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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

The whittakerM function Ma, b(z) is related to the confluent hypergeometric function by the formula:

,

The WhittakerM 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.

    Note:   MuPAD® defines for all complex numbers a. As a consequence, the MuPAD whittakerM differs from the corresponding function in M. Abramowitz and I. A. Stegun, "Handbook of Mathematical Functions" when and 1 + 2 b are negative integers and . Some of the formulas in Chapter 13 of the "Handbook of Mathematical Functions" do not hold for the MuPAD whittakerM with such arguments. Cf. 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

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.

Overloaded By

z

Algorithms

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

.

See Also

MuPAD Functions

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