WOLFRAM

gives the derivative of the Airy function .

Details

Examples

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Basic Examples  (5)Summary of the most common use cases

Evaluate numerically:

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Plot over a subset of the reals:

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Plot over a subset of the complexes:

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Series expansion at the origin:

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Series expansion at Infinity:

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Scope  (40)Survey of the scope of standard use cases

Numerical Evaluation  (5)

Evaluate numerically to high precision:

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

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Evaluate for complex arguments:

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Evaluate AiryAiPrime efficiently at high precision:

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

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Or compute average-case statistical intervals using Around:

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Compute the elementwise values of an array:

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Or compute the matrix AiryBiPrime function using MatrixFunction:

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Specific Values  (3)

Simple exact values are generated automatically:

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Limiting value at infinity:

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Find a zero of AiryAiPrime using Solve:

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Visualization  (3)

Plot the AiryAiPrime function:

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Plot the real part of TemplateBox[{z}, AiryAiPrime]:

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Plot the imaginary part of TemplateBox[{z}, AiryAiPrime]:

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

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Function Properties  (9)

AiryAiPrime is defined for all real and complex values:

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Function range of AiryAiPrime:

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AiryAiPrime is an analytic function of x:

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AiryAiPrime is neither non-increasing nor non-decreasing:

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AiryAiPrime is not injective:

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AiryAiPrime is surjective:

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AiryAiPrime is neither non-negative nor non-positive:

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AiryAiPrime has no singularities or discontinuities:

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AiryAiPrime is neither convex nor concave:

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Differentiation  (3)

First derivative:

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Higher derivatives:

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Formula for the ^(th) derivative:

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Integration  (3)

Integral of AiryAiPrime gives back AiryAi:

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Definite integral of AiryAiPrime:

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More integrals:

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Series Expansions  (4)

Taylor expansion for AiryAiPrime:

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Plot the first three approximations for AiryAiPrime around :

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General term in the series expansion of AiryAiPrime:

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Find the series expansion at infinity:

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The behavior at negative infinity is quite different:

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AiryAiPrime can be applied to power series:

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Integral Transforms  (3)

Compute the Fourier transform using FourierTransform:

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MellinTransform:

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HankelTransform:

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Function Identities and Simplifications  (3)

Functional identity:

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Simplify the expression to AiryAiPrime:

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FunctionExpand tries to simplify the argument of AiryAiPrime:

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Function Representations  (4)

Relationship to Bessel functions:

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AiryAiPrime can be represented as a DifferentialRoot:

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AiryAiPrime can be represented in terms of MeijerG:

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TraditionalForm formatting:

Applications  (4)Sample problems that can be solved with this function

Solve differential equations in terms of AiryAiPrime:

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Solution of the timeindependent Schrödinger equation in a linear cone potential:

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The normalizable states are determined through the zeros of AiryAiPrime:

Plot the normalizable states:

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An integral kernel related to the Gaussian unitary ensembles:

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A convolution integral solving the modified linearized KortewegdeVries equation for any function :

Verify solution:

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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:

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Compare with the output of Wronskian:

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FunctionExpand tries to simplify the argument of AiryAiPrime:

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Airy functions are generated as solutions by DSolve:

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Obtain AiryAiPrime from sums:

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AiryAiPrime appears in special cases of several mathematical functions:

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Possible Issues  (3)Common pitfalls and unexpected behavior

Machine-precision input is insufficient to give a correct answer:

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Use arbitrary-precision evaluation instead:

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A larger setting for $MaxExtraPrecision can be needed:

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Machine-number inputs can give highprecision results:

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Neat Examples  (1)Surprising or curious use cases

Nested integrals of the square of AiryAiPrime:

Wolfram Research (1991), AiryAiPrime, Wolfram Language function, https://reference.wolfram.com/language/ref/AiryAiPrime.html (updated 2022).
Wolfram Research (1991), AiryAiPrime, Wolfram Language function, https://reference.wolfram.com/language/ref/AiryAiPrime.html (updated 2022).

Text

Wolfram Research (1991), AiryAiPrime, Wolfram Language function, https://reference.wolfram.com/language/ref/AiryAiPrime.html (updated 2022).

Wolfram Research (1991), AiryAiPrime, Wolfram Language function, https://reference.wolfram.com/language/ref/AiryAiPrime.html (updated 2022).

CMS

Wolfram Language. 1991. "AiryAiPrime." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/AiryAiPrime.html.

Wolfram Language. 1991. "AiryAiPrime." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/AiryAiPrime.html.

APA

Wolfram Language. (1991). AiryAiPrime. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/AiryAiPrime.html

Wolfram Language. (1991). AiryAiPrime. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/AiryAiPrime.html

BibTeX

@misc{reference.wolfram_2025_airyaiprime, author="Wolfram Research", title="{AiryAiPrime}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/AiryAiPrime.html}", note=[Accessed: 10-July-2025 ]}

@misc{reference.wolfram_2025_airyaiprime, author="Wolfram Research", title="{AiryAiPrime}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/AiryAiPrime.html}", note=[Accessed: 10-July-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_airyaiprime, organization={Wolfram Research}, title={AiryAiPrime}, year={2022}, url={https://reference.wolfram.com/language/ref/AiryAiPrime.html}, note=[Accessed: 10-July-2025 ]}

@online{reference.wolfram_2025_airyaiprime, organization={Wolfram Research}, title={AiryAiPrime}, year={2022}, url={https://reference.wolfram.com/language/ref/AiryAiPrime.html}, note=[Accessed: 10-July-2025 ]}