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