- 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 .
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 (4)
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)
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 WhittakerM types give different symbolic forms:
WhittakerM threads elementwise over lists:
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:
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:
The bound-state Coulomb eigenfunction in parabolic coordinates:
Decompose the eigenfunction in terms of spherical eigenfunctions:
Parabolic coordinates relate to radial coordinates as and :
Properties & Relations (4)