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Math Help - Determining the Period of a function

  1. #1
    Ase
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    Determining the Period of a function

    Hi

    The question is whether the function [cos(pi*x)]^n is periodic with period 1?

    I cannot remember how to determine the period of a function. However, I do remember the definition:

    f(x+h)=f(x), the h is the period.

    I just thought that the period of, for instanse, cosinus was 2Pi. So how can the question be if the period is 1 or not?

    Really appreciate the help
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  2. #2
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    Quote Originally Posted by Ase View Post
    Hi

    The question is whether the function [cos(pi*x)]^n is periodic with period 1?

    I cannot remember how to determine the period of a function. However, I do remember the definition:

    f(x+h)=f(x), the h is the period.

    I just thought that the period of, for instanse, cosinus was 2Pi. So how can the question be if the period is 1 or not?

    Really appreciate the help

    The answer ( the period ) depends on whether n is even or odd , do you know why ?
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  3. #3
    Ase
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    Hi

    I am not sure. Could you elaborate?
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  4. #4
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    we know that  \cos[ \pi (x + 1 )] = \cos(\pi x + \pi ) = - \cos(x)

    However , if n is even , we have  \cos^n[ \pi (x + 1 )]

     =  [- \cos(x)]^n = (-1)^n \cos^n(\pi x)  = \cos^n(\pi x )

    so it has a period 1 for n is even , but if n is odd , it only has a period 2 .
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  5. #5
    Ase
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    