WhittakerM
✖
WhittakerM
Details

- Mathematical function, suitable for both symbolic and numerical manipulation.
- WhittakerM is related to the Kummer confluent 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
open allclose allBasic Examples (6)Summary of the most common use cases

https://wolfram.com/xid/0j49rir4-nkyuf

Use FunctionExpand to expand in terms of hypergeometric functions:

https://wolfram.com/xid/0j49rir4-5fexz

Plot over a subset of the reals :

https://wolfram.com/xid/0j49rir4-fknucq

Plot over a subset of the complexes:

https://wolfram.com/xid/0j49rir4-kiedlx

Series expansion at the origin:

https://wolfram.com/xid/0j49rir4-t1nfp

Series expansion at Infinity:

https://wolfram.com/xid/0j49rir4-cugjvu

Scope (35)Survey of the scope of standard use cases
Numerical Evaluation (6)

https://wolfram.com/xid/0j49rir4-l274ju


https://wolfram.com/xid/0j49rir4-cksbl4


https://wolfram.com/xid/0j49rir4-b0wt9

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

https://wolfram.com/xid/0j49rir4-y7k4a


https://wolfram.com/xid/0j49rir4-hfml09

Evaluate efficiently at high precision:

https://wolfram.com/xid/0j49rir4-di5gcr


https://wolfram.com/xid/0j49rir4-bq2c6r

WhittakerM can be used with Interval and CenteredInterval objects:

https://wolfram.com/xid/0j49rir4-h0d6g


https://wolfram.com/xid/0j49rir4-dj6d9x

Compute the elementwise values of an array:

https://wolfram.com/xid/0j49rir4-thgd2

Or compute the matrix WhittakerM function using MatrixFunction:

https://wolfram.com/xid/0j49rir4-o5jpo

Specific Values (7)
WhittakerM for symbolic parameters:

https://wolfram.com/xid/0j49rir4-fc9m8o


https://wolfram.com/xid/0j49rir4-bmqd0y


https://wolfram.com/xid/0j49rir4-e41pf2

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

https://wolfram.com/xid/0j49rir4-f2hrld


https://wolfram.com/xid/0j49rir4-fkkzx

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

https://wolfram.com/xid/0j49rir4-klij8s

Compute the associated WhittakerM function for half-integer parameters:

https://wolfram.com/xid/0j49rir4-hfz8z6

Different cases of WhittakerM give different symbolic forms:

https://wolfram.com/xid/0j49rir4-chhice

WhittakerM threads elementwise over lists:

https://wolfram.com/xid/0j49rir4-m2hznn

Visualization (3)
Plot the WhittakerM function for various orders:

https://wolfram.com/xid/0j49rir4-ecj8m7


https://wolfram.com/xid/0j49rir4-zpq38


https://wolfram.com/xid/0j49rir4-f87e

Plot as real parts of two parameters vary:

https://wolfram.com/xid/0j49rir4-elqrq8

Function Properties (11)

https://wolfram.com/xid/0j49rir4-cl7ele

Complex domain of WhittakerM:

https://wolfram.com/xid/0j49rir4-de3irc


https://wolfram.com/xid/0j49rir4-pugwvl

WhittakerM may reduce to simpler functions:

https://wolfram.com/xid/0j49rir4-bm44p7


https://wolfram.com/xid/0j49rir4-b8hemb

is not an analytic function of
for integer values of
:

https://wolfram.com/xid/0j49rir4-h5x4l2


https://wolfram.com/xid/0j49rir4-5fz4kz

It is analytic for other values of :

https://wolfram.com/xid/0j49rir4-1tvuzk

is neither non-decreasing nor non-increasing:

https://wolfram.com/xid/0j49rir4-o1mnqv


https://wolfram.com/xid/0j49rir4-fxi9f9


https://wolfram.com/xid/0j49rir4-zf7zy


https://wolfram.com/xid/0j49rir4-la4so0


https://wolfram.com/xid/0j49rir4-unklza

is neither non-negative nor non-positive on its real domain:

https://wolfram.com/xid/0j49rir4-7w275t

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

https://wolfram.com/xid/0j49rir4-1zvsit


https://wolfram.com/xid/0j49rir4-y8p27s

is neither convex nor concave on its real domain:

https://wolfram.com/xid/0j49rir4-xza3wn

TraditionalForm formatting:

https://wolfram.com/xid/0j49rir4-k6az4l

Differentiation (3)
First derivative with respect to z:

https://wolfram.com/xid/0j49rir4-krpoah

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

https://wolfram.com/xid/0j49rir4-z33jv

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

https://wolfram.com/xid/0j49rir4-fxwmfc

Formula for the derivative with respect to z:

https://wolfram.com/xid/0j49rir4-cb5zgj

Series Expansions (5)
Find the Taylor expansion using Series:

https://wolfram.com/xid/0j49rir4-ewr1h8

Plots of the first three approximations around :

https://wolfram.com/xid/0j49rir4-binhar

General term in the series expansion using SeriesCoefficient:

https://wolfram.com/xid/0j49rir4-dznx2j

Find the series expansion at Infinity:

https://wolfram.com/xid/0j49rir4-syq

Find series expansion for an arbitrary symbolic direction :

https://wolfram.com/xid/0j49rir4-t5t

Taylor expansion at a generic point:

https://wolfram.com/xid/0j49rir4-jwxla7

Applications (2)Sample problems that can be solved with this function
The bound-state Coulomb eigenfunction in parabolic coordinates:

https://wolfram.com/xid/0j49rir4-bvcaux
Decompose the eigenfunction in terms of spherical eigenfunctions:

https://wolfram.com/xid/0j49rir4-gmpxzy
Parabolic coordinates relate to radial coordinates as and
:

https://wolfram.com/xid/0j49rir4-et42z9


https://wolfram.com/xid/0j49rir4-dndrzs

Green's function of the 3D Coulomb potential:

https://wolfram.com/xid/0j49rir4-edo33e

https://wolfram.com/xid/0j49rir4-fz0c01

Properties & Relations (4)Properties of the function, and connections to other functions
Use FunctionExpand to expand WhittakerM into other functions:

https://wolfram.com/xid/0j49rir4-zktra


https://wolfram.com/xid/0j49rir4-g6xk8e


https://wolfram.com/xid/0j49rir4-e0dc9e

Integrate expressions involving Whittaker functions:

https://wolfram.com/xid/0j49rir4-fcfb3b

WhittakerM can be represented as a DifferentialRoot:

https://wolfram.com/xid/0j49rir4-burqaq

WhittakerM can be represented as a DifferenceRoot:

https://wolfram.com/xid/0j49rir4-dn1ebz


https://wolfram.com/xid/0j49rir4-hn6zh1

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
]}
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
]}