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Proving a function is peridoic

  1. Nov 3, 2013 #1
    Show that cos(x) + cos([itex]\alpha[/itex]x) is periodic if [itex]\alpha[/itex] is a rational number.


    Ok So I don't think I have ever done a question proving periodicity. But by definition a function f is periodic if:

    f(x + p) = f(x)

    so then:

    f(x+p) = cos(x +p) + cos([itex]\alpha[/itex](x+p))
    = [cos(x)cos(p)-sin(x)sin(p)] + [cos([itex]\alpha[/itex]x)cos([itex]\alpha[/itex]p)-sin([itex]\alpha[/itex]x)sin([itex]\alpha[/itex]p)


    and this is where I am stuck, so from what I've learned I have to solve for p is some fashion, I'm just not sure how,
     
  2. jcsd
  3. Nov 3, 2013 #2

    tiny-tim

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    hi trap101! :smile:

    hint: cos(0) + cos([itex]\alpha[/itex]0) = 2 :wink:
     
  4. Nov 3, 2013 #3

    pasmith

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    The first thing to do with a rational number is write it as [itex]\alpha = r/s[/itex] for co-prime integers r and s.

    You already know that [itex]\cos x[/itex] has period [itex]2\pi[/itex] and [itex]\cos(\alpha x)[/itex] has period [itex](2\pi/\alpha)[/itex]. So you need to find [itex]p[/itex] such that [itex]x + p = x + 2n\pi[/itex] and [itex]x + p = x + 2m\pi/\alpha[/itex] for integers n and m.
     
  5. Nov 3, 2013 #4

    UltrafastPED

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    You have expressions for f(x) [given], and f(x+p) [derived above].
    Now assume that alpha is rational, m/n, and then see if there is any period p which satisfies the condition f(x)=f(x+p).
     
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