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WhittakerM

The Whittaker M function

WhittakerW

The Whittaker W function

 

Calling Sequence

Parameters

Description

Examples

References

Calling Sequence

WhittakerM(mu, nu, z)

WhittakerW(mu, nu, z)

Parameters

mu

-

algebraic expression

nu

-

algebraic expression

z

-

algebraic expression

Description

• 

The Whittaker functions WhittakerM(mu, nu, z) and WhittakerW(mu, nu, z) solve the differential equation

`y''`+14+μz+14ν2z2y=0

• 

They can be defined in terms of the hypergeometric and Kummer functions as follows:

WhittakerMμ,ν,z=ⅇ12zz12+νhypergeom12+νμ,1+2ν,z

WhittakerWμ,ν,z=ⅇ12zz12+νKummerU12+νμ,1+2ν,z

Examples

WhittakerM1,2,0.5

0.1606687379

(1)

zWhittakerWμ,ν,z

12μzWhittakerWμ,ν,zWhittakerWμ+1,ν,zz

(2)

seriesWhittakerM2,3,x,x

x7/227x9/2+23448x11/2+Ox13/2

(3)

seriesWhittakerW12,13,x,x

323Γ232x1/6ππ3x5/6Γ232+943Γ232x7/6π310π3x11/6Γ232+9163Γ232x13/6π340π3x17/6Γ232+272243Γ232x19/6π9880π3x23/6Γ232+2717923Γ232x25/6π97040π3x29/6Γ232+81465923Γ232x31/6π27239360π3x35/6Γ232+Ox37/6

(4)

simplifyWhittakerWμ+73,ν,x

2μ+83x2μ+23xWhittakerWμ+13,ν,x16μ+νν+μ16WhittakerWμ23,ν,x56μ+νν+μ+56WhittakerWμ+13,ν,x

(5)

References

  

Abramowitz, M., and Stegun I. Handbook of Mathematical Functions. New York: Dover Publications.

  

Luke, Y. The Special Functions and Their Approximations. Vol 1. Academic Press, 1969.

See Also

hypergeom

inifcns

KummerU

 


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