Re: troublesome integral

• To: mathgroup at smc.vnet.net
• Subject: [mg126033] Re: troublesome integral
• From: danl at wolfram.com
• Date: Fri, 13 Apr 2012 04:52:18 -0400 (EDT)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com
• References: <jm64ib\$6la\$1@smc.vnet.net>

```On Thursday, April 12, 2012 3:42:51 AM UTC-5, 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

You can do this by translating so that the cosine in the exponential has a simple argument of 'b', recognizing that by periodicity you need not change the range of integration.

In[190]:= Integrate[Cos[b + a] Exp[I*z*Cos[b]], {b, 0, 2*Pi},
Assumptions -> {Element[{a, z}, Reals]}]

Out[190]= 2*I*Pi*BesselJ[1, z]*Cos[a]

Daniel Lichtblau
Wolfram Research

```

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