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 TemplateBox[{{x, +, {ⅈ,  , y}}}, AiryAiPrime]:

Plot the imaginary part of TemplateBox[{{x, +, {ⅈ,  , y}}}, AiryAiPrime]:

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 ^(th) 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 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:

Use arbitrary-precision evaluation instead:

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:

Introduced in 1991
 (2.0)