# AiryAiPrime

AiryAiPrime[z]

gives the derivative of the Airy function .

# Examples

open allclose all

## 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(40)

### Numerical Evaluation(5)

Evaluate numerically to high precision:

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

Evaluate for complex arguments:

Evaluate AiryAiPrime efficiently at high precision:

AiryAiPrime threads elementwise over lists and matrices:

AiryAiPrime can be used with Interval and CenteredInterval objects:

### Specific Values(3)

Simple exact values are generated automatically:

Limiting value at infinity:

Find a zero of AiryAiPrime using Solve:

### Visualization(3)

Plot the AiryAiPrime function:

Plot the real part of :

Plot the imaginary part of :

A plot of the absolute value of AiryAiPrime over the complex plane:

### Function Properties(9)

AiryAiPrime is defined for all real and complex values:

Function range of AiryAiPrime:

AiryAiPrime is an analytic function of x:

AiryAiPrime is neither non-increasing nor non-decreasing:

AiryAiPrime is not injective:

AiryAiPrime is surjective:

AiryAiPrime is neither non-negative nor non-positive:

AiryAiPrime has no singularities or discontinuities:

AiryAiPrime is neither convex nor concave:

### Differentiation(3)

First derivative:

Higher derivatives:

Formula for the derivative:

### Integration(3)

Integral of AiryAiPrime gives back AiryAi:

Definite integral of AiryAiPrime:

More integrals:

### Series Expansions(4)

Taylor expansion for AiryAiPrime:

Plot the first three approximations for AiryAiPrime around :

General term in the series expansion of AiryAiPrime:

Find the series expansion at infinity:

The behavior at negative infinity is quite different:

AiryAiPrime can be applied to power series:

### Integral Transforms(3)

Compute the Fourier transform using FourierTransform:

### Function Identities and Simplifications(3)

Functional identity:

Simplify the expression to AiryAiPrime:

FunctionExpand tries to simplify the argument of AiryAiPrime:

### Function Representations(4)

Relationship to Bessel functions:

AiryAiPrime can be represented as a DifferentialRoot:

AiryAiPrime can be represented in terms of MeijerG:

## Applications(4)

Solve differential equations in terms of AiryAiPrime:

Solution of the timeindependent Schrödinger equation in a linear cone potential:

The normalizable states are determined through the zeros of AiryAiPrime:

Plot the normalizable states:

An integral kernel related to the Gaussian unitary ensembles:

A convolution integral solving the modified linearized KortewegdeVries equation for any function :

Verify solution:

## 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 AiryAiPrime:

Airy functions are generated as solutions by DSolve:

Obtain AiryAiPrime from sums:

AiryAiPrime 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 AiryAiPrime:

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

#### Text

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

#### CMS

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

#### APA

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

#### BibTeX

@misc{reference.wolfram_2024_airyaiprime, author="Wolfram Research", title="{AiryAiPrime}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/AiryAiPrime.html}", note=[Accessed: 20-July-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_airyaiprime, organization={Wolfram Research}, title={AiryAiPrime}, year={2022}, url={https://reference.wolfram.com/language/ref/AiryAiPrime.html}, note=[Accessed: 20-July-2024 ]}