# AiryAi

AiryAi[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 .
• 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

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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(42)

### Numerical Evaluation(5)

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:

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

Or compute average-case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix AiryAi function using MatrixFunction:

### 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 :

Plot the imaginary part of :

### Function Properties(9)

AiryAi is defined for all real and complex values:

Approximate function range of AiryAi:

AiryAi is an analytic function of x:

AiryAi is neither non-increasing nor non-decreasing:

AiryAi is not injective:

AiryAi is not surjective:

AiryAi is neither non-negative nor non-positive:

AiryAi has no singularities or discontinuities:

AiryAi is neither convex nor concave:

### Differentiation(3)

First derivative:

Higher derivatives:

Formula for the 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:

## Applications(4)

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:

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

Find the location of the mode:

## 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:

Integral:

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:

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:

## Neat Examples(1)

Play a vibrato sound made from a linear combination of AiryAi functions:

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).

#### CMS

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

#### BibTeX

@misc{reference.wolfram_2024_airyai, author="Wolfram Research", title="{AiryAi}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/AiryAi.html}", note=[Accessed: 14-September-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_airyai, organization={Wolfram Research}, title={AiryAi}, year={2022}, url={https://reference.wolfram.com/language/ref/AiryAi.html}, note=[Accessed: 14-September-2024 ]}