# AiryAiPrime

AiryAiPrime[z]

gives the derivative of the Airy function .

# Details # 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(39)

### 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(2)

Plot the AiryAiPrime function:

Plot the real part of :

Plot the imaginary part of :

### 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(5)

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

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: 