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 > >