# 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 :

Plot the imaginary part of :

### 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  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:

### 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:

## 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: 