Anger Function

Author

Eric W. Weisstein
August 26, 2008

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

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

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

Notation

AngerJ[ν,x]//TraditionalForm

Definition

V7

(1/π)Integrate[Cos[ν θ-z Sin[θ]],{θ,0,π}]//InputForm

FunctionExpand[AngerJ[ν,x]]

FullSimplify[%]

TraditionalForm[%]

Table[AngerJ[ν,x],{ν,0,4}]

Table[Cos[((π ν)/2)] HypergeometricPFQRegularized[{1},{1-(ν/2),1+(ν/2)},-((x^2)/4)]+(1/2) x HypergeometricPFQRegularized[{1},{((3-ν)/2),((3+ν)/2)},-((x^2)/4)] Sin[((π ν)/2)],{ν,0,4}]//FullSimplify

(1/π)Integrate[Cos[ν θ-z Sin[θ]],{θ,0,π},Assumptions->z∈Reals]

Table[AngerJ[ν,x],{ν,0,4}]

Table[(1/π)Integrate[Cos[ν θ-z Sin[θ]],{θ,0,π},Assumptions->z∈Reals],{ν,0,4}]//FullSimplify

FullSimplify[BesselJ[n,-z],z∈Reals&&n∈Integers&&Mod[n,2]==0]

FullSimplify[BesselJ[n,-z]-BesselJ[n,z],z∈Reals&&Mod[n,2]==0]

With[{ν=2,z=-.2},{BesselJ[ν,z],BesselJ[ν,-z],(1/π)NIntegrate[Cos[ν θ-z Sin[θ]],{θ,0,π}],AngerJ[ν,z]}]

FullSimplify[%]

{BesselJ[0,Abs[z]],BesselJ[1,z],BesselJ[2,Abs[z]],BesselJ[3,z],((z (-24+(z^2)) BesselJ[0,z]-8 (-6+(z^2)) BesselJ[1,z])/(z^3))}

Series

Series[AngerJ[n,z],{z,0,4}]

SeriesCoefficient[AngerJ[n,z],{z,0,k}]