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Re: troublesome integral
*To*: mathgroup at smc.vnet.net
*Subject*: [mg126010] Re: troublesome integral
*From*: "Dr. Wolfgang Hintze" <weh at snafu.de>
*Date*: Fri, 13 Apr 2012 04:44:21 -0400 (EDT)
*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com
*References*: <jm64ib$6la$1@smc.vnet.net>
On 12 Apr., 10:42, peter lindsay <p... at me.com> 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]}=
,Assumptions->z\[Element]Reals]
>
> 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
In[44]:=
FullSimplify[Integrate[Cos[g - a]*Exp[I*z*Cos[g]],
{g, 0, 2*Pi}], z =E2=88=88 Reals]
Out[44]=
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
In[48]:=
Distribute[
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}]/
(I*Pi*Cos[a])]
Out[48]=
z - z^3/8 + z^5/192 - z^7/9216 + z^9/737280
and comparing this to the suspected result
In[45]:=
Normal[2*Series[BesselJ[1, z], {z, 0, 10}]]
Out[45]=
z - z^3/8 + z^5/192 - z^7/9216 + z^9/737280
No stict proof as 10 != inf but very plausible.
Regards,
Wolfgang
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