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# whittakerW

The Whittaker W function

### Use only in the MuPAD Notebook Interface.

This functionality does not run in MATLAB.

## Syntax

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

## Description

The whittakerW function Wa, b(z) is related to the confluent hypergeometric function kummerU(a, b, z) = U(a, b, z) by the formula:

.

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. Cf. Example 1.

Explicit symbolic expressions are returned for some particular values of the parameters. Cf. 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

Unevaluated calls are returned for exact or symbolic arguments:

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

Floating point values are returned for floating-point arguments:

`whittakerM(-2, 0.5, -50), whittakerW(-3/2, 1/2, 0.0)`

### Example 2

Explicit expressions are returned for some specific values of the parameters:

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

### Example 3

The functions diff, float, limit, and series handle expressions involving the Whittaker functions

`diff(whittakerM(a,b,z),z), float(whittakerW(-3/2,1/2,0))`

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

### Example 4

For some values of the input parameters, recurrence and differential relations in Chapter 13 of M. Abramowitz and I. A. Stegun, "Handbook of Mathematical Functions" may not hold for the MuPAD® whittakerM functions. For example, Formula 13.4.32

is not satisfied for a = 0 and :

```expand(x*diff(whittakerM(0, -3/2, x), x) <>
x/2*whittakerM(0, -3/2, x) - whittakerM(1, -3/2, x))```

## Parameters

 a, b, z arithmetical expressions

## Return Values

Arithmetical expression.

z

## Algorithms

Ma, b(z) and Wa, b(z) satisfy Whittaker's differential equation:

.