# Documentation

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# `airyBi`

Airy function of the second kind

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## Syntax

```airyBi(`z`)
airyBi(`z`, `n`)
```

## Description

`airyBi(z)` represents the Airy function of the second kind. The Airy functions of the first and second kind are linearly independent solutions of the differential equation .

`airyBi(z, n)` represents the `n`-th derivative of `airyBi(z)` with respect to `z`.

`airyBi(z)` is equivalent to ```airyBi(z, 0)```.

For `n ≥ 2`, derivatives of the Airy functions are automatically expressed in terms of the Airy functions and their first derivative. See Example 1.

`airyBi` returns special values for ```z = 0``` and `z = ±∞`. For all other symbolic values of z, unevaluated function calls are returned. See Example 2.

## Environment Interactions

When called with floating-point arguments, this functions is sensitive to the environment variable `DIGITS` which determines the numerical working precision.

## Examples

### Example 1

Second and higher derivatives of Airy functions are rewritten in terms of Airy functions and their first derivatives:

`airyBi(x), airyBi(x, 1), airyBi(sin(x), 3)`

### Example 2

For `z = 0`, special values are returned:

`airyBi(0), airyBi(0, 1), airyBi(0, 27)`

For `n = 0`, `n = 1` and any symbolic `z ≠ 0`, `z ≠ ±∞`, a symbolic call is returned:

`airyBi(-1), airyBi(x, 1)`

Floating-point values are returned for floating-point arguments:

`airyBi(0.0), airyBi(-3.24819, 1), airyBi(-3.45 + 2.75*I)`

### Example 3

`diff`, `float`, `limit`, `series`, and other functions handle expressions involving the Airy functions:

`diff(airyBi(x^2), x)`

`float(airyBi(PI))`

`limit(airyBi(-x), x = infinity)`

`series(airyBi(x, 1), x = infinity)`

## Parameters

 `z` Arithmetical expression `n` Arithmetical expression representing a nonnegative integer

## Return Values

Arithmetical expression.

`z`