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The Whittaker M function

MuPAD® notebooks are not recommended. Use MATLAB® live scripts instead.

MATLAB live scripts support most MuPAD functionality, though there are some differences. For more information, see Convert MuPAD Notebooks to MATLAB Live Scripts.


whittakerM(a, b, z)


whittakerM returns the Whittaker M function, Ma,b(z).

The Whittaker functions Ma,b(z) and Wa,b(z) are linearly independent solutions of the following differential equation:


The Whittaker M function is defined via the confluent hypergeometric function pFq(a,b,z)=Φ(a,b,z) as follows:


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.


MuPAD® defines F11(a,a,z)=ex 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 ba+12 and 1+2b are negative integers and ba+121+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.


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


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))


a, b, z

Arithmetical expressions

Return Values

Arithmetical expression.

Overloaded By


See Also

MuPAD Functions

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