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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 Wa,b(z).

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

d2wdz2+(14+az+14b2z2)w=0

The Whittaker W function is defined via the confluent hypergeometric Kummer U function U(a,b,z) as follows:

Wa,b(z)=ez/2zb+1/2U(ba+12,1+2b,z)

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.

Overloaded By

z

See Also

MuPAD Functions

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