AiryBiPrime
✖
AiryBiPrime
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
open allclose allBasic Examples (5)Summary of the most common use cases

https://wolfram.com/xid/01y6njlcsy8-z756a

Plot over a subset of the reals:

https://wolfram.com/xid/01y6njlcsy8-4nxyp

Plot over a subset of the complexes:

https://wolfram.com/xid/01y6njlcsy8-kiedlx

Series expansion at the origin:

https://wolfram.com/xid/01y6njlcsy8-hqxev

Series expansion at Infinity:

https://wolfram.com/xid/01y6njlcsy8-rmazn

Scope (38)Survey of the scope of standard use cases
Numerical Evaluation (5)
Evaluate numerically to high precision:

https://wolfram.com/xid/01y6njlcsy8-hj17gk

The precision of the output tracks the precision of the input:

https://wolfram.com/xid/01y6njlcsy8-id8vzn

Evaluate for complex arguments:

https://wolfram.com/xid/01y6njlcsy8-c2s97x

Evaluate AiryBiPrime efficiently at high precision:

https://wolfram.com/xid/01y6njlcsy8-di5gcr


https://wolfram.com/xid/01y6njlcsy8-bq2c6r

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

https://wolfram.com/xid/01y6njlcsy8-gnclzo


https://wolfram.com/xid/01y6njlcsy8-czjvbq

Or compute average-case statistical intervals using Around:

https://wolfram.com/xid/01y6njlcsy8-cw18bq

Compute the elementwise values of an array:

https://wolfram.com/xid/01y6njlcsy8-thgd2

Or compute the matrix AiryBiPrime function using MatrixFunction:

https://wolfram.com/xid/01y6njlcsy8-o5jpo

Specific Values (3)
Simple exact values are generated automatically:

https://wolfram.com/xid/01y6njlcsy8-fbq8zf


https://wolfram.com/xid/01y6njlcsy8-ciezym

Find a zero of AiryBiPrime using Solve:

https://wolfram.com/xid/01y6njlcsy8-f2hrld


https://wolfram.com/xid/01y6njlcsy8-jx0ev5

Visualization (2)
Plot the AiryBiPrime function:

https://wolfram.com/xid/01y6njlcsy8-ecj8m7


https://wolfram.com/xid/01y6njlcsy8-ouu484


https://wolfram.com/xid/01y6njlcsy8-e4mwbj

Function Properties (9)
AiryBiPrime is defined for all real and complex values:

https://wolfram.com/xid/01y6njlcsy8-cl7ele


https://wolfram.com/xid/01y6njlcsy8-de3irc

Function range of AiryBiPrime:

https://wolfram.com/xid/01y6njlcsy8-evf2yr

AiryBiPrime is an analytic function of x:

https://wolfram.com/xid/01y6njlcsy8-h5x4l2

AiryBiPrime is neither non-increasing nor non-decreasing:

https://wolfram.com/xid/01y6njlcsy8-g6kynf

AiryBiPrime is not injective:

https://wolfram.com/xid/01y6njlcsy8-gi38d7


https://wolfram.com/xid/01y6njlcsy8-ctca0g

AiryBiPrime is surjective:

https://wolfram.com/xid/01y6njlcsy8-hkqec4


https://wolfram.com/xid/01y6njlcsy8-hdm869

AiryBiPrime is neither non-negative nor non-positive:

https://wolfram.com/xid/01y6njlcsy8-84dui

AiryBiPrime has no singularities or discontinuities:

https://wolfram.com/xid/01y6njlcsy8-mdtl3h


https://wolfram.com/xid/01y6njlcsy8-mn5jws

AiryBiPrime is neither convex nor concave:

https://wolfram.com/xid/01y6njlcsy8-kdss3

Differentiation (3)

https://wolfram.com/xid/01y6njlcsy8-mmas49


https://wolfram.com/xid/01y6njlcsy8-nfbe0l


https://wolfram.com/xid/01y6njlcsy8-fxwmfc


https://wolfram.com/xid/01y6njlcsy8-odmgl1

Integration (3)
Indefinite integral of AiryBiPrime gives back AiryBi:

https://wolfram.com/xid/01y6njlcsy8-bponid

Definite integral of AiryBiPrime:

https://wolfram.com/xid/01y6njlcsy8-jskynr


https://wolfram.com/xid/01y6njlcsy8-s9hg4


https://wolfram.com/xid/01y6njlcsy8-gmwef2

Series Expansions (4)
Taylor expansion for AiryBiPrime:

https://wolfram.com/xid/01y6njlcsy8-ewr1h8

Plot the first three approximations for AiryBiPrime around :

https://wolfram.com/xid/01y6njlcsy8-binhar

General term in the series expansion of AiryBiPrime:

https://wolfram.com/xid/01y6njlcsy8-dznx2j

Find the series expansion at infinity:

https://wolfram.com/xid/01y6njlcsy8-jcanng

The behavior at negative infinity is quite different:

https://wolfram.com/xid/01y6njlcsy8-ec62o0

AiryBiPrime can be applied to power series:

https://wolfram.com/xid/01y6njlcsy8-eqj3mh

Integral Transforms (2)
Compute the Fourier cosine transform using FourierCosTransform:

https://wolfram.com/xid/01y6njlcsy8-bdev9a


https://wolfram.com/xid/01y6njlcsy8-fj60io

Function Identities and Simplifications (3)

https://wolfram.com/xid/01y6njlcsy8-bbvqog

Simplify the expression to AiryBiPrime:

https://wolfram.com/xid/01y6njlcsy8-bh9t4q

FunctionExpand tries to simplify the argument of AiryBiPrime:

https://wolfram.com/xid/01y6njlcsy8-neugr


https://wolfram.com/xid/01y6njlcsy8-e6ayls

Function Representations (4)
Relationship to Bessel functions:

https://wolfram.com/xid/01y6njlcsy8-c333o7

AiryBiPrime can be represented as a DifferentialRoot:

https://wolfram.com/xid/01y6njlcsy8-heb2f7

Represent in terms of MeijerG using MeijerGReduce:

https://wolfram.com/xid/01y6njlcsy8-bc9t2n


https://wolfram.com/xid/01y6njlcsy8-rtm7m

TraditionalForm formatting:

https://wolfram.com/xid/01y6njlcsy8-c22ecr

Applications (3)Sample problems that can be solved with this function
Solve differential equations in terms of AiryBiPrime:

https://wolfram.com/xid/01y6njlcsy8-f730b5

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

https://wolfram.com/xid/01y6njlcsy8-mhdy00

https://wolfram.com/xid/01y6njlcsy8-f4zwy6

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

https://wolfram.com/xid/01y6njlcsy8-dpzcv3

The normalizable states are determined through the zeros of AiryAiPrime:

https://wolfram.com/xid/01y6njlcsy8-dc0a99

https://wolfram.com/xid/01y6njlcsy8-d3h04o

https://wolfram.com/xid/01y6njlcsy8-k1ldz2

Properties & Relations (5)Properties of the function, and connections to other functions
Use FullSimplify to simplify Airy functions, here in the Wronskian of the Airy equation:

https://wolfram.com/xid/01y6njlcsy8-mwg5g


https://wolfram.com/xid/01y6njlcsy8-0o90z

Compare with the output of Wronskian:

https://wolfram.com/xid/01y6njlcsy8-ht9xpg

Generate Airy functions from differential equations:

https://wolfram.com/xid/01y6njlcsy8-r550r


https://wolfram.com/xid/01y6njlcsy8-7l451

Obtain AiryBiPrime from sums:

https://wolfram.com/xid/01y6njlcsy8-j6r4gb

AiryBiPrime appears in special cases of several mathematical functions:

https://wolfram.com/xid/01y6njlcsy8-cj6ex7

Possible Issues (3)Common pitfalls and unexpected behavior
Machine-precision input is insufficient to give a correct answer:

https://wolfram.com/xid/01y6njlcsy8-g2sowt

Use arbitrary-precision evaluation instead:

https://wolfram.com/xid/01y6njlcsy8-h84pak

A larger setting for $MaxExtraPrecision can be needed:

https://wolfram.com/xid/01y6njlcsy8-en1hnr



https://wolfram.com/xid/01y6njlcsy8-bsj9sk

Machine-number inputs can give high‐precision results:

https://wolfram.com/xid/01y6njlcsy8-fvmejx


https://wolfram.com/xid/01y6njlcsy8-iifp35

Neat Examples (1)Surprising or curious use cases
Nested integrals of the square of AiryBiPrime:

https://wolfram.com/xid/01y6njlcsy8-b92m1m

Wolfram Research (1991), AiryBiPrime, Wolfram Language function, https://reference.wolfram.com/language/ref/AiryBiPrime.html (updated 2022).
Text
Wolfram Research (1991), AiryBiPrime, Wolfram Language function, https://reference.wolfram.com/language/ref/AiryBiPrime.html (updated 2022).
Wolfram Research (1991), AiryBiPrime, Wolfram Language function, https://reference.wolfram.com/language/ref/AiryBiPrime.html (updated 2022).
CMS
Wolfram Language. 1991. "AiryBiPrime." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/AiryBiPrime.html.
Wolfram Language. 1991. "AiryBiPrime." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/AiryBiPrime.html.
APA
Wolfram Language. (1991). AiryBiPrime. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/AiryBiPrime.html
Wolfram Language. (1991). AiryBiPrime. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/AiryBiPrime.html
BibTeX
@misc{reference.wolfram_2025_airybiprime, author="Wolfram Research", title="{AiryBiPrime}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/AiryBiPrime.html}", note=[Accessed: 29-March-2025
]}
BibLaTeX
@online{reference.wolfram_2025_airybiprime, organization={Wolfram Research}, title={AiryBiPrime}, year={2022}, url={https://reference.wolfram.com/language/ref/AiryBiPrime.html}, note=[Accessed: 29-March-2025
]}