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.
AiryBiPrime can be used with Interval and CenteredInterval objects. »

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 (38)

Numerical Evaluation (5)

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:

Compute worst-case guaranteed intervals using Interval and CenteredInterval objects:

Or compute average-case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix AiryBiPrime function using MatrixFunction:

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 :

Plot the imaginary part of :

Function Properties (9)

AiryBiPrime is defined for all real and complex values:

Function range of AiryBiPrime:

AiryBiPrime is an analytic function of x:

AiryBiPrime is neither non-increasing nor non-decreasing:

AiryBiPrime is not injective:

AiryBiPrime is surjective:

AiryBiPrime is neither non-negative nor non-positive:

AiryBiPrime has no singularities or discontinuities:

AiryBiPrime is neither convex nor concave:

Differentiation (3)

First derivative:

Higher derivatives:

Formula for the 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 (3)

Solve differential equations in terms of AiryBiPrime:

Solution of the modified linearized Korteweg–deVries equation for any function :

Verify the solution:

Solution of the time‐independent Schrödinger equation in a linear cone potential, represented with AiryAiPrime and AiryBiPrime:

The normalizable states are determined through the zeros of AiryAiPrime:

Plot the normalizable states:

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 high‐precision results:

Neat Examples (1)

Nested integrals of the square of AiryBiPrime:

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