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Re: troublesome integral

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  • Subject: [mg126042] Re: troublesome integral
  • From: David Bailey <dave at>
  • Date: Fri, 13 Apr 2012 04:55:27 -0400 (EDT)
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  • References: <jm64ib$6la$>

On 12/04/2012 09:42, peter lindsay 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]]  [ I think ]
> thanks
> Peter Lindsay

I can confirm the integral seems to loop.

I found that this integral will evaluate:

aa=Integrate[Exp[I \[Beta]] Exp[I z Cos[\[Beta]-\[Alpha]]],{\[Beta],0,2 

and (of course)

bb=Integrate[Exp[-I \[Beta]] Exp[I z Cos[\[Beta]-\[Alpha]]],{\[Beta],0,2 

Evaluating (aa+bb)/2 and simplifying, I get

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

This is not the answer you expect, and does not depend on \[Alpha] but I 
am not sure if splitting the integral in that way is sound - there might 
be convergence issues.

David Bailey

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