WOLFRAM

WhittakerM[k,m,z]

gives the Whittaker function TemplateBox[{k, m, z}, WhittakerM].

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].
  • TemplateBox[{k, m, z}, WhittakerM] 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)Summary of the most common use cases

Evaluate numerically:

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Use FunctionExpand to expand in terms of hypergeometric functions:

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Plot over a subset of the reals :

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Plot over a subset of the complexes:

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Series expansion at the origin:

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Series expansion at Infinity:

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Scope  (35)Survey of the scope of standard use cases

Numerical Evaluation  (6)

Evaluate numerically:

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Evaluate to high precision:

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

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Complex number input:

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Evaluate efficiently at high precision:

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WhittakerM can be used with Interval and CenteredInterval objects:

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Compute the elementwise values of an array:

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Or compute the matrix WhittakerM function using MatrixFunction:

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Specific Values  (7)

WhittakerM for symbolic parameters:

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Values at zero:

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

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

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Compute the associated WhittakerM function for half-integer parameters:

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Different cases of WhittakerM give different symbolic forms:

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WhittakerM threads elementwise over lists:

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Visualization  (3)

Plot the WhittakerM function for various orders:

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Plot the real part of :

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Plot the imaginary part of :

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Plot as real parts of two parameters vary:

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Function Properties  (11)

Real domain of TemplateBox[{2, 0, z}, WhittakerM]:

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Complex domain of WhittakerM:

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Approximate range of TemplateBox[{2, 0, z}, WhittakerM]:

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WhittakerM may reduce to simpler functions:

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TemplateBox[{k, m, x}, WhittakerM] is not an analytic function of for integer values of :

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Nor is it meromorphic:

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It is analytic for other values of :

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TemplateBox[{2, 0, x}, WhittakerM] is neither non-decreasing nor non-increasing:

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

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TemplateBox[{2, {1, /, 2}, x}, WhittakerM] is not surjective:

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

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

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TemplateBox[{2, 0, x}, WhittakerM] is neither convex nor concave on its real domain:

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TraditionalForm formatting:

Differentiation  (3)

First derivative with respect to z:

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

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

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Formula for the ^(th) derivative with respect to z:

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Series Expansions  (5)

Find the Taylor expansion using Series:

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Plots of the first three approximations around :

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General term in the series expansion using SeriesCoefficient:

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Find the series expansion at Infinity:

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Find series expansion for an arbitrary symbolic direction :

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Taylor expansion at a generic point:

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Applications  (2)Sample problems that can be solved with this function

The bound-state Coulomb eigenfunction in parabolic coordinates:

Decompose the eigenfunction in terms of spherical eigenfunctions:

Parabolic coordinates relate to radial coordinates as and :

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Green's function of the 3D Coulomb potential:

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Properties & Relations  (4)Properties of the function, and connections to other functions

Use FunctionExpand to expand WhittakerM into other functions:

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Integrate expressions involving Whittaker functions:

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WhittakerM can be represented as a DifferentialRoot:

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WhittakerM can be represented as a DifferenceRoot:

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Neat Examples  (1)Surprising or curious use cases

Plot the Riemann surface of TemplateBox[{{3, /, 5}, {1, /, 3}, z}, WhittakerM]:

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

Text

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

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

CMS

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

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

APA

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

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

BibTeX

@misc{reference.wolfram_2025_whittakerm, author="Wolfram Research", title="{WhittakerM}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/WhittakerM.html}", note=[Accessed: 08-June-2025 ]}

@misc{reference.wolfram_2025_whittakerm, author="Wolfram Research", title="{WhittakerM}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/WhittakerM.html}", note=[Accessed: 08-June-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_whittakerm, organization={Wolfram Research}, title={WhittakerM}, year={2007}, url={https://reference.wolfram.com/language/ref/WhittakerM.html}, note=[Accessed: 08-June-2025 ]}

@online{reference.wolfram_2025_whittakerm, organization={Wolfram Research}, title={WhittakerM}, year={2007}, url={https://reference.wolfram.com/language/ref/WhittakerM.html}, note=[Accessed: 08-June-2025 ]}