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:

MellinTransform:

HankelTransform:

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 (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 Map–Airy 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:

Use arbitrary-precision evaluation instead:

A larger setting for $MaxExtraPrecision can be needed:

Machine-number inputs can give high‐precision 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:

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