# AiryBiPrime

AiryBiPrime[z]

gives the derivative of the Airy function .

# Details • Mathematical function, suitable for both symbolic and numerical manipulation.
• For certain special arguments, AiryBiPrime automatically evaluates to exact values.
• AiryBiPrime can be evaluated to arbitrary numerical precision.
• AiryBiPrime 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(30)

### Numerical Evaluation(4)

Evaluate numerically to high precision:

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

Evaluate for complex arguments:

Evaluate AiryBiPrime efficiently at high precision:

### Specific Values(3)

Simple exact values are generated automatically:

Limiting value at infinity:

Find a zero of AiryBiPrime using Solve:

### Visualization(2)

Plot the AiryBiPrime function:

Plot the real part of :

Plot the imaginary part of :

### Function Properties(2)

AiryBiPrime is defined for all real and complex values:

Function range of AiryBiPrime:

### Differentiation(3)

First derivative:

Higher derivatives:

Formula for the  derivative:

### Integration(3)

Indefinite integral of AiryBiPrime gives back AiryBi:

Definite integral of AiryBiPrime:

More integrals:

### Series Expansions(4)

Taylor expansion for AiryBiPrime:

Plot the first three approximations for AiryBiPrime around :

General term in the series expansion of AiryBiPrime:

Find the series expansion at infinity:

The behavior at negative infinity is quite different:

AiryBiPrime can be applied to power series:

### Integral Transforms(2)

Compute the Fourier cosine transform using FourierCosTransform:

### Function Identities and Simplifications(3)

Functional identity:

Simplify the expression to AiryBiPrime:

FunctionExpand tries to simplify the argument of AiryBiPrime:

### Function Representations(4)

Relationship to Bessel functions:

AiryBiPrime can be represented as a DifferentialRoot:

Represent in terms of MeijerG using:

## Applications(2)

Solve differential equations in terms of AiryBiPrime:

Solution of the modified linearized KortewegdeVries equation for any function :

Verify the solution:

## Properties & Relations(5)

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

Compare with the output of Wronskian:

Generate Airy functions from differential equations:

Integral transforms:

Obtain AiryBiPrime from sums:

AiryBiPrime appears in special cases of several mathematical functions:

## Possible Issues(3)

Machine-precision input is insufficient to give a correct answer: 