# WhittakerW

WhittakerW[k,m,z]

gives the Whittaker function .

# Details • Mathematical function, suitable for both symbolic and numerical manipulation.
• WhittakerW is related to the confluent hypergeometric function by .
• is infinite at for integer .
• For certain special arguments, WhittakerW automatically evaluates to exact values.
• WhittakerW can be evaluated to arbitrary numerical precision.
• WhittakerW automatically threads over lists.
• WhittakerW[k,m,z] has a branch cut discontinuity in the complex plane running from to .
• WhittakerW can be used with Interval and CenteredInterval objects. »

# Examples

open allclose all

## Basic Examples(6)

Evaluate numerically:

Use FunctionExpand to expand in terms of hypergeometric functions:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Series expansion at Infinity:

## Scope(35)

### Numerical Evaluation(5)

Evaluate numerically:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Complex number input:

Evaluate efficiently at high precision:

WhittakerW can be used with Interval and CenteredInterval objects:

### Specific Values(7)

WhittakerW for symbolic parameters:

Values at zero:

Evaluate symbolically at the origin:

Find the first positive maximum of WhittakerW[3,1/2,x]:

Compute the associated WhittakerW[3,1/2,x] function:

Compute the associated WhittakerW function for half-integer parameters:

Different WhittakerW types give different symbolic forms:

### Visualization(4)

Plot the WhittakerW function for various orders:

Plot the real part of :

Plot the imaginary part of :

Plot as real parts of two parameters vary:

Types 2 and 3 of WhittakerW function have different branch cut structures:

### Function Properties(11)

Real domain of :

Complex domain of WhittakerW:

Approximate range of :

WhittakerW may reduce to simpler functions:

WhittakerW is not an analytic function:

Nor is it meromorphic: is neither non-decreasing nor non-increasing on its real domain: is not injective: is neither non-negative nor non-positive on its real domain:

WhittakerW has both singularity and discontinuity in (-,0]: is neither convex nor concave on its real domain:

### Differentiation(3)

First derivative with respect to z:

Higher derivatives with respect to z when k=1/3 and m=1/2:

Plot the higher derivatives with respect to z when k=1/3 and m=1/2:

Formula for the  derivative with respect to z:

### Series Expansions(5)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

General term in the series expansion using SeriesCoefficient:

Find the series expansion at Infinity:

Find series expansion for an arbitrary symbolic direction :

Taylor expansion at a generic point:

## Applications(1)

Green's function of the 3D Coulomb potential:

## Properties & Relations(4)

Use FunctionExpand to expand WhittakerW into other functions:

Integrate expressions involving Whittaker functions:

WhittakerW can be represented as a DifferentialRoot:

WhittakerW can be represented as a DifferenceRoot: