gives the Airy function .


  • 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.


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Basic Examples  (5)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Series expansion at Infinity:

Scope  (34)

Numerical Evaluation  (4)

Evaluate numerically to high precision:

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

Evaluate for complex arguments:

Evaluate AiryAi efficiently at high precision:

AiryAi threads elementwise over lists and matrices:

Specific Values  (4)

Simple exact values are generated automatically:

Limiting values at infinity:

The first three zeros:

Find a zero of AiryAi using Solve:

Visualization  (2)

Plot the AiryAi function:

Plot the real part of TemplateBox[{{x, +, {ⅈ,  , y}}}, AiryAi]:

Plot the imaginary part of TemplateBox[{{x, +, {ⅈ,  , y}}}, AiryAi]:

Function Properties  (2)

AiryAi is defined for all real and complex values:

Approximate function range of AiryAi:

Differentiation  (3)

First derivative:

Higher derivatives:

Formula for the ^(th) derivative:

Integration  (3)

Indefinite integral of AiryAi:

Verify the anti-derivative:

Definite integral of AiryAi:

More integrals:

Series Expansions  (5)

Taylor expansion for AiryAi:

Plot the first three approximations for AiryAi around :

General term in the series expansion of AiryAi:

Find the series expansion at infinity:

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

AiryAi can be applied to power series:

Integral Transforms  (3)

Compute the Fourier transform using FourierTransform:



Function Identities and Simplifications  (3)

Simplify the expression to AiryAi:

FunctionExpand tries to simplify the argument of AiryAi:

Functional identity:

Function Representations  (5)

Integral representation for real argument:

Relationship to Bessel functions:

AiryAi can be represented as a DifferentialRoot:

AiryAi can be represented in terms of MeijerG:

TraditionalForm formatting:

Applications  (3)

Solve the Schrödinger equation in a linear potential (e.g. uniform electric field):

Plot the absolute value in the complex plane:

Nested integrals of the square of AiryAi:

Properties & Relations  (8)

Use FullSimplify to simplify expressions involving Airy functions:

Compare with the output of Wronskian:

FunctionExpand tries to simplify the argument of AiryAi:

Solve the Airy differential equation:

Find a numerical root:

Compare with built-in function AiryAiZero:


Verify the anti-derivative:

Integral transforms:

AiryAi can be represented as a DifferentialRoot:

AiryAi can be represented in terms of MeijerG:

Possible Issues  (5)

Machine-precision input is insufficient to get a correct answer:

Use arbitrary-precision evaluation instead:

A larger setting for $MaxExtraPrecision can be needed:

Machine-number inputs can give highprecision results:

Simplifications sometimes hold only in parts of the complex plane:

Parentheses are required when inputting in the traditional form:

Introduced in 1988