AiryBiPrime[z]
gives the derivative of the Airy function
.
AiryBiPrime
AiryBiPrime[z]
gives the derivative of the Airy function
.
Details
- Mathematical function, suitable for both symbolic and numerical manipulation.
- For certain special arguments, AiryBiPrime automatically evaluates to exact values.
- AiryBiPrime can be evaluated to arbitrary numerical precision.
- AiryBiPrime automatically threads over lists.
- AiryBiPrime can be used with Interval and CenteredInterval objects. »
Examples
open all close allBasic Examples (5)
AiryBiPrime[1.8]Plot over a subset of the reals:
Plot[AiryBiPrime[x], {x, -10, 2}]Plot over a subset of the complexes:
ComplexPlot3D[AiryBiPrime[z], {z, -2 - 2I, 2 + 2I}, PlotLegends -> Automatic]Series expansion at the origin:
Series[AiryBiPrime[x], {x, 0, 5}]Series expansion at Infinity:
Series[AiryBiPrime[x], {x, ∞, 2}]//NormalScope (38)
Numerical Evaluation (5)
Evaluate numerically to high precision:
N[AiryBiPrime[5 / 2], 50]The precision of the output tracks the precision of the input:
AiryBiPrime[2.50000000000000000000000]Evaluate for complex arguments:
AiryBiPrime[2.5 + I]Evaluate AiryBiPrime efficiently at high precision:
AiryBiPrime[0.5`500]//TimingAiryBiPrime[0.5`5000];//TimingCompute worst-case guaranteed intervals using Interval and CenteredInterval objects:
AiryBiPrime[Interval[{1.9, 2}]]AiryBiPrime[CenteredInterval[1, 0.1]]Or compute average-case statistical intervals using Around:
AiryBiPrime[ Around[2, 0.01]]Compute the elementwise values of an array:
AiryBiPrime[{{-1.2, 0}, {0, 1.}}]Or compute the matrix AiryBiPrime function using MatrixFunction:
MatrixFunction[AiryBiPrime, {{-1.2, 0}, {0, 1.}}]Specific Values (3)
Simple exact values are generated automatically:
AiryBiPrime[0]Limit[AiryBiPrime[x], x -> Infinity]Find a zero of AiryBiPrime using Solve:
xzero = N@Solve[AiryBiPrime[x] == 0 && -3 < x < -2, x][[1, 1, 2]]//QuietPlot[AiryBiPrime[x], {x, -5, 2}, Epilog -> Style[Point[{xzero, AiryBiPrime[xzero]}], PointSize[Large], Red]]Visualization (2)
Plot the AiryBiPrime function:
Plot[AiryBiPrime[x], {x, -6, 2}]ComplexContourPlot[Re[AiryBiPrime[z]], {z, -4 - 4I, 4 + 4I}, Contours -> 20]ComplexContourPlot[Im[AiryBiPrime[z]], {z, -4 - 4I, 4 + 4I}, Contours -> 20]Function Properties (9)
AiryBiPrime is defined for all real and complex values:
FunctionDomain[AiryBiPrime[x], x]FunctionDomain[AiryBiPrime[z], z, Complexes]Function range of AiryBiPrime:
FunctionRange[AiryBiPrime[x], x, y]AiryBiPrime is an analytic function of x:
FunctionAnalytic[AiryBiPrime[x], x]AiryBiPrime is neither non-increasing nor non-decreasing:
FunctionMonotonicity[AiryBiPrime[x], x]AiryBiPrime is not injective:
FunctionInjective[AiryBiPrime[x], x]Plot[{AiryBiPrime[x], .3}, {x, -10, 5}]AiryBiPrime is surjective:
FunctionSurjective[AiryBiPrime[x], x]Plot[{AiryBiPrime[x], -1.5}, {x, -50, 5}]AiryBiPrime is neither non-negative nor non-positive:
FunctionSign[AiryBiPrime[x], x]AiryBiPrime has no singularities or discontinuities:
FunctionSingularities[AiryBiPrime[x], x]FunctionDiscontinuities[AiryBiPrime[x], x]AiryBiPrime is neither convex nor concave:
FunctionConvexity[AiryBiPrime[x], x]Differentiation (3)
D[AiryBiPrime[x], x]Table[D[AiryBiPrime[x], {x, n}], {n, 1, 4}]Plot[Evaluate[%], {x, -1, 1}, PlotLegends -> {"First Derivative", "Second Derivative", "Third Derivative", "Fourth Derivative"}]D[AiryBiPrime[x], {x, n}]Integration (3)
Indefinite integral of AiryBiPrime gives back AiryBi:
Integrate[AiryBiPrime[x], x]Definite integral of AiryBiPrime:
Integrate[AiryBiPrime[x], {x, -Infinity, 0}]Integrate[AiryBi[a x]AiryBiPrime[a x], x]Integrate[x ^ a AiryBiPrime[x] ^ 2, x]//TraditionalFormSeries Expansions (4)
Taylor expansion for AiryBiPrime:
Series[AiryBiPrime[x], {x, 0, 7}]Plot the first three approximations for AiryBiPrime around
:
terms = Normal@{Series[AiryBiPrime[x], {x, 0, 2}], Series[AiryBiPrime[x], {x, 0, 3}], Series[AiryBiPrime[x], {x, 0, 5}]};
Plot[{AiryBiPrime[x], terms}, {x, -2, 2}]General term in the series expansion of AiryBiPrime:
SeriesCoefficient[AiryBiPrime[x], {x, 0, n}]Find the series expansion at infinity:
Series[AiryBiPrime[x], {x, Infinity, 3}]The behavior at negative infinity is quite different:
Series[AiryBiPrime[x], {x, -Infinity, 1}]AiryBiPrime can be applied to power series:
AiryBiPrime[Log[1 + x] + O[x] ^ 4]Integral Transforms (2)
Compute the Fourier cosine transform using FourierCosTransform:
FourierCosTransform[AiryBiPrime[-t], t, s]//TraditionalFormHankelTransform[AiryBiPrime[r], r, s ]Function Identities and Simplifications (3)
AiryBiPrime[z] + E^(2 I π/3) AiryBiPrime[E^-(2 I π/3) z] + E^-(2 I π/3) AiryBiPrime[E^(2 I π/3) z]//FullSimplifySimplify the expression to AiryBiPrime:
(z^2 Hypergeometric0F1[(5/3), (z^3/9)]/2 3^1 / 6 Gamma[(2/3)]) + (3^1 / 6 Hypergeometric0F1[(1/3), (z^3/9)]/ Gamma[(1/3)])//FullSimplifyFunctionExpand tries to simplify the argument of AiryBiPrime:
FunctionExpand [AiryBiPrime[z Exp[2Pi I / 3]]] // SimplifyFunctionExpand[AiryBiPrime[(z^3)^1 / 3]]Function Representations (4)
Relationship to Bessel functions:
(1/Sqrt[3])(z^2(z^3 / 2)^-(2/3) BesselI[(2/3), (2 z^3 / 2/3)] + (z^3 / 2)^2 / 3 BesselI[-(2/3), (2 z^3 / 2/3)])//FullSimplifyAiryBiPrime can be represented as a DifferentialRoot:
DifferentialRootReduce[AiryBiPrime[x], x]Represent in terms of MeijerG using MeijerGReduce:
MeijerGReduce[AiryBiPrime[x], x]Activate[%]//FullSimplifyTraditionalForm formatting:
AiryBiPrime[z]//TraditionalFormApplications (3)
Solve differential equations in terms of AiryBiPrime:
DSolve[-9 z ^ 2 w[z] + 9 z ^ 5 w[z] - 15 w'[z] + 20 z ^ 3 w'[z] + 15 z w''[z] - 10 z ^ 4 w''[z] - 6 z ^ 2 w'''[z] + z ^ 3 w''''[z] == 0, w, z]Solution of the modified linearized Korteweg–deVries equation for any function
:
u[x_, t_] := (3 t) ^ (-2 / 3) Integrate[AiryBiPrime[(x - y) (3 t) ^ (-1 / 3)] f[y], {y, -Infinity, Infinity}]D[u[x, t], t, x] + D[u[x, t], x, x, x, x] //. α_.Integrate[a_, i_] + β_.Integrate[b_, i_] :> Integrate[α a + β b, i]Solution of the time‐independent Schrödinger equation in a linear cone potential, represented with AiryAiPrime and AiryBiPrime:
DSolve[{-ψ''[z] + Piecewise[{{z, z > 0}, {-z, z < 0}}] ψ[z] == ε ψ[z],
ψ[0] == 1, ψ'[0] == 0}, ψ[z], z]The normalizable states are determined through the zeros of AiryAiPrime:
b[n_] := x /. FindRoot[AiryAiPrime[x], {x, -(3Pi / 8(Max[0, 4n - 1])) ^ (2 / 3), -(3Pi / 8(4n + 3)) ^ (2 / 3)}]ψ[n_, x_] := 1 / Sqrt[-b[n]] / AiryAi[b[n]]AiryAi[Sqrt[x ^ 2] + b[n]]Plot[Evaluate[Prepend[Table[-b[n] + ψ[n, x], {n, 0, 12}], Abs[x]]], {x, -16, 16}, Frame -> True, Axes -> False]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]Generate Airy functions from differential equations:
DSolve[z w''[z] - w'[z] - z ^ 2w[z] == 0, w, z]FourierCosTransform[AiryBiPrime[-t], t, s]//TraditionalFormObtain AiryBiPrime from sums:
(3^1 / 6 /Gamma[(1/3)])Underoverscript[∑, k = 0, ∞](1/Pochhammer[(1/3), k] k!)((z^3/9))^k + (z^2 /2 3^1 / 6 Gamma[(2/3)])Underoverscript[∑, k = 0, ∞](1/Pochhammer[(5/3), k] k!)((z^3/9))^k//FullSimplifyAiryBiPrime appears in special cases of several mathematical functions:
{(z^2 Hypergeometric0F1[(5/3), (z^3/9)]/2 3^1 / 6 Gamma[(2/3)]) + (3^1 / 6 Hypergeometric0F1[(1/3), (z^3/9)]/ Gamma[(1/3)]),
MeijerG[{{}, {-(1/6), (1/3)}}, {{0, (2/3)}, {-(1/6), (1/3)}}, 3^-2 / 3 z, (1/3)]}//FunctionExpandPossible Issues (3)
Machine-precision input is insufficient to give a correct answer:
AiryBiPrime[-10. ^ 12 ]Use arbitrary-precision evaluation instead:
N[AiryBiPrime[-10 ^ 12 ], 10]A larger setting for $MaxExtraPrecision can be needed:
N[AiryBiPrime[-10 ^ 100], 20]Block[{$MaxExtraPrecision = 200}, N[AiryBiPrime[-10 ^ 100], 20]]Machine-number inputs can give high‐precision results:
AiryBiPrime[10. ^ 3]MachineNumberQ[%]Neat Examples (1)
Nested integrals of the square of AiryBiPrime:
NestList[Integrate[#, z]&, AiryBiPrime[z] ^ 2, 4]//Simplify//TraditionalFormSee 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), AiryBiPrime, Wolfram Language function, https://reference.wolfram.com/language/ref/AiryBiPrime.html (updated 2022).
CMS
Wolfram Language. 1991. "AiryBiPrime." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/AiryBiPrime.html.
APA
Wolfram Language. (1991). AiryBiPrime. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/AiryBiPrime.html
BibTeX
@misc{reference.wolfram_2026_airybiprime, author="Wolfram Research", title="{AiryBiPrime}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/AiryBiPrime.html}", note=[Accessed: 13-June-2026]}
BibLaTeX
@online{reference.wolfram_2026_airybiprime, organization={Wolfram Research}, title={AiryBiPrime}, year={2022}, url={https://reference.wolfram.com/language/ref/AiryBiPrime.html}, note=[Accessed: 13-June-2026]}