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Simple Fourier coefficient exponential

  1. Dec 24, 2016 #1
    1. The problem statement, all variables and given/known data

    I think I am being stupid, I am trying to show that

    ## \int^{T}_{0} e^i\frac{2\pi(n-m)t}{T} dt = 0 ## [1] if ## n \neq m##
    ## = T ## if ##n=m##, ##T## the period.



    2. Relevant equations

    I am using the following ##cos## and ##sin## orthogonal identities:

    1) ##\int^{2 \pi / g}_0 sin ( n g x) sin ( m g x ) dx = \delta_{nm} ##
    2) ##\int^{2 \pi / g}_0 cos ( n g x) cos ( m g x ) dx = \delta_{nm} ##
    3) ##\int^{2 \pi / g}_0 sin ( n g x) cos ( m g x ) dx = \delta_{nm} ##

    3. The attempt at a solution

    ## e^i\frac{2\pi(n-m)t}{T} dt = e^i\frac{2\pi(n)t}{T} . e^i\frac{2\pi(n-m)t}{T} ##
    (Let ##g=\frac { 2 \pi i}{T} ##)
    ##= (cos ( g n t) + i sin (gnt) ).(cos ( g m t) - i sin ( g m t))##

    So using the orthogonal identities [1] the contribution to [1] from the ##coscos## and ##sinsin## term gives ##2 \delta _{nm} ##, and then from the cross ##sin## and ##cos## terms I am getting

    ## i \delta_{nm} - i \delta_{nm} = 0 ##.

    So I am getting ## 2 \delta _{nm} ##

    Which I can see straight off is wrong since I am not dividing by the period to get the coefficient. (independent of it)

    Many thanks in advance.
     
  2. jcsd
  3. Dec 24, 2016 #2

    Ssnow

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    mmmhh, I think ##g=\frac{2\pi}{T}##, using the orthogonal relations is good, in fact another way can be calculating directing the integral treating ##i## as a constant, in this way you can see the rersult that is still immediate ...

    Ssnow
     
  4. Dec 24, 2016 #3

    Ray Vickson

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    If ##a \neq 0## we have
    $$\int_0^T e^{i a t/T} \, dt = \frac{1}{a} iT \left(1-e^{ia}\right)$$.
    Put ##a = 2 \pi (n-m)## and simplify, assuming ##n,m## are unequal integers.

    Of course, when ##a=0## the integral is just ##\int_0^T 1 \, dt. ##.
     
  5. Dec 24, 2016 #4
    so where have I gone wrong?
     
  6. Dec 24, 2016 #5
    but what is the error in my working anybody please?
     
  7. Dec 24, 2016 #6

    Ray Vickson

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    Your equations (1), (2), (3) are incorrect. ##\int_0^{2\pi/g} \cos(n g t) \cos(m g t) \, dt = c \delta_{m,n} ,## but ##c \neq 1##.
     
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