AiryAiPrime

AiryAiPrime[z]

gives the derivative of the Airy function .

Details

Mathematical function, suitable for both symbolic and numerical manipulation.
For certain special arguments, AiryAiPrime automatically evaluates to exact values.
AiryAiPrime can be evaluated to arbitrary numerical precision.
AiryAiPrime 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 (31)

Numerical Evaluation (4)

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:

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 (2)

AiryAiPrime is defined for all real and complex values:

Function range of AiryAiPrime:

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:

MellinTransform:

HankelTransform:

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:

TraditionalForm formatting:

Applications (5)

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

Solve differential equations in terms of AiryAiPrime:

Solution of the time‐independent 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 Korteweg–deVries 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:

Use arbitrary-precision evaluation instead:

A larger setting for $MaxExtraPrecision can be needed:

Machine-number inputs can give high‐precision results:

Neat Examples (1)

Nested integrals of the square of AiryAiPrime:

Introduced in 1991

(2.0)

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