# AiryBiPrime

AiryBiPrime[z]

gives the derivative of the Airy function .

# 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(38)

### Numerical Evaluation(5)

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:

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 AiryBiPrime function using MatrixFunction:

### 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(9)

AiryBiPrime is defined for all real and complex values:

Function range of AiryBiPrime:

AiryBiPrime is an analytic function of x:

AiryBiPrime is neither non-increasing nor non-decreasing:

AiryBiPrime is not injective:

AiryBiPrime is surjective:

AiryBiPrime is neither non-negative nor non-positive:

AiryBiPrime has no singularities or discontinuities:

AiryBiPrime is neither convex nor concave:

### 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(3)

Solve differential equations in terms of AiryBiPrime:

Solution of the modified linearized KortewegdeVries equation for any function :

Verify the solution:

Solution of the timeindependent Schrödinger equation in a linear cone potential, represented with AiryAiPrime and AiryBiPrime:

The normalizable states are determined through the zeros of AiryAiPrime:

Plot the normalizable states:

## 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:

A larger setting for \$MaxExtraPrecision can be needed:

Machine-number inputs can give highprecision results:

## Neat Examples(1)

Nested integrals of the square of AiryBiPrime:

Wolfram Research (1991), AiryBiPrime, Wolfram Language function, https://reference.wolfram.com/language/ref/AiryBiPrime.html (updated 2022).

#### Text

Wolfram Research (1991), AiryBiPrime, Wolfram Language function, https://reference.wolfram.com/language/ref/AiryBiPrime.html (updated 2022).

#### CMS

Wolfram Language. 1991. "AiryBiPrime." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/AiryBiPrime.html.

#### APA

Wolfram Language. (1991). AiryBiPrime. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/AiryBiPrime.html

#### BibTeX

@misc{reference.wolfram_2024_airybiprime, author="Wolfram Research", title="{AiryBiPrime}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/AiryBiPrime.html}", note=[Accessed: 19-September-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_airybiprime, organization={Wolfram Research}, title={AiryBiPrime}, year={2022}, url={https://reference.wolfram.com/language/ref/AiryBiPrime.html}, note=[Accessed: 19-September-2024 ]}