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. > > 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 \[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