# Documentation

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# `whittakerW`

The Whittaker W function

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## Syntax

```whittakerW(`a`, `b`, `z`)
```

## Description

`whittakerW` returns the Whittaker W function ${W}_{a,b}\left(z\right)$.

The Whittaker functions ${M}_{a,b}\left(z\right)$ and ${W}_{a,b}\left(z\right)$ are linearly independent solutions of the following differential equation:

`$\frac{{d}^{2}w}{d{z}^{2}}+\left(-\frac{1}{4}+\frac{a}{z}+\frac{\frac{1}{4}-{b}^{2}}{{z}^{2}}\right)w=0$`

The Whittaker W function is defined via the confluent hypergeometric Kummer U function $U\left(a,\text{\hspace{0.17em}}b,\text{\hspace{0.17em}}z\right)$ as follows:

`${W}_{a,b}\left(z\right)={e}^{-z/2}\text{\hspace{0.17em}}{z}^{b+1/2}\text{\hspace{0.17em}}U\left(b-a+\frac{1}{2},\text{\hspace{0.17em}}1+2b,\text{\hspace{0.17em}}z\right)$`

The WhittakerW function is defined for complex arguments `a`, `b`, and `z`.

For most of the values of the parameters, an unevaluated function call is returned. See Example 1.

Explicit symbolic expressions are returned for some particular values of the parameters. See Example 2.

## Environment Interactions

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

## Examples

### Example 1

For exact or symbolic arguments, `whittakerW` returns unevaluated calls:

```whittakerW(a, b, x); whittakerW(-3/2, 1/2, 1)```

For floating-point arguments, `whittakerW` returns floating-point results:

```whittakerW(2, 0.5, -5), whittakerW(-3/2, 1/2, 0.0)```

### Example 2

For some specific values of the parameters, `whittakerW` returns explicit expressions:

```whittakerW(0, b, x); whittakerW(-3/2, 1/2, 0); whittakerW(-3/2, 0 ,x); whittakerW(a, -a + 1/2, x)```

### Example 3

`diff`, `float`, `limit`, `series`, and other functions handle expressions involving the Whittaker W function:

`diff(whittakerW(a, b, z), z)`

`float(whittakerW(-3/2, 1/2, 0))`

`series(whittakerW(-3/2, 1/2, x), x, 2)`

## Parameters

 `a`, `b`, `z` Arithmetical expressions

## Return Values

Arithmetical expression.

`z`