Lommel Function

Author

Eric W. Weisstein
February 17, 2008

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

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

©2008 Wolfram Research, Inc. except for portions noted otherwise

One Variable

LommelS1[μ_,ν_,z_]:=(z^(μ+1)/((μ+1)^2-ν^2))*HypergeometricPFQ[{1},{(3+μ+ν)/2,(3+μ-ν)/2},-(z^2/4)]

LommelS2[μ_,ν_,z_]:=(z^(μ+1)/((μ+1)^2-ν^2))*HypergeometricPFQ[{1},{(3+μ+ν)/2,(3+μ-ν)/2},-(z^2/4)]+((2^(-1+μ+ν)*Gamma[ν]*Gamma[(1/2)*(1+μ+ν)])/(z^ν*Gamma[(1/2)*(1-μ+ν)]))*Hypergeometric0F1[1-ν,-(z^2/4)]+((2^(-1+μ-ν)*z^ν*Gamma[-ν]*Gamma[(1/2)*(1+μ-ν)])/Gamma[(1/2)*(1-μ-ν)])*Hypergeometric0F1[1+ν,-(z^2/4)]

FullSimplify[(1/2)π(BesselY[ν,z]Integrate[(z^μ)BesselJ[ν,z],{z,0,z},GenerateConditions->False]-BesselJ[ν,z]Integrate[(z^μ)BesselY[ν,z],{z,0,z},GenerateConditions->False])]

FullSimplify[BesselI[ν,z]Integrate[(z^μ)BesselK[ν,z],{z,0,c1},GenerateConditions->False]-BesselJ[ν,z]Integrate[(z^μ)BesselI[ν,z],{z,c2,z},GenerateConditions->False]]

Two Variables

(Subscript[U, n])(w,z)

Sum[(-1)^m((((w/z)))^(n+2m))BesselJ[n+2m,z],{m,0,∞}]

(Subscript[V, n])(w,z)

Sum[(-1)^m((((w/z)))^(-n-2m))BesselJ[-n-2m,z],{m,0,∞}]

Table[Subtract@@{BesselJ[n,x],(-1)^n BesselJ[-n,x]}/.Thread[{n,x}->{Random[Integer,{1,5}],Random[Complex]}],{10}]//Chop