AiryBiPrime

AiryBiPrime[z]

gives the derivative of the Airy function .

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • For certain special arguments, AiryBiPrime automatically evaluates to exact values.
  • AiryBiPrime can be evaluated to arbitrary numerical precision.
  • AiryBiPrime 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  (30)

Numerical Evaluation  (4)

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:

AiryBiPrime threads elementwise over lists:

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

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

Function Properties  (2)

AiryBiPrime is defined for all real and complex values:

Function range of AiryBiPrime:

Differentiation  (3)

First derivative:

Higher derivatives:

Formula for the ^(th) 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:

HankelTransform:

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 MeijerGReduce:

TraditionalForm formatting:

Applications  (2)

Solve differential equations in terms of AiryBiPrime:

Solution of the modified linearized KortewegdeVries equation for any function :

Verify the solution:

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:

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 AiryBiPrime:

Introduced in 1991
 (2.0)