-1.332531755*I
Test Values: {W = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, u = 1/2*3^(1/2)+1/2*I, xi = 1/2*3^(1/2)+1/2*I}


Result: -.7165063513-.4910254040*I
Test Values: {W = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, u = 1/2*3^(1/2)+1/2*I, xi = -1/2+1/2*I*3^(1/2)}


Result: -.2834936493-.7410254042*I
Test Values: {W = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, u = 1/2*3^(1/2)+1/2*I, xi = 1/2-1/2*I*3^(1/2)}


Result: -.3750000004-.8995190529*I
Test Values: {W = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, u = 1/2*3^(1/2)+1/2*I, xi = -1/2*3^(1/2)-1/2*I}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-0.6250000000000002, -1.3325317547305482]
Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[W, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ξ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ψ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.7410254037844384, -0.9665063509461098]
Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[W, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ξ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ψ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
2.8.E32 J ν ⁡ ( x ) + Y ν ⁡ ( x ) = 0 Bessel-J 𝜈 𝑥 Bessel-Y-Weber 𝜈 𝑥 0 {\displaystyle{\displaystyle J_{\nu}(x)+Y_{\nu}(x)=0}}
\BesselJ{\nu}(x)+\BesselY{\nu}(x) = 0		 

BesselJ(nu, x)+ BesselY(nu, (x) ) = 0

BesselJ[\[Nu], x]+ BesselY[\[Nu], (x) ] == 0
Translation Error Translation Error - -
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