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 .

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  (35)

Numerical Evaluation  (4)

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:

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  (12)

Real domain of TemplateBox[{2, 0, x}, WhittakerW]:

Complex domain of WhittakerW:

Approximate range of TemplateBox[{2, 0, x}, WhittakerW]:

WhittakerW may reduce to simpler functions:

WhittakerW threads elementwise over lists:

WhittakerW is not an analytic function:

Nor is it meromorphic:

TemplateBox[{2, 0, x}, WhittakerW] is neither non-decreasing nor non-increasing on its real domain:

TemplateBox[{2, 0, x}, WhittakerW] is not injective:

TemplateBox[{2, 0, x}, WhittakerW] is not surjective:

TemplateBox[{2, 0, x}, WhittakerW] is neither non-negative nor non-positive on its real domain:

WhittakerW has both singularity and discontinuity in (-,0]:

TemplateBox[{2, 0, x}, WhittakerW] 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 ^(th) 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:

Wolfram Research (2007), WhittakerW, Wolfram Language function, https://reference.wolfram.com/language/ref/WhittakerW.html.

Text

Wolfram Research (2007), WhittakerW, Wolfram Language function, https://reference.wolfram.com/language/ref/WhittakerW.html.

BibTeX

@misc{reference.wolfram_2021_whittakerw, author="Wolfram Research", title="{WhittakerW}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/WhittakerW.html}", note=[Accessed: 23-July-2021 ]}

BibLaTeX

@online{reference.wolfram_2021_whittakerw, organization={Wolfram Research}, title={WhittakerW}, year={2007}, url={https://reference.wolfram.com/language/ref/WhittakerW.html}, note=[Accessed: 23-July-2021 ]}

CMS

Wolfram Language. 2007. "WhittakerW." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/WhittakerW.html.

APA

Wolfram Language. (2007). WhittakerW. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/WhittakerW.html