AiryBi[z]
gives the Airy function
.
AiryBi
AiryBi[z]
gives the Airy function
.
Details
- Mathematical function, suitable for both symbolic and numerical manipulation.
- The Airy function
is a solution to the differential equation
.
increases exponentially as
. - AiryBi[z] is an entire function of z with no branch cut discontinuities.
- For certain special arguments, AiryBi automatically evaluates to exact values.
- AiryBi can be evaluated to arbitrary numerical precision.
- AiryBi automatically threads over lists.
- AiryBi can be used with Interval and CenteredInterval objects. »
Examples
open all close allBasic Examples (5)
AiryBi[1.8]Plot over a subset of the reals:
Plot[AiryBi[x], {x, -10, 3}]Plot over a subset of the complexes:
ComplexPlot3D[AiryBi[z], {z, -2 - 2I, 2 + 2I}, PlotLegends -> Automatic]Series expansion at the origin:
Series[AiryBi[x], {x, 0, 4}]Series expansion at Infinity:
Series[AiryBi[x], {x, ∞, 2}]//NormalScope (40)
Numerical Evaluation (5)
Evaluate numerically to high precision:
N[AiryBi[2], 50]The precision of the output tracks the precision of the input:
AiryBi[2.00000000000000000000000]Evaluate for complex arguments:
AiryBi[2.5 + I]Evaluate AiryBi efficiently at high precision:
AiryBi[0.5`500]//TimingAiryBi[0.5`5000];//TimingCompute worst-case guaranteed intervals using Interval and CenteredInterval objects:
AiryBi[Interval[{1.9, 2}]]AiryBi[CenteredInterval[2, 0.1]]Or compute average-case statistical intervals using Around:
AiryBi[ Around[2, 0.01]]Compute the elementwise values of an array:
AiryBi[{{-1.2, 0}, {0, 1.}}]Or compute the matrix AiryBi function using MatrixFunction:
MatrixFunction[AiryBi, {{-1.2, 0}, {0, 1.}}]Specific Values (4)
Simple exact values are generated automatically:
AiryBi[0]{Limit[AiryBi[x], x -> Infinity], Limit[AiryBi[x], x -> -Infinity]}{AiryBiZero[1], AiryBiZero[2], AiryBiZero[3]}//NFind a zero of AiryBi using Solve:
xzero = Solve[AiryBi[x] == 0 && -4 < x < -2, x][[1, 1, 2]]Plot[AiryBi[x], {x, -4, 1}, Epilog -> Style[Point[{xzero, AiryBi[xzero]}], PointSize[Large], Red]]Visualization (2)
Plot the AiryBi function:
Plot[AiryBi[x], {x, -8, 3}]ComplexContourPlot[Re[AiryBi[z]], {z, -4 - 4I, 4 + 4I}, Contours -> 20]ComplexContourPlot[Im[AiryBi[z]], {z, -4 - 4I, 4 + 4I}, Contours -> 20]Function Properties (9)
AiryBi is defined for all real and complex values:
FunctionDomain[AiryBi[x], x]FunctionDomain[AiryBi[z], z, Complexes]Approximate function range of AiryBi:
FunctionRange[AiryBi[x], x, y]//NAiryBi is an analytic function of x:
FunctionAnalytic[AiryBi[x], x]AiryBi is neither non-increasing nor non-decreasing:
FunctionMonotonicity[AiryBi[x], x]AiryBi is not injective:
FunctionInjective[AiryBi[x], x]Plot[{AiryBi[x], .3}, {x, -10, 5}]AiryBi is not surjective:
FunctionSurjective[AiryBi[x], x]Plot[{AiryBi[x], -2}, {x, -10, 5}]AiryBi is neither non-negative nor non-positive:
FunctionSign[AiryBi[x], x]AiryBi has no singularities or discontinuities:
FunctionSingularities[AiryBi[x], x]FunctionDiscontinuities[AiryBi[x], x]AiryBi is neither convex nor concave:
FunctionConvexity[AiryBi[x], x]Differentiation (3)
D[AiryBi[x], x]Table[D[AiryBi[x], {x, n}], {n, 1, 4}]Plot[Evaluate[%], {x, -1, 1}, PlotLegends -> {"First Derivative", "Second Derivative", "Third Derivative", "Fourth Derivative"}]D[AiryBi[x], {x, n}]Integration (3)
Indefinite integral of AiryBi:
Integrate[AiryBi[x], x]//FullSimplifyDefinite integral of AiryBi:
Integrate[AiryBi[x], {x, -Infinity, 0}]∫ z^αAiryBi[z]^2ⅆz//TraditionalForm∫z^2 AiryAi[z] AiryBi[z]ⅆz//FullSimplify//TraditionalFormSeries Expansions (5)
Taylor expansion for AiryBi:
Series[AiryBi[x], {x, 0, 7}]Plot the first three approximations for AiryBi around
:
terms = Normal@Table[Series[AiryBi[x], {x, 0, m}], {m, 1, 5, 2}];
Plot[{AiryBi[x], terms}, {x, -2, 2}]General term in the series expansion of AiryBi:
SeriesCoefficient[AiryBi[x], {x, 0, n}]Find series expansions at infinity:
Series[AiryBi[x], {x, Infinity, 5}]Series[AiryBi[x], {x, -Infinity, 2}]Expansion at infinity for an arbitrary symbolic direction
:
Series[AiryBi[x], {x, DirectedInfinity[z], 1}]//NormalAiryBi can be applied to power series:
AiryBi[x + (x^2/2) + (x^3/9) + O[x]^4]Integral Transforms (2)
Compute the Fourier cosine transform using FourierCosTransform:
FourierCosTransform[AiryBi[-t], t, ω]//TraditionalFormHankelTransform[AiryBi[r], r, s ]Function Identities and Simplifications (3)
Simplify the expression to AiryBi:
(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)])//FullSimplifyFunctionExpand tries to simplify the argument of AiryBi:
FunctionExpand[AiryBi[z E^(2 π I/3)] ]// SimplifyFunctionExpand[AiryBi[(z^3)^1 / 3]]AiryBi[z] + E^(2 π I/3) AiryBi[z E^(2 π I/3)] + E^-(2 π I/3) AiryBi[z E^-(2 π I/3)]//FullSimplifyFunction Representations (4)
Relationship to Bessel functions:
AiryBi[z] == (1/Sqrt[3])((z^3 / 2)^1 / 3 BesselI[-(1/3), (2 z^3 / 2/3)] + z (z^3 / 2)^-(1/3) BesselI[(1/3), (2 z^3 / 2/3)])//FullSimplifyAiryBi can be represented as a DifferentialRoot:
DifferentialRootReduce[AiryBi[x], x]AiryBi can be represented in terms of MeijerG:
MeijerGReduce[AiryBi[x], x]Activate[%]//FullSimplifyTraditionalForm formatting:
AiryBi[z]//TraditionalFormApplications (2)
Solve the Schrödinger equation in a linear potential (e.g. uniform electric field):
DSolve[-ψ''[x] + x ψ[x] == ε ψ[x], ψ[x], x]Check the Sommerfeld radiation condition for a combination of Airy functions:
Series[AiryBi[z] - I AiryAi[z], {z, -∞, 2}]There is only an outgoing plane wave:
Simplify[Normal[%]//TrigToExp]Properties & Relations (5)
Use FullSimplify to simplify Airy functions, here in the Wronskian of the Airy equation:
Det[Outer[D[#1, {x, #2}]&, {AiryAi[x], AiryBi[x]}, {0, 1}]]FullSimplify[%]Compare with the output of Wronskian:
Wronskian[{AiryAi[x], AiryBi[x]}, x]FunctionExpand tries to simplify the argument of AiryBi:
FunctionExpand[AiryBi[z E^(2 π I/3)] ]// SimplifyGenerate Airy functions from differential equations:
DSolve[w''[z] == z w[z], w[z], z]DSolve[w'''[z] - 4z w'[z] - 2w[z] == 0, w[z], z]FindRoot[AiryBi[z] == 0, {z, -2}]Compare with the built-in function AiryBiZero:
AiryBiZero[1]//N∫ z^αAiryBi[z]^2ⅆz//TraditionalForm∫z^2 AiryAi[z] AiryBi[z]ⅆz//FullSimplify//TraditionalFormPossible Issues (5)
Machine-precision input is insufficient to get a correct answer:
AiryBi[-10. ^ 12 ]Use arbitrary-precision evaluation instead:
N[AiryBi[-10 ^ 12 ], 10]A larger setting for $MaxExtraPrecision can be needed:
N[AiryBi[-10 ^ 100], 20]Block[{$MaxExtraPrecision = 200}, N[AiryBi[-10 ^ 100], 20]]Machine-number inputs can give high‐precision results:
AiryBi[10. ^ 3]MachineNumberQ[%]Simplifications sometimes hold only in parts of the complex plane:
FullSimplify[Sqrt[-(z/3)] (BesselJ[-(1/3), (2/3) (-z)^3 / 2] - BesselJ[(1/3), (2/3) (-z)^3 / 2]), z < 0]{Sqrt[-(z/3)] (BesselJ[-(1/3), (2/3) (-z)^3 / 2] - BesselJ[(1/3), (2/3) (-z)^3 / 2]), AiryBi[z]} /. z -> 2.Parentheses are required for correct parsing in the traditional form:
Bi xBi(x)See Also
Tech Notes
Related Guides
Related Links
History
Introduced in 1991 (2.0) | Updated in 2021 (13.0) ▪ 2022 (13.1)
Text
Wolfram Research (1991), AiryBi, Wolfram Language function, https://reference.wolfram.com/language/ref/AiryBi.html (updated 2022).
CMS
Wolfram Language. 1991. "AiryBi." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/AiryBi.html.
APA
Wolfram Language. (1991). AiryBi. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/AiryBi.html
BibTeX
@misc{reference.wolfram_2026_airybi, author="Wolfram Research", title="{AiryBi}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/AiryBi.html}", note=[Accessed: 13-June-2026]}
BibLaTeX
@online{reference.wolfram_2026_airybi, organization={Wolfram Research}, title={AiryBi}, year={2022}, url={https://reference.wolfram.com/language/ref/AiryBi.html}, note=[Accessed: 13-June-2026]}