# WhittakerM

WhittakerM[k,m,z]

gives the Whittaker function .

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

# Examples

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## 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(34)

### 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:

WhittakerM can be used with Interval and CenteredInterval objects:

### Specific Values(7)

WhittakerM for symbolic parameters:

Values at zero:

Find the first positive maximum of WhittakerM[5,1/2,x]:

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

Compute the associated WhittakerM function for half-integer parameters:

Different WhittakerM types give different symbolic forms:

### Visualization(3)

Plot the WhittakerM function for various orders:

Plot the real part of :

Plot the imaginary part of :

Plot as real parts of two parameters vary:

### Function Properties(11)

Real domain of :

Complex domain of WhittakerM:

Approximate range of :

WhittakerM may reduce to simpler functions: is not an analytic function of for integer values of :

Nor is it meromorphic:

It is analytic for other values of : is neither non-decreasing nor non-increasing: is not injective: is not surjective: is neither non-negative nor non-positive on its real domain:

WhittakerM 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(2)

The bound-state Coulomb eigenfunction in parabolic coordinates:

Decompose the eigenfunction in terms of spherical eigenfunctions:

Parabolic coordinates relate to radial coordinates as and :

Green's function of the 3D Coulomb potential:

## Properties & Relations(4)

Use FunctionExpand to expand WhittakerM into other functions:

Integrate expressions involving Whittaker functions:

WhittakerM can be represented as a DifferentialRoot:

WhittakerM can be represented as a DifferenceRoot: