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Prove integral of complex exp

  1. Oct 4, 2013 #1
    1. The problem statement, all variables and given/known data
    Prove that the integral of a complex exponential over an integer number of periods is zero.


    2. Relevant equations

    [itex] \int_{0}^{T_{0}}e^{j (2\pi /T_{0}) kt} dt = 0 , k = integer[/itex]

    3. The attempt at a solution

    I am never sure how to work a proof. In this case, i can see that it would be true but not sure how you go about "proving" it. That the area from 0 to 1/2 T0 would zero out the area from 1/2 T0 to T0. Can someone point me to a good example on how to work this type of proof? or help me through this one?

    [itex]\frac{1}{e^{j(2\pi /T_{0})k}}e^{j(2\pi /T_{0})kt} \mid ^{0}_{T_{0}}[/itex]
     
  2. jcsd
  3. Oct 4, 2013 #2

    mfb

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    I would start with substitution to get rid of the variable T0, and consider the case k=1 first.
    You can split the integral in two parts and show that they are equal apart from their sign.
     
  4. Oct 4, 2013 #3
    So would you just say consider T0 = 1, but i would worry that would be like saying that since 2+2 = 4 and 2*2 = 4 therefor addition and multiplication are the same...
     
  5. Oct 4, 2013 #4
    [itex]\frac{e^{j(2\pi /T_{0})kT_{0}}}{j(2\pi/T_{0})k}- \frac{e^0}{j(2\pi/T_{0})k}[/itex]

    [itex]e^j\theta = cos\theta +j sin\theta [/itex]

    [itex]\frac{cos(2\pi k)+j sin(2\pi k)}{j(2\pi /T_{0})k} - \frac{1}{j(2\pi /T_{0})k}[/itex]


    [itex]cos(2\pi k) = 1[/itex]

    [itex]j sin(2\pi k) = 0[/itex]

    [itex] \frac{1}{j(2\pi /T_{0})k} - \frac{1}{j(2\pi /T_{0})k} = 0[/itex]

    Is this sufficient?
     
  6. Oct 4, 2013 #5

    Dick

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    Yes, that's exactly what you want. Though it is true that ##e^{j 2 \pi k}=1## for k an integer, right? Probably no need to go through the sines and cosines.
     
    Last edited: Oct 4, 2013
  7. Oct 5, 2013 #6

    mfb

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    Staff: Mentor

    Hmm right, my idea was more complicated than necessary. You can simply compute the integral.

    No, it is like saying T0/T0=1 for all real T0 (apart from 0). A sound mathematical proof that T0 does not influence the result.
     
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