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Integral operation

  1. Mar 17, 2015 #1
    My question concerns a integral and why it vanishes:
    -nηIJ 1/2π ∫0 dσ ei(m+n)σ=-nηIJ deltam+n=0


    Just to justify why this should be on the beyond the standard model forum, this is part of a calulation concerning the comutators of the alpha modes.
     
  2. jcsd
  3. Mar 17, 2015 #2

    RUber

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    I assume m and n are integers?
    If ##m+n \neq 0## then, ##m+n = a \in \mathbb{Z}## and
    ## \int_0^{2\pi} e^{i (m+n) \sigma} d\sigma = \frac{1}{i(m+n)} e^{i (m+n) \sigma} |_0^{2\pi} = \frac{1}{i(m+n)} - \frac{1}{i(m+n)} = 0.##
    If ##m+n=0## then ## \int_0^{2\pi} e^{i (m+n) \sigma} d\sigma = \int_0^{2\pi} 1 d\sigma= 2\pi.##
     
  4. Mar 17, 2015 #3
    Thankyou.
     
  5. Mar 17, 2015 #4

    RUber

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    No problem--this idea is fundamental to Fourier transforms and a host of other applications requiring orthogonality of sines and cosines.
     
  6. Mar 17, 2015 #5
    Oh... thats an FT I missed that :) would've made that easyer,thanks.
     
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