AiryAi

AiryAi[z]

gives the Airy function TemplateBox[{z}, AiryAi].

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • The Airy function TemplateBox[{z}, AiryAi] is a solution to the differential equation .
  • TemplateBox[{z}, AiryAi] 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:

AiryAi threads elementwise over lists and matrices:

AiryAi can be used with Interval and CenteredInterval objects:

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[{z}, AiryAi]:

Plot the imaginary part of TemplateBox[{z}, AiryAi]:

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 ^(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:

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 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:

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:

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_2023_airyai, author="Wolfram Research", title="{AiryAi}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/AiryAi.html}", note=[Accessed: 18-March-2024 ]}

BibLaTeX

@online{reference.wolfram_2023_airyai, organization={Wolfram Research}, title={AiryAi}, year={2022}, url={https://reference.wolfram.com/language/ref/AiryAi.html}, note=[Accessed: 18-March-2024 ]}