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Re: troublesome integral

  • To: mathgroup at smc.vnet.net
  • Subject: [mg126011] Re: troublesome integral
  • From: "Stephen Luttrell" <steve at _removemefirst_stephenluttrell.com>
  • Date: Fri, 13 Apr 2012 04:44:42 -0400 (EDT)
  • Delivered-to: l-mathgroup@mail-archive0.wolfram.com
  • References: <jm64ib$6la$1@smc.vnet.net>

You can rewrite the integral as

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

which can be rearranged as (the Sin[\[Alpha]] Sin[\[Beta]] part integrates 
to zero)

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

which evaluates to

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

Over the years I too have encountered this problem with evaluating integral 
representations of Bessel functions, and the only solution I have found is 
to give mathematica a helping hand, as above.

-- 
Stephen Luttrell
West Malvern, UK


"peter lindsay" <pl0 at me.com> wrote in message 
news:jm64ib$6la$1 at smc.vnet.net...
>
> 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
>
> 




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