gives the Airy function .
- Mathematical function, suitable for both symbolic and numerical manipulation.
- The Airy function is a solution to the differential equation .
- tends to zero as .
- AiryAi[z] is an entire function of z with no branch cut discontinuities.
- For certain special arguments, AiryAi automatically evaluates to exact values.
- AiryAi can be evaluated to arbitrary numerical precision.
- AiryAi automatically threads over lists.
Examplesopen allclose all
Basic Examples (5)
Plot over a subset of the reals:
Plot over a subset of the complexes:
Series expansion at the origin:
Series expansion at Infinity:
Numerical Evaluation (4)
Evaluate numerically to high precision:
The precision of the output tracks the precision of the input:
Evaluate for complex arguments:
Evaluate AiryAi efficiently at high precision:
AiryAi threads elementwise over lists and matrices:
Specific Values (4)
Simple exact values are generated automatically:
Limiting values at infinity:
The first three zeros:
Find a zero of AiryAi using Solve:
Plot the AiryAi function:
Plot the real part of :
Plot the imaginary part of :
Function Properties (2)
AiryAi is defined for all real and complex values:
Approximate function range of AiryAi:
Formula for the derivative:
Indefinite integral of AiryAi:
Verify the anti-derivative:
Definite integral of AiryAi:
Series Expansions (5)
Taylor expansion for AiryAi:
Plot the first three approximations for AiryAi around :
General term in the series expansion of AiryAi:
Find the series expansion at infinity:
Find the series expansion at infinity for an arbitrary symbolic direction :
AiryAi can be applied to power series:
Function Identities and Simplifications (3)
Function Representations (5)
Solve the Schrödinger equation in a linear potential (e.g. uniform electric field):
Plot the absolute value in the complex plane:
Nested integrals of the square of AiryAi:
Properties & Relations (8)
Possible Issues (5)
Machine-precision input is insufficient to get a correct answer:
Use arbitrary-precision evaluation instead:
A larger setting for $MaxExtraPrecision can be needed:
Machine-number inputs can give high‐precision results:
Simplifications sometimes hold only in parts of the complex plane:
Parentheses are required when inputting in the traditional form: