Re: troublesome integral

• To: mathgroup at smc.vnet.net
• Subject: [mg126042] Re: troublesome integral
• From: David Bailey <dave at removedbailey.co.uk>
• Date: Fri, 13 Apr 2012 04:55:27 -0400 (EDT)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com
• References: <jm64ib\$6la\$1@smc.vnet.net>

```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.
>
>
> 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
\[Pi]},Assumptions->z\[Element]Reals]

and (of course)

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

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
http://www.dbaileyconsultancy.co.uk

```

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