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MathGroup Archive 2011

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Integral representation of Bessel functions

  • To: mathgroup at smc.vnet.net
  • Subject: [mg116551] Integral representation of Bessel functions
  • From: John Travolta Sardus <pireddag at gmail.com>
  • Date: Sat, 19 Feb 2011 05:14:12 -0500 (EST)

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


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