AiryAi
✖
AiryAi
Details

- Mathematical function, suitable for both symbolic and numerical manipulation.
- The Airy function
is a solution to the differential equation
.
tends to zero as
.
- AiryAi[z] is an entire function of z with no branch cut discontinuities.
- For certain special arguments, AiryAi automatically evaluates to exact values.
- AiryAi can be evaluated to arbitrary numerical precision.
- AiryAi automatically threads over lists.
- AiryAi 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/0tzzpwk6-qjs

Plot over a subset of the reals:

https://wolfram.com/xid/0tzzpwk6-dud

Plot over a subset of the complexes:

https://wolfram.com/xid/0tzzpwk6-kiedlx

Series expansion at the origin:

https://wolfram.com/xid/0tzzpwk6-e0

Series expansion at Infinity:

https://wolfram.com/xid/0tzzpwk6-rmazn

Scope (42)Survey of the scope of standard use cases
Numerical Evaluation (5)
Evaluate numerically to high precision:

https://wolfram.com/xid/0tzzpwk6-kzf

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

https://wolfram.com/xid/0tzzpwk6-u4h

Evaluate for complex arguments:

https://wolfram.com/xid/0tzzpwk6-fu5

Evaluate AiryAi efficiently at high precision:

https://wolfram.com/xid/0tzzpwk6-di5gcr


https://wolfram.com/xid/0tzzpwk6-bq2c6r

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

https://wolfram.com/xid/0tzzpwk6-gnclzo


https://wolfram.com/xid/0tzzpwk6-onkex4

Or compute average-case statistical intervals using Around:

https://wolfram.com/xid/0tzzpwk6-cw18bq

Compute the elementwise values of an array:

https://wolfram.com/xid/0tzzpwk6-thgd2

Or compute the matrix AiryAi function using MatrixFunction:

https://wolfram.com/xid/0tzzpwk6-o5jpo

Specific Values (4)
Simple exact values are generated automatically:

https://wolfram.com/xid/0tzzpwk6-e5v


https://wolfram.com/xid/0tzzpwk6-ciezym


https://wolfram.com/xid/0tzzpwk6-e3n9bq

Find a zero of AiryAi using Solve:

https://wolfram.com/xid/0tzzpwk6-f2hrld


https://wolfram.com/xid/0tzzpwk6-pneqel

Visualization (2)
Plot the AiryAi function:

https://wolfram.com/xid/0tzzpwk6-ecj8m7


https://wolfram.com/xid/0tzzpwk6-ouu484


https://wolfram.com/xid/0tzzpwk6-hkvpe5

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

https://wolfram.com/xid/0tzzpwk6-cl7ele


https://wolfram.com/xid/0tzzpwk6-de3irc

Approximate function range of AiryAi:

https://wolfram.com/xid/0tzzpwk6-evf2yr

AiryAi is an analytic function of x:

https://wolfram.com/xid/0tzzpwk6-h5x4l2

AiryAi is neither non-increasing nor non-decreasing:

https://wolfram.com/xid/0tzzpwk6-g6kynf

AiryAi is not injective:

https://wolfram.com/xid/0tzzpwk6-gi38d7


https://wolfram.com/xid/0tzzpwk6-ctca0g

AiryAi is not surjective:

https://wolfram.com/xid/0tzzpwk6-hkqec4


https://wolfram.com/xid/0tzzpwk6-hdm869

AiryAi is neither non-negative nor non-positive:

https://wolfram.com/xid/0tzzpwk6-84dui

AiryAi has no singularities or discontinuities:

https://wolfram.com/xid/0tzzpwk6-mdtl3h


https://wolfram.com/xid/0tzzpwk6-mn5jws

AiryAi is neither convex nor concave:

https://wolfram.com/xid/0tzzpwk6-kdss3

Differentiation (3)

https://wolfram.com/xid/0tzzpwk6-mmas49


https://wolfram.com/xid/0tzzpwk6-nfbe0l


https://wolfram.com/xid/0tzzpwk6-fxwmfc


https://wolfram.com/xid/0tzzpwk6-odmgl1

Integration (3)
Indefinite integral of AiryAi:

https://wolfram.com/xid/0tzzpwk6-bponid


https://wolfram.com/xid/0tzzpwk6-kodj3u

Definite integral of AiryAi:

https://wolfram.com/xid/0tzzpwk6-b9jw7l


https://wolfram.com/xid/0tzzpwk6-qix


https://wolfram.com/xid/0tzzpwk6-qu5

Series Expansions (5)
Taylor expansion for AiryAi:

https://wolfram.com/xid/0tzzpwk6-ewr1h8

Plot the first three approximations for AiryAi around :

https://wolfram.com/xid/0tzzpwk6-binhar

General term in the series expansion of AiryAi:

https://wolfram.com/xid/0tzzpwk6-dznx2j

Find the series expansion at infinity:

https://wolfram.com/xid/0tzzpwk6-syq

Find the series expansion at infinity for an arbitrary symbolic direction :

https://wolfram.com/xid/0tzzpwk6-fgf8cw

AiryAi can be applied to power series:

https://wolfram.com/xid/0tzzpwk6-c6gcbt

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

https://wolfram.com/xid/0tzzpwk6-cxw1ng


https://wolfram.com/xid/0tzzpwk6-i3uco


https://wolfram.com/xid/0tzzpwk6-fj60io

Function Identities and Simplifications (3)
Simplify the expression to AiryAi:

