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Re: troublesome integral

You can rewrite the integral as

 Cos[\[Alpha] + \[Beta]] Exp[I z Cos[\[Beta]]], {\[Beta], 0, 2 \[Pi]},
  Assumptions -> z \[Element] Reals]

which can be rearranged as (the Sin[\[Alpha]] Sin[\[Beta]] part integrates 
to zero)

 Cos[\[Alpha]] Cos[\[Beta]] Exp[I z Cos[\[Beta]]], {\[Beta], 0,
  2 \[Pi]}, Assumptions -> z \[Element] Reals]

which evaluates to

2 I \[Pi] BesselJ[1, z] Cos[\[Alpha]]

Over the years I too have encountered this problem with evaluating integral 
representations of Bessel functions, and the only solution I have found is 
to give mathematica a helping hand, as above.

Stephen Luttrell
West Malvern, UK

"peter lindsay" <pl0 at> wrote in message 
news:jm64ib$6la$1 at
> A couple of colleagues wondered about this. I've sent it on to support @ 
> wolfram who are escalating it to the developers. Possibly someone here has 
> an answer though ?
> Integrate[Cos[\[Beta]] Exp[I z Cos[\[Beta]-\[Alpha]]],{\[Beta],0,2 
> \[Pi]},Assumptions->z\[Element]Reals]
> doesn't seem to run.
> Answer should be
> 2 I \[Pi] BesselJ[1,z] Cos[\[Alpha]]  [ I think ]
> thanks
> Peter Lindsay

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