# whittakerM

The Whittaker M function

### Use only in the MuPAD Notebook Interface.

This functionality does not run in MATLAB.

## Syntax

```whittakerM(`a`, `b`, `z`)
```

## Description

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

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 M function is defined via the confluent hypergeometric function ${}_{p}{F}_{q}\left(a,\text{\hspace{0.17em}}b,\text{\hspace{0.17em}}z\right)=\Phi \left(a,\text{\hspace{0.17em}}b,\text{\hspace{0.17em}}z\right)$ as follows:

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

The Whittaker M 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.

 Note:   MuPAD® defines ${}_{1}F{}_{1}\left(a,\text{\hspace{0.17em}}a,\text{\hspace{0.17em}}z\right)={e}^{x}$ for all complex numbers $a$. As a consequence, the MuPAD `whittakerM` function differs from the corresponding function in M. Abramowitz and I. A. Stegun, "Handbook of Mathematical Functions" when $b-a+\frac{1}{2}$ and $1+2b$ are negative integers and $b-a+\frac{1}{2}\ge 1+2b$. Some of the formulas in Chapter 13 of the "Handbook of Mathematical Functions" do not hold for the MuPAD `whittakerM` with such arguments. See Example 4.

## 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, `whittakerM` returns unevaluated calls:

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

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

```whittakerM(-2, 0.5, -50), whittakerM(-3/2, 1/2, 1.0)```

### Example 2

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

```whittakerM(0, b, x); whittakerM(-3/2, 1/2, 0); whittakerM(-3/2, 0, x)```

### Example 3

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

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

`float(whittakerM(-3/2, 1/2, 1))`

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

### 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" do not hold for the MuPAD `whittakerM` functions. For example, Formula 13.4.32

$z\frac{\partial }{\partial z}{M}_{a,b}\left(z\right)=\left(\frac{z}{2}-a\right){M}_{a,b}\left(z\right)+\left(a+b+\frac{1}{2}\right){M}_{a+1,\text{\hspace{0.17em}}b}\left(z\right)$

is not satisfied for `a = 0` and ```b = -3/2```:

```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`

## See Also

### MuPAD Functions

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