Built-in Mathematica Symbol

AiryAi

AiryAi[z]
gives the Airy function

.

MORE INFORMATION

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.

EXAMPLES

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Basic Examples (3)

Evaluate numerically:

Scope (6)

Evaluate for complex arguments:

Evaluate to high precision:

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

AiryAi threads element-wise over lists:

Simple exact values are generated automatically:

Find series expansions at infinity:

TraditionalForm formatting:

Generalizations & Extensions (2)

AiryAi can be applied to power series:

Find series expansions at infinity for an arbitrary symbolic direction

:

Applications (4)

Solve the Schrödinger equation in a linear potential (e.g. uniform electric field):

Plot the absolute value in the complex plane:

Plot the imaginary part in the complex plane:

Nested integrals of the square of AiryAi:

Properties & Relations (6)

Use FullSimplify to simplify expressions involving Airy functions:

Compare with the output of Wronskian:

FunctionExpand tries to simplify the argument of AiryAi:

Solve the Airy differential equation:

Find a numerical root:

Compare with built-in function AiryAiZero:

Integral:

Verify the anti-derivative:

Integral transforms:

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: