AiryAiPrime
Details

- Mathematical function, suitable for both symbolic and numerical manipulation.
- For certain special arguments, AiryAiPrime automatically evaluates to exact values.
- AiryAiPrime can be evaluated to arbitrary numerical precision.
- AiryAiPrime automatically threads over lists.
- AiryAiPrime can be used with Interval and CenteredInterval objects. »
Examples
open allclose allBasic Examples (5)Summary of the most common use cases

https://wolfram.com/xid/0n4qgwri2j4-qi32q

Plot over a subset of the reals:

https://wolfram.com/xid/0n4qgwri2j4-dl1p7

Plot over a subset of the complexes:

https://wolfram.com/xid/0n4qgwri2j4-v3wu45

Series expansion at the origin:

https://wolfram.com/xid/0n4qgwri2j4-b0fz11

Series expansion at Infinity:

https://wolfram.com/xid/0n4qgwri2j4-rmazn

Scope (40)Survey of the scope of standard use cases
Numerical Evaluation (5)
Evaluate numerically to high precision:

https://wolfram.com/xid/0n4qgwri2j4-iwqulx

The precision of the output tracks the precision of the input:

https://wolfram.com/xid/0n4qgwri2j4-mtq8j7

Evaluate for complex arguments:

https://wolfram.com/xid/0n4qgwri2j4-beuxep

Evaluate AiryAiPrime efficiently at high precision:

https://wolfram.com/xid/0n4qgwri2j4-di5gcr


https://wolfram.com/xid/0n4qgwri2j4-bq2c6r

Compute worst-case guaranteed intervals using Interval and CenteredInterval objects:

https://wolfram.com/xid/0n4qgwri2j4-gnclzo


https://wolfram.com/xid/0n4qgwri2j4-5ofmc

Or compute average-case statistical intervals using Around:

https://wolfram.com/xid/0n4qgwri2j4-cw18bq

Compute the elementwise values of an array:

https://wolfram.com/xid/0n4qgwri2j4-thgd2

Or compute the matrix AiryBiPrime function using MatrixFunction:

https://wolfram.com/xid/0n4qgwri2j4-o5jpo

Specific Values (3)
Simple exact values are generated automatically:

https://wolfram.com/xid/0n4qgwri2j4-fbou2t


https://wolfram.com/xid/0n4qgwri2j4-ciezym

Find a zero of AiryAiPrime using Solve:

https://wolfram.com/xid/0n4qgwri2j4-f2hrld


https://wolfram.com/xid/0n4qgwri2j4-one2fb

Visualization (3)
Plot the AiryAiPrime function:

https://wolfram.com/xid/0n4qgwri2j4-ecj8m7


https://wolfram.com/xid/0n4qgwri2j4-ouu484


https://wolfram.com/xid/0n4qgwri2j4-pesily

A plot of the absolute value of AiryAiPrime over the complex plane:

https://wolfram.com/xid/0n4qgwri2j4-ydxx0

Function Properties (9)
AiryAiPrime is defined for all real and complex values:

https://wolfram.com/xid/0n4qgwri2j4-cl7ele


https://wolfram.com/xid/0n4qgwri2j4-de3irc

Function range of AiryAiPrime:

https://wolfram.com/xid/0n4qgwri2j4-evf2yr

AiryAiPrime is an analytic function of x:

https://wolfram.com/xid/0n4qgwri2j4-h5x4l2

AiryAiPrime is neither non-increasing nor non-decreasing:

https://wolfram.com/xid/0n4qgwri2j4-g6kynf

AiryAiPrime is not injective:

https://wolfram.com/xid/0n4qgwri2j4-gi38d7


https://wolfram.com/xid/0n4qgwri2j4-ctca0g

AiryAiPrime is surjective:

https://wolfram.com/xid/0n4qgwri2j4-hkqec4


https://wolfram.com/xid/0n4qgwri2j4-hdm869

AiryAiPrime is neither non-negative nor non-positive:

https://wolfram.com/xid/0n4qgwri2j4-84dui

AiryAiPrime has no singularities or discontinuities:

https://wolfram.com/xid/0n4qgwri2j4-mdtl3h


https://wolfram.com/xid/0n4qgwri2j4-mn5jws

AiryAiPrime is neither convex nor concave:

https://wolfram.com/xid/0n4qgwri2j4-kdss3

Differentiation (3)

https://wolfram.com/xid/0n4qgwri2j4-mmas49


https://wolfram.com/xid/0n4qgwri2j4-nfbe0l


https://wolfram.com/xid/0n4qgwri2j4-fxwmfc


https://wolfram.com/xid/0n4qgwri2j4-odmgl1

Integration (3)
Integral of AiryAiPrime gives back AiryAi:

https://wolfram.com/xid/0n4qgwri2j4-bponid

Definite integral of AiryAiPrime:

https://wolfram.com/xid/0n4qgwri2j4-jskynr


https://wolfram.com/xid/0n4qgwri2j4-5qvb3


https://wolfram.com/xid/0n4qgwri2j4-s9hg4

Series Expansions (4)
Taylor expansion for AiryAiPrime:

https://wolfram.com/xid/0n4qgwri2j4-ewr1h8

Plot the first three approximations for AiryAiPrime around :

https://wolfram.com/xid/0n4qgwri2j4-binhar

General term in the series expansion of AiryAiPrime:

https://wolfram.com/xid/0n4qgwri2j4-dznx2j

Find the series expansion at infinity:

https://wolfram.com/xid/0n4qgwri2j4-81hzc

The behavior at negative infinity is quite different:

https://wolfram.com/xid/0n4qgwri2j4-t5t

AiryAiPrime can be applied to power series:

https://wolfram.com/xid/0n4qgwri2j4-c1qefo

Integral Transforms (3)
Compute the Fourier transform using FourierTransform:

https://wolfram.com/xid/0n4qgwri2j4-cxw1ng


https://wolfram.com/xid/0n4qgwri2j4-r7z


https://wolfram.com/xid/0n4qgwri2j4-fj60io

Function Identities and Simplifications (3)

https://wolfram.com/xid/0n4qgwri2j4-bbvqog

Simplify the expression to AiryAiPrime:

https://wolfram.com/xid/0n4qgwri2j4-bh9t4q

FunctionExpand tries to simplify the argument of AiryAiPrime:

https://wolfram.com/xid/0n4qgwri2j4-dslz4s


https://wolfram.com/xid/0n4qgwri2j4-e6ayls

Function Representations (4)
Relationship to Bessel functions:

https://wolfram.com/xid/0n4qgwri2j4-c333o7

AiryAiPrime can be represented as a DifferentialRoot:

https://wolfram.com/xid/0n4qgwri2j4-czbpv1

AiryAiPrime can be represented in terms of MeijerG:

https://wolfram.com/xid/0n4qgwri2j4-gj3zr4


https://wolfram.com/xid/0n4qgwri2j4-e9uj64

TraditionalForm formatting:

https://wolfram.com/xid/0n4qgwri2j4-g9l2l0

Applications (4)Sample problems that can be solved with this function
Solve differential equations in terms of AiryAiPrime:

https://wolfram.com/xid/0n4qgwri2j4-eq9tb0

Solution of the time‐independent Schrödinger equation in a linear cone potential:

https://wolfram.com/xid/0n4qgwri2j4-dpzcv3

The normalizable states are determined through the zeros of AiryAiPrime:

https://wolfram.com/xid/0n4qgwri2j4-dc0a99

https://wolfram.com/xid/0n4qgwri2j4-d3h04o

https://wolfram.com/xid/0n4qgwri2j4-k1ldz2

An integral kernel related to the Gaussian unitary ensembles:

https://wolfram.com/xid/0n4qgwri2j4-fwbg6v

A convolution integral solving the modified linearized Korteweg–deVries equation for any function :

https://wolfram.com/xid/0n4qgwri2j4-l7nedv

https://wolfram.com/xid/0n4qgwri2j4-c1nx8z

Properties & Relations (5)Properties of the function, and connections to other functions
Use FullSimplify to simplify Airy functions, here in the Wronskian of the Airy equation:

https://wolfram.com/xid/0n4qgwri2j4-xdrla


https://wolfram.com/xid/0n4qgwri2j4-jm4iqr

Compare with the output of Wronskian:

https://wolfram.com/xid/0n4qgwri2j4-forewx

FunctionExpand tries to simplify the argument of AiryAiPrime:

https://wolfram.com/xid/0n4qgwri2j4-si79p

Airy functions are generated as solutions by DSolve:

https://wolfram.com/xid/0n4qgwri2j4-bboit3

Obtain AiryAiPrime from sums:

https://wolfram.com/xid/0n4qgwri2j4-cel1f9

AiryAiPrime appears in special cases of several mathematical functions:

https://wolfram.com/xid/0n4qgwri2j4-0w5hl

Possible Issues (3)Common pitfalls and unexpected behavior
Machine-precision input is insufficient to give a correct answer:

https://wolfram.com/xid/0n4qgwri2j4-bzgqm1

Use arbitrary-precision evaluation instead:

https://wolfram.com/xid/0n4qgwri2j4-bzyu2c

A larger setting for $MaxExtraPrecision can be needed:

https://wolfram.com/xid/0n4qgwri2j4-d5dzw7



https://wolfram.com/xid/0n4qgwri2j4-fly685

Machine-number inputs can give high‐precision results:

https://wolfram.com/xid/0n4qgwri2j4-ea5iyb


https://wolfram.com/xid/0n4qgwri2j4-lnd7og

Neat Examples (1)Surprising or curious use cases
Nested integrals of the square of AiryAiPrime:

https://wolfram.com/xid/0n4qgwri2j4-f854xr

Wolfram Research (1991), AiryAiPrime, Wolfram Language function, https://reference.wolfram.com/language/ref/AiryAiPrime.html (updated 2022).
Text
Wolfram Research (1991), AiryAiPrime, Wolfram Language function, https://reference.wolfram.com/language/ref/AiryAiPrime.html (updated 2022).
Wolfram Research (1991), AiryAiPrime, Wolfram Language function, https://reference.wolfram.com/language/ref/AiryAiPrime.html (updated 2022).
CMS
Wolfram Language. 1991. "AiryAiPrime." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/AiryAiPrime.html.
Wolfram Language. 1991. "AiryAiPrime." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/AiryAiPrime.html.
APA
Wolfram Language. (1991). AiryAiPrime. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/AiryAiPrime.html
Wolfram Language. (1991). AiryAiPrime. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/AiryAiPrime.html
BibTeX
@misc{reference.wolfram_2025_airyaiprime, author="Wolfram Research", title="{AiryAiPrime}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/AiryAiPrime.html}", note=[Accessed: 10-July-2025
]}
BibLaTeX
@online{reference.wolfram_2025_airyaiprime, organization={Wolfram Research}, title={AiryAiPrime}, year={2022}, url={https://reference.wolfram.com/language/ref/AiryAiPrime.html}, note=[Accessed: 10-July-2025
]}