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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

x722x927+23x112448+Ox132

(3)

seriesWhittakerW12,13,x,x

33Γ232x162ππ3x56Γ232+93Γ232x764π3π3x11610Γ232+93Γ232x13616π3π3x17640Γ232+273Γ232x196224π9π3x236880Γ232+273Γ232x2561792π9π3x2967040Γ232+813Γ232x31646592π27π3x356239360Γ232+Ox376

(4)

simplifyWhittakerWμ+73,ν,x

ν+μ16x2μ8316ν+μWhittakerWμ23,ν,x+WhittakerWμ+13,ν,x5μ2+4x+253μ+x2ν210x3+8936

(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