Whittaker Function

Author

Eric W. Weisstein
January 10, 2009

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/WhittakerFunction.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/WhittakerFunction.html.

©2009 Wolfram Research, Inc. except for portions noted otherwise

Code

MyWhittakerM[k_,m_,z_]:=Exp[-z/2]z^(1/2+m)Hypergeometric1F1[1/2+m-k,1+2m,z]
MyWhittakerW[k_,m_,z_]:=Exp[-z/2]z^(m+1/2)HypergeometricU[1/2+m-k,1+2m,z]

WhittakerW

Notation

WhittakerW[k,m,z]//TraditionalForm

Sums

(z^(m+1/2))exp(-(z/2)) (∑_(n=0)^∞)(((z^n) (Pochhammer[-k+m+(1/2), n]))/(n! (Pochhammer[2 m+1, n])))==(WhittakerM[k, m, z])

Product[-(1/2)+m-k+l,{l,n}]

Table[Pochhammer[(1/2)-k+m,n],{n,0,3}]

Product[2m+k,{k,n}]

Pochhammer[1+2 m,n]

Table[n!Pochhammer[1+2 m,n],{n,0,3}]

FullSimplify[((Pochhammer[(1/2)-k+m,n])/(n!Pochhammer[1+2 m,n])),n∈Integers&&n>0]

(z^(1/2+m))Exp[-z/2]Sum[((Pochhammer[(1/2)-k+m,n])/(n! Pochhammer[1+2 m,n]))(z^n),{n,0,∞}]

(E^(-z/2)) (z^((1/2)+m)) Hypergeometric1F1[(1/2)-k+m,1+2 m,z]

FullSimplify[%]

FullSimplify[%-WhittakerM[k,m,z]]

Integrals

(WhittakerW[k, m, z])==(∫_0^∞)(e^(-t)) (t^(-k+m-(1/2))) ((((t/z)+1))^(k+m-(1/2)))dt

Whittaker and Watson p. 340

WhittakerWW[k_,m_,z_,opts___]:=(((E^(-z/2))(z^k))/(Gamma[(1/2)-k+m]))Integrate[(t^(-k-1/2+m))(((1+(t/z)))^(k-1/2+m))(E^(-t)),{t,0,∞},opts]

(((E^(-z/2))(z^k))/(Gamma[(1/2)-k+m]))Integrate[(t^(-k-1/2+m))(((1+(t/z)))^(k-1/2+m))(E^(-t)),{t,0,∞}]

WhittakerWW[k,m,z]

NWhittakerW[k_,m_,z_,opts___]:=(((E^(-z/2))(z^k))/(Gamma[(1/2)-k+m]))NIntegrate[(t^(-k-1/2+m))(((1+(t/z)))^(k-1/2+m))(E^(-t)),{t,0,∞},opts,Method->GaussKronrod]

vals=Table[{Random[Integer,{1,10}],Random[Integer,{1,10}],Random[Complex,{2-2I,2+2I}]},{20}];

Length[vals=Select[Table[Random[Complex,(I+1){-1,1},20],{20},{3}],Re[#[[2]]-1/2-#[[1]]]≤0&]]

(WhittakerW[m,k,z]-NWhittakerW[m,k,z]/.Thread[{m,k,z}->#]&)/@vals//Chop

Through Hypergeometrics

v6

FunctionExpand[WhittakerW[k,m,z]]

expr=FullSimplify[Subtract@@{WhittakerW[k,m,z],Exp[-z/2]z^(m+1/2)HypergeometricU[1/2+m-k,1+2m,z]}]

Through ParabolicCylinderD

FunctionExpand[ParabolicCylinderD[n,x]==((1 )/(Sqrt[x]))(2^((n/2)+(1/4)))WhittakerW[(n/2)+(1/4),-(1/4),((x^2)/2)]]

Particular Cases

v5.2

FullSimplify[(π^(-1/2))(x^(-1/2))(E^(-(x^2)/2))MyWhittakerW[-1/4,1/4,x^2,Assumptions->x>0],x>0]

FullSimplify[(π^(-1/2))(x^(-1/2))(E^(-(x^2)/2))FunctionExpand[WhittakerWW[-1/4,1/4,x^2],x>0],x>0]

v6

FullSimplify[(π^(-1/2))(x^(-1/2))(E^(-(x^2)/2))WhittakerW[-1/4,1/4,x^2]]

WhittakerM

Notation

WhittakerM[k,m,z]//TraditionalForm

Through Hypergeometrics

v6

FunctionExpand[WhittakerM[k,m,z]]

expr=FullSimplify[Subtract@@{WhittakerM[k,m,z],(E^(-z/2)) (z^((1/2)+m)) Hypergeometric1F1[(1/2)-k+m,1+2 m,z]}]