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Is cosx+cos(sqrt(2)x) periodic?

  1. Nov 24, 2012 #1
    1. The problem statement, all variables and given/known data

    Is the function cosx + cos(sqrt(2)x) is periodic?

    2. Relevant equations

    cos(x)=cos(x+2pi)

    3. The attempt at a solution

    For the above function to be periodic:
    cosx + cos(sqrt(2)x) = cos(x+T) + cos(sqrt(2)(x + T))
    Does that imply that 2pi = T AND 2pi = sqrt(2)T, ergo there is no such T?
     
  2. jcsd
  3. Nov 24, 2012 #2
    It doesn't exactly imply that. However, by the definition of periodicity, you need integer T to satisfy the equation. For example, the function cos(2x)+cos(3x) is periodic, one has period [itex]\pi[/itex] and the other has period [itex]2\pi/3[/itex]. It is not hard to see that both of these functions are also periodic by [itex]2\pi[/itex], and hence their sum is also periodic. Returning to your question, your answer is correct, the function is not periodic. However, your equations seem a bit fallacious to me. All you need is that the period of the sum is the least common multiple of the periods of the summands.

    Can you see such a least common multiple?
     
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