Enable JavaScript to interact with content and submit forms on Wolfram websites. Learn how

WhittakerM

WhittakerM[k,m,z]

gives the Whittaker function .

Details

Mathematical function, suitable for both symbolic and numerical manipulation.
WhittakerM is related to the Kummer confluent hypergeometric function by $TemplateBox[{k, m, z}, WhittakerM]=e^(-z/2)z^(m+1/2) TemplateBox[{{m, -, k, +, {1, /, 2}}, {{2, , m}, +, 1}, z}, Hypergeometric1F1]$ .
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

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 (6)

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:

Compute the elementwise values of an array:

Or compute the matrix WhittakerM function using MatrixFunction:

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 cases of WhittakerM give different symbolic forms:

WhittakerM threads elementwise over lists:

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:

TraditionalForm formatting:

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:

Neat Examples (1)

Plot the Riemann surface of :

Top