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Re: troublesome integral

On 12 Apr., 10:42, peter lindsay <p... at> wrote:
> 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]}=
> doesn't seem to run.
> Answer should be
> 2 I \[Pi] BesselJ[1,z] Cos[\[Alpha]] =C2 [ I think ]
> thanks
> Peter Lindsay
Your result is correct.
As it happens frequently, Mathematica likes to accept some help:

1) let b-a=g and consider

FullSimplify[Integrate[Cos[g - a]*Exp[I*z*Cos[g]],
   {g, 0, 2*Pi}], z =E2=88=88 Reals]

2*I*Pi*BesselJ[1, z]*Cos[a]

Here an error has crept in: the limits of integration are not correct,
they should read {g,-a,2 Pi-a}

2) The result is nevertheless correct as can be seen with the original
integral by expanding the exponential function, integrating term by
term, summing up

  Plus @@ Table[Integrate[Cos[b]*(I*z*Cos[b - a])^k,
       {b, 0, 2*Pi}, Assumptions -> {z =E2=88=88 Reals,
         k =E2=88=88 Integers}]/k!, {k, 0, 10}]/

z - z^3/8 + z^5/192 - z^7/9216 + z^9/737280

and comparing this to the suspected result

Normal[2*Series[BesselJ[1, z], {z, 0, 10}]]

z - z^3/8 + z^5/192 - z^7/9216 + z^9/737280

No stict proof as 10 != inf but very plausible.


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