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Re: Integral representation of Bessel functions

  • To: mathgroup at smc.vnet.net
  • Subject: [mg116755] Re: Integral representation of Bessel functions
  • From: Daniel Lichtblau <danl at wolfram.com>
  • Date: Sat, 26 Feb 2011 06:07:29 -0500 (EST)

John Travolta Sardus wrote:
> I have tried an integral (version 8), I find a result that I do not
> agree with
> 
> Assuming[x \[Element] Reals && \[Psi] \[Element] Reals,
>   Integrate[
>    Exp[I*(\[Phi] - \[Psi])]*
>     Exp[I*x*Cos[(\[Phi] - \[Psi])]], {\[Phi], -\[Pi], \[Pi]}]]
> 
> I obtain
> 
>   \[Pi] BesselJ[1, x] (I Cos[\[Psi]] + Sin[\[Psi]])
> 
> While according to me it should be
> 
> \[Pi] BesselJ[1, x]
> 
> Or at least no dependence on \[Psi] must be present, as I can do a
> change of variable eliminating \[Psi] (the integral is on the whole
> circle and the function is periodic with period 2*\[Pi]). Do I make a
> mistake with the input? Do I read the output incorrectly? Or what
> else?
> 
> Thanks in advance for any answer.
> 
> Giovanni
> 

Just to confirm, this is indeed a bug. Will get filed and fixed (either 
to return a correct result, or, more likely, unevalauted).

Daniel Lichtblau
Wolfram Research



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