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Fourier Integral

  1. Dec 15, 2008 #1
    Hello, I want to calculate the integral

    [tex]\int^{2\pi}_{0} \exp(i(k t + \cos(t+\delta)))dt[/tex]

    where k and [tex]\delta[/tex] are integer and real numbers, respectivily.

    I know with [tex]\delta[/tex]=0 the result is given in terms of Bessel functions, but I don't know what to do if [tex]\delta[/tex][tex]\neq[/tex]0.

    Any help would be appreciate, thanks in advance.

    ACM
     
  2. jcsd
  3. Dec 16, 2008 #2

    tiny-tim

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    Welcome to PF!

    Hi arcmed ! Welcome to PF! :smile:

    (have a delta: δ :wink:)

    Hint: substitute u = t + δ. :smile:
     
  4. Dec 16, 2008 #3
    Hello tiny-tim, thanks to take time for answer.

    I had already considered this substitution, but in that case, the integral limits change from 0 and [tex]2\pi[/tex] to [tex]\delta[/tex] and [tex]\delta+2\pi[/tex], respectivily. So, again, I can't use the fact the original integral with [tex]\delta=0[/tex] can be given in terms of Bessel functions.

    ACM
     
  5. Dec 16, 2008 #4

    Ben Niehoff

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    The function

    [tex]e^{i \cos \theta + i m \theta[/tex]

    is periodic with period [itex]2\pi[/itex]. Therefore the integral over any period ought to be the same.
     
  6. Dec 16, 2008 #5
    Hello Ben, thank you, you are right.
    I had never participated in a forum, but now I realize it is an useful tool, sometimes one can forget very obvious things.
    Again, thanks to tiny-tim also.
    ACM
     
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