https://wolfram.com/xid/0tzzpwk6-bh9t4q

FunctionExpand tries to simplify the argument of AiryAi:

https://wolfram.com/xid/0tzzpwk6-dbxl2u


https://wolfram.com/xid/0tzzpwk6-h57qby


https://wolfram.com/xid/0tzzpwk6-bbvqog

Function Representations (5)
Integral representation for real argument:

https://wolfram.com/xid/0tzzpwk6-n5inaf

Relationship to Bessel functions:

https://wolfram.com/xid/0tzzpwk6-c333o7

AiryAi can be represented as a DifferentialRoot:

https://wolfram.com/xid/0tzzpwk6-jn8ef

AiryAi can be represented in terms of MeijerG:

https://wolfram.com/xid/0tzzpwk6-bx4i42


https://wolfram.com/xid/0tzzpwk6-qgtynb

TraditionalForm formatting:

https://wolfram.com/xid/0tzzpwk6-f2tj0x

Applications (4)Sample problems that can be solved with this function
Solve the Schrödinger equation in a linear potential (e.g. uniform electric field):

https://wolfram.com/xid/0tzzpwk6-yg2

Plot the absolute value in the complex plane:

https://wolfram.com/xid/0tzzpwk6-wp4

Nested integrals of the square of AiryAi:

https://wolfram.com/xid/0tzzpwk6-mk9

Compute the probability density of Map–Airy distribution [MathWorld] in closed form, represented with AiryAi and AiryAiPrime functions:

https://wolfram.com/xid/0tzzpwk6-d9dqr0


https://wolfram.com/xid/0tzzpwk6-jq38zm

Find the location of the mode:

https://wolfram.com/xid/0tzzpwk6-db1d7g

Properties & Relations (8)Properties of the function, and connections to other functions
Use FullSimplify to simplify expressions involving Airy functions:

https://wolfram.com/xid/0tzzpwk6-bnk


https://wolfram.com/xid/0tzzpwk6-tko

Compare with the output of Wronskian:

https://wolfram.com/xid/0tzzpwk6-forewx

FunctionExpand tries to simplify the argument of AiryAi:

https://wolfram.com/xid/0tzzpwk6-ihp

Solve the Airy differential equation:

https://wolfram.com/xid/0tzzpwk6-khu


https://wolfram.com/xid/0tzzpwk6-wey


https://wolfram.com/xid/0tzzpwk6-lpf

Compare with built-in function AiryAiZero:

https://wolfram.com/xid/0tzzpwk6-ldj7mn


https://wolfram.com/xid/0tzzpwk6-uml


https://wolfram.com/xid/0tzzpwk6-k98357


https://wolfram.com/xid/0tzzpwk6-r7z

AiryAi can be represented as a DifferentialRoot:

https://wolfram.com/xid/0tzzpwk6-byxm20

AiryAi can be represented in terms of MeijerG:

https://wolfram.com/xid/0tzzpwk6-ialr1l


https://wolfram.com/xid/0tzzpwk6-exa0zh

Possible Issues (5)Common pitfalls and unexpected behavior
Machine-precision input is insufficient to get a correct answer:

https://wolfram.com/xid/0tzzpwk6-s2r

Use arbitrary-precision evaluation instead:

https://wolfram.com/xid/0tzzpwk6-cc9

A larger setting for $MaxExtraPrecision can be needed:

https://wolfram.com/xid/0tzzpwk6-fti



https://wolfram.com/xid/0tzzpwk6-bwy

Machine-number inputs can give high‐precision results:

https://wolfram.com/xid/0tzzpwk6-lf9g4


https://wolfram.com/xid/0tzzpwk6-g2v

Simplifications sometimes hold only in parts of the complex plane:

https://wolfram.com/xid/0tzzpwk6-hg6


https://wolfram.com/xid/0tzzpwk6-l6x

Parentheses are required when inputting in the traditional form:

https://wolfram.com/xid/0tzzpwk6-usk


https://wolfram.com/xid/0tzzpwk6-bzk

Neat Examples (1)Surprising or curious use cases
Play a vibrato sound made from a linear combination of AiryAi functions:

https://wolfram.com/xid/0tzzpwk6-kvx0fs

Wolfram Research (1988), AiryAi, Wolfram Language function, https://reference.wolfram.com/language/ref/AiryAi.html (updated 2022).
Text
Wolfram Research (1988), AiryAi, Wolfram Language function, https://reference.wolfram.com/language/ref/AiryAi.html (updated 2022).
Wolfram Research (1988), AiryAi, Wolfram Language function, https://reference.wolfram.com/language/ref/AiryAi.html (updated 2022).
CMS
Wolfram Language. 1988. "AiryAi." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/AiryAi.html.
Wolfram Language. 1988. "AiryAi." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/AiryAi.html.
APA
Wolfram Language. (1988). AiryAi. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/AiryAi.html
Wolfram Language. (1988). AiryAi. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/AiryAi.html
BibTeX
@misc{reference.wolfram_2025_airyai, author="Wolfram Research", title="{AiryAi}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/AiryAi.html}", note=[Accessed: 01-June-2025
]}
BibLaTeX
@online{reference.wolfram_2025_airyai, organization={Wolfram Research}, title={AiryAi}, year={2022}, url={https://reference.wolfram.com/language/ref/AiryAi.html}, note=[Accessed: 01-June-2025
]}