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AiryAiZero[k]

represents the k^(th) zero of the Airy function TemplateBox[{x}, AiryAi].

AiryAiZero[k,x0]

represents the k^(th) zero less than x0.

Details
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Numerical Evaluation  
Specific Values  
Visualization  
Series Expansion  
Applications  
See Also
Tech Notes
Related Guides
History
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AiryAiZero

AiryAiZero[k]

represents the k^(th) zero of the Airy function TemplateBox[{x}, AiryAi].

AiryAiZero[k,x0]

represents the k^(th) zero less than x0.

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • N[AiryAiZero[k]] gives a numerical approximation so long as the specified zero exists.
  • AiryAiZero[k] represents the k^(th) zero less than 0.
  • AiryAiZero can be evaluated to arbitrary numerical precision.
  • AiryAiZero automatically threads over lists. »

Examples

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Basic Examples  (4)

Evaluate numerically:

Wolfram Language code: N[AiryAiZero[1]]

AiryAiZero gives the zeros of AiryAi:

Wolfram Language code: AiryAi[AiryAiZero[1]]

Display zeros of the AiryAi function over a subset of the reals:

Wolfram Language code: Plot[AiryAi[z], {z, -8, 5}, Epilog -> {PointSize[0.03], Red, Point[Table[{AiryAiZero[k], 0}, {k, 5}]]}]

Series expansion at Infinity:

Wolfram Language code: Series[AiryAiZero[ x], {x, ∞, 2}]

Scope  (12)

Numerical Evaluation  (5)

Find the third zero of TemplateBox[{x}, AiryAi]:

Wolfram Language code: N[AiryAiZero[3]]

Find the second zero of TemplateBox[{x}, AiryAi] less than :

Wolfram Language code: N[AiryAiZero[2, -500]]

Evaluate numerically to high precision:

Wolfram Language code: N[AiryAiZero[1], 25]

Evaluate efficiently at high precision:

Wolfram Language code: AiryAiZero[5.2`10000];//Timing

Compute the elementwise values of an array using automatic threading:

Wolfram Language code: AiryAiZero[2, {{2, 1}, {3, 2}}]//N

Or compute the matrix AiryAiZero function using MatrixFunction:

Wolfram Language code: MatrixFunction[AiryAiZero[2, #]&, {{2, 1}, {3, 2}}]//FullSimplify

Compute average-case statistical intervals using Around:

Wolfram Language code: AiryAiZero[ 1, Around[2, 0.01]]

Specific Values  (4)

Limiting value at infinity:

Wolfram Language code: Limit[AiryAiZero[x], x -> Infinity]

The first three zeros:

Wolfram Language code: {AiryAiZero[1], AiryAiZero[2], AiryAiZero[3]}//N

Find the second zero of AiryAi using Solve:

Wolfram Language code: xzero = x /. Solve[AiryAi[x] == 0 && -6 < x < -4, x][[1]]
Wolfram Language code: Plot[AiryAi[x], {x, -7, 3}, Epilog -> Style[Point[{xzero, AiryAi[xzero]}], PointSize[Large], Red]]

AiryAiZero threads elementwise over lists:

Wolfram Language code: AiryAiZero[{1, 2, 3}]//N

Visualization  (2)

Display zeros of AiryAi function:

Wolfram Language code: Plot[AiryAi[z], {z, -10, 5}, Epilog -> {PointSize[0.03], Point[Table[{AiryAiZero[k], 0}, {k, 7}]]}]

Show the second zero less than :

Wolfram Language code: Plot[AiryAi[z], {z, -7, 4}, Epilog -> {PointSize[0.03], Red, Point[{AiryAiZero[2, -4], 0}]}]

Series Expansion  (1)

Asymptotic behavior of AiryAiZero[k] for large k:

Wolfram Language code: Series[AiryAiZero[k], {k, ∞, 5}]
Wolfram Language code: Series[AiryAiZero[k], {k, ∞, 2}]

Applications  (1)

Display zeros on the plot:

Wolfram Language code: Plot[AiryAi[z], {z, -20, 5}, Prolog -> {PointSize[0.03], Point[Table[{AiryAiZero[k], 0}, {k, 19}]]}]
Wolfram Research (2007), AiryAiZero, Wolfram Language function, https://reference.wolfram.com/language/ref/AiryAiZero.html.

Text

Wolfram Research (2007), AiryAiZero, Wolfram Language function, https://reference.wolfram.com/language/ref/AiryAiZero.html.

CMS

Wolfram Language. 2007. "AiryAiZero." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/AiryAiZero.html.

APA

Wolfram Language. (2007). AiryAiZero. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/AiryAiZero.html

BibTeX

@misc{reference.wolfram_2026_airyaizero, author="Wolfram Research", title="{AiryAiZero}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/AiryAiZero.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_airyaizero, organization={Wolfram Research}, title={AiryAiZero}, year={2007}, url={https://reference.wolfram.com/language/ref/AiryAiZero.html}, note=[Accessed: 12-June-2026]}