AiryBi

AiryBi[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 .
  • increases exponentially as .
  • AiryBi[z] is an entire function of z with no branch cut discontinuities.
  • For certain special arguments, AiryBi automatically evaluates to exact values.
  • AiryBi can be evaluated to arbitrary numerical precision.
  • AiryBi automatically threads over lists.

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  (32)

Numerical Evaluation  (4)

Evaluate numerically to high precision:

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

Evaluate for complex arguments:

Evaluate AiryBi efficiently at high precision:

AiryBi 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 AiryBi using Solve:

Visualization  (2)

Plot the AiryBi function:

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

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

Function Properties  (2)

AiryBi is defined for all real and complex values:

Approximate function range of AiryBi:

Differentiation  (3)

First derivative:

Higher derivatives:

Formula for the ^(th) derivative:

Integration  (3)

Indefinite integral of AiryBi:

Definite integral of AiryBi:

More integrals:

Series Expansions  (5)

Taylor expansion for AiryBi:

Plot the first three approximations for AiryBi around :

General term in the series expansion of AiryBi:

Find series expansions at infinity:

Expansion at infinity for an arbitrary symbolic direction :

AiryBi can be applied to power series:

Integral Transforms  (2)

Compute the Fourier cosine transform using FourierCosTransform:

HankelTransform:

Function Identities and Simplifications  (3)

Simplify the expression to AiryBi:

FunctionExpand tries to simplify the argument of AiryBi:

Functional identity:

Function Representations  (4)

Relationship to Bessel functions:

AiryBi can be represented as a DifferentialRoot:

AiryBi can be represented in terms of MeijerG:

TraditionalForm formatting:

Applications  (2)

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

Check the Sommerfeld radiation condition for a combination of Airy functions:

There is only an outgoing plane wave:

Properties & Relations  (5)

Use FullSimplify to simplify Airy functions, here in the Wronskian of the Airy equation:

Compare with the output of Wronskian:

FunctionExpand tries to simplify the argument of AiryBi:

Generate Airy functions from differential equations:

Find a numerical root:

Compare with the built-in function AiryBiZero:

Integrals:

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 for correct parsing in the traditional form:

Introduced in 1991
 (2.0)