gives the Airy function .
- Mathematical function, suitable for both symbolic and numerical manipulation.
- The Airy function is a solution to the differential equation .
- increases exponentially as .
- AiryBi[z] is an entire function of z with no branch cut discontinuities.
- For certain special arguments, AiryBi automatically evaluates to exact values.
- AiryBi can be evaluated to arbitrary numerical precision.
- AiryBi 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 AiryBi efficiently at high precision:
AiryBi 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 AiryBi using Solve:
Plot the AiryBi function:
Plot the real part of :
Plot the imaginary part of :
Function Properties (2)
AiryBi is defined for all real and complex values:
Approximate function range of AiryBi:
Formula for the derivative:
Indefinite integral of AiryBi:
Definite integral of AiryBi:
Series Expansions (5)
Taylor expansion for AiryBi:
Plot the first three approximations for AiryBi around :
General term in the series expansion of AiryBi:
Find series expansions at infinity:
Expansion at infinity for an arbitrary symbolic direction :
AiryBi can be applied to power series:
Function Identities and Simplifications (3)
Function Representations (4)
Solve the Schrödinger equation in a linear potential (e.g. uniform electric field):
Check the Sommerfeld radiation condition for a combination of Airy functions:
There is only an outgoing plane wave:
Properties & Relations (5)
Use FullSimplify to simplify Airy functions, here in the Wronskian of the Airy equation:
Compare with the output of Wronskian:
FunctionExpand tries to simplify the argument of AiryBi:
Generate Airy functions from differential equations:
Find a numerical root:
Compare with the built-in function AiryBiZero:
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 for correct parsing in the traditional form: