Airy Functions

Author

Eric W. Weisstein
January 24, 2007

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/AiryFunctions.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/AiryFunctions.html.

©2007 Wolfram Research, Inc. except for portions noted otherwise

Definition

Ai(z)

FullSimplify[(1/3)(Sqrt[z])(BesselI[-(1/3),(2/3)(z^(3/2))]-BesselI[(1/3),(2/3)(z^(3/2))])]

FullSimplify[(((Sqrt[z]) BesselK[(1/3),((2 (z^(3/2)))/3)])/((Sqrt[3]) π)),z>0]

FullSimplify[Hypergeometric0F1[2/3,z^3/9]/(3^(2/3) Gamma[2/3])-(z Hypergeometric0F1[4/3,z^3/9])/(3^(1/3) Gamma[1/3])]

Table[{
AiryAi[z],
Hypergeometric0F1[2/3,z^3/9]/(3^(2/3) Gamma[2/3])-(z Hypergeometric0F1[4/3,z^3/9])/(3^(1/3) Gamma[1/3]),
(((Sqrt[z]) BesselK[(1/3),((2 (z^(3/2)))/3)])/((Sqrt[3]) π))
}/.z->Random[Real,{-1,1},10],{5}]//ColumnForm

Bi(z)

FullSimplify[(Sqrt[(z/3)])(BesselI[-(1/3),(2/3)(z^(3/2))]+BesselI[(1/3),(2/3)(z^(3/2))])]

FullSimplify[(Sqrt[(z/3)])(BesselI[-(1/3),(2/3)(z^(3/2))]+BesselI[(1/3),(2/3)(z^(3/2))]),z>0]

FullSimplify[Hypergeometric0F1[2/3,z^3/9]/(3^(1/6) Gamma[2/3])+(3^(1/6) z Hypergeometric0F1[4/3,z^3/9])/Gamma[1/3]]

Table[{
AiryBi[z],
Hypergeometric0F1[2/3,z^3/9]/(3^(1/6) Gamma[2/3])+(3^(1/6) z Hypergeometric0F1[4/3,z^3/9])/Gamma[1/3],
(Sqrt[(z/3)])(BesselI[-(1/3),(2/3)(z^(3/2))]+BesselI[(1/3),(2/3)(z^(3/2))])
}/.z->Random[Real,{-1,1},10],{5}]//ColumnForm

Ai(x) and Bi(x) Along Real Axis

<<Utilities`Plot`
<<Utilities`Typesetting`

Special Values

{AiryAi[0],AiryBi[0]}

Special Values of Derivatives

{AiryAi'[0],AiryBi'[0]}

AiryAi

Plots

ComplexPlot3D[AiryAi[z],{z,-5,5},PlotPoints->200]//Timing

ComplexContourPlot[AiryAi[z],{z,-5,5}]//Timing

AiryBi

Plots

ComplexPlot3D[AiryBi[z],{z,-3,3},PlotPoints->100]//Timing

ComplexContourPlot[AiryBi[z],{z,-3,3}]//Timing

AiryGi

Plot

ComplexPlot3D[AiryGi[z],{z,-3,3},PlotPoints->50]//Timing

ComplexPlot3D[AiryGi[z],{z,-3,3},PlotPoints->100]//Timing

ComplexContourPlot[AiryGi[z],{z,-3,3}]//Timing

(((∫_0^∞)sin(((t^3)/3)+t z)dt)/π)

v4.0

1/Pi Integrate[Sin[1/3 t^3+z t],{t,0,Infinity}]

v5

1/Pi Integrate[Sin[1/3 t^3+z t],{t,0,Infinity}]

Expand[FullSimplify[%,z∈Reals]]//TraditionalForm

1/Pi Integrate[Sin[1/3 t^3+z t],{t,0,Infinity},Assumptions->x∈Reals]

1/Pi Integrate[Sin[1/3 t^3+z t],{t,0,Infinity},Assumptions->z∈Reals]

FullSimplify[%,z∈Reals]

v6

1/Pi Integrate[Sin[1/3 t^3+z t],{t,0,Infinity},Assumptions->x∈Reals]

1/Pi Integrate[Sin[1/3 t^3+z t],{t,0,Infinity},Assumptions->z∈Reals]

1/Pi Integrate[Sin[1/3 t^3+z t],{t,0,Infinity}]

1/Pi Integrate[Sin[1/3 t^3+z t],{t,0,Infinity},Assumptions->Re[z]<0]

V7

1/Pi Integrate[Sin[1/3 t^3+z t],{t,0,Infinity},Assumptions->z∈Reals]

%//FullSimplify//Expand//TraditionalForm

expr1=-(((z^5) (Subscript[, 1])(Subscript[F, 4])(1;(7/6),(4/3),(5/3),(11/6);((z^6)/1296)))/(40 π))-(((z^2) (Subscript[, 1])(Subscript[F, 4])(1;(2/3),(5/6),(7/6),(4/3);((z^6)/1296)))/(2 π))+((AiryBi[z])/3);

(AiryAi[z]) ∫(AiryBi[t])dt-(AiryBi[z]) ∫(AiryAi[t])dt==((AiryBi[z])/3)-(((z^2) (Subscript[, 1])(Subscript[F, 2])(1;(4/3),(5/3);((z^3)/9)))/(2 π))

(1/3)AiryAi[z]+Integrate[AiryBi[z]AiryBi[t]-AiryAi[t]AiryBi[z], {t,0,z},Assumptions->z>0]//FullSimplify//Expand//TraditionalForm

FullSimplify[%,z>0]

i=AiryAi[z]*Integrate[AiryBi[t], t] - AiryBi[z]*Integrate[AiryAi[t], t]

expr2=FullSimplify[(1/3)AiryBi[z]+Subtract@@{i/.t->z,i/.t->0}]

expr2//Expand//TraditionalForm

Series[expr1-expr2,{z,0,10}]

Series[Subtract@@{expr2,(1/3)AiryBi[z]-((z^2)/(2π))HypergeometricPFQ[{1},{(4/3),(5/3)},((z^3)/9)]},{z,0,10}]

AiryHi

Plots

ComplexPlot3D[AiryHi[z],{z,-3,3},PlotPoints->50]//Timing

ComplexPlot3D[AiryHi[z],{z,-3,3},PlotPoints->100]//Timing

ComplexContourPlot[AiryHi[z],{z,-3,3}]//Timing

Computation

v4.0

1/Pi Integrate[Sin[-1/3 t^3+z t],{t,0,Infinity}]

v5

1/Pi Integrate[Exp[-1/3 t^3+z t],{t,0,Infinity}]

1/Pi Integrate[Exp[-1/3 t^3+z t],{t,0,Infinity},Assumptions->z∈Reals]

1/Pi Integrate[Exp[-1/3 t^3+z t],{t,0,Infinity},Assumptions->z>0]

v6

1/Pi Integrate[Exp[-1/3 t^3+z t],{t,0,Infinity}]

1/Pi Integrate[Exp[-1/3 t^3+z t],{t,0,Infinity},Assumptions->Re[z]>0]

1/Pi Integrate[Exp[-1/3 t^3+z t],{t,0,Infinity},Assumptions->z∈Reals]

1/Pi Integrate[Exp[-1/3 t^3+z t],{t,0,Infinity},Assumptions->z>0]

Expand[%]//TraditionalForm

Series

<<DiscreteMath`RSolve`

Ai

Series[AiryAi[z],{z,0,5}]

SeriesTerm[AiryAi[z],{z,0,n}]

l1=CoefficientList[Series[AiryAi[x],{x,0,4}],x]

l2=CoefficientList[(1/((3^(2/3))π))Sum[((Gamma[(n+1)/3])/(n!))((((3^(1/3))x))^n)Sin[((2(n+1)π)/3)],{n,0,4}],x]

FullSimplify[l2]

FullSimplify[l1-l2]

Bi

Series[AiryBi[z],{z,0,5}]

SeriesTerm[AiryBi[z],{z,0,n}]

l1=CoefficientList[Series[AiryBi[x],{x,0,4}],x]

l2=CoefficientList[(1/((3^(1/6))π))Sum[((Gamma[(n+1)/3])/(n!))((((3^(1/3))x))^n)Abs[Sin[((2(n+1)π)/3)]],{n,0,4}],x]

FullSimplify[l1-l2]

Integral Representations

AiryAi

ai=Integrate[Exp[I(z t+t^3/3)],{t,-Infinity,Infinity}]/(2Pi)

FullSimplify[ai]

Sums

1/3^(2/3)/Pi Sum[Gamma[(n+1)/3]/n! Sin[2(n+1)Pi/3](3^(1/3) x)^n,{n,0,Infinity}]

FullSimplify[%1]

Differential Equation

y[x_]:=A AiryAi[k^(2/3) x]+B AiryBi[k^(2/3) x]

Simplify[y''[x]-k^2 x y[x]]

Zero Argument

AiryAi[0]

AiryBi[0]

General Differential Equation

y[x_]:=1/3 Sqrt[x](A BesselI[-1/3,2/3 k x^(3/2)]-B BesselI[1/3,2/3 k x^(3/2)])

z[k0_,x0_]:=N[y''[x]-k^2 x y[x]/.{k->k0,x->x0,A->1,B->1}]

Table[z[1,x],{x,-2,2,.2}]

Table[z[k,1],{k,-2,2,.2}]

y[x_]:=a0(1+Sum[x^(3n)/(3n)!!!/(3n-1)!!!,{n,1,k}])

Simplify[y''[x]- x y[x]] /. k->5

AiryGi

FullSimplify[AiryGi[z]]//Timing

FullSimplify[AiryGi[z,GenerateConditions->False]]//Timing

FullSimplify[(1/(360 π (z^5) ((Abs[z])^6)))(-9) z^7 Abs[z]^6 20 ]

FullSimplify[(1/(360 π (z^5) ((Abs[z])^6)))(z^3) ]

Simplify[(1/(360 π (z^5) ((Abs[z])^6)))60 π z {1,(z^10) ,((Abs[z])^9) }]

Gi[z_?NumericQ]:=(1/(360 π (z^5) ((Abs[z])^6)))(-9 (z^7) ((Abs[z])^6) (20 HypergeometricPFQ[{1},{(2/3),(5/6),(7/6),(4/3)},((z^6)/1296)]+(z^3) HypergeometricPFQ[{1},{(7/6),(4/3),(5/3),(11/6)},((z^6)/1296)])+60 π z ((z^10) (AiryBi[-Abs[z]]+AiryBi[Abs[z]])+((Abs[z])^9) (-I (Sqrt[3]) (AiryAi[-Abs[z]]-AiryAi[Abs[z]]) Im[z]+(-AiryBi[-Abs[z]]+AiryBi[Abs[z]]) Re[z])))

AiryHi

FullSimplify[AiryHi[z]]

FullSimplify[AiryHi[z,GenerateConditions->False]]

Hi[z_?NumericQ]:=((2 (Sqrt[-z]) (BesselJ[-(1/3),(2/3) (((-z))^(3/2))]-BesselJ[(1/3),(2/3) (((-z))^(3/2))]))/(3 (Sqrt[3])))+(((z^2) HypergeometricPFQ[{1},{(4/3),(5/3)},((z^3)/9)])/(2 π))