gives the Whittaker function .
- 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. »
Examplesopen allclose all
Basic Examples (6)
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:
Numerical Evaluation (5)
The precision of the output tracks the precision of the input:
Evaluate efficiently at high precision:
WhittakerM can be used with Interval and CenteredInterval objects:
Specific Values (7)
WhittakerM for symbolic parameters:
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:
WhittakerM threads elementwise over lists:
Plot the WhittakerM function for various orders:
Function Properties (11)
Complex domain of WhittakerM:
WhittakerM may reduce to simpler functions:
is not an analytic function of for integer values of :
It is analytic for other values of :
is neither non-decreasing nor non-increasing:
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:
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 :
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:
Wolfram Research (2007), WhittakerM, Wolfram Language function, 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.
Wolfram Language. (2007). WhittakerM. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/WhittakerM.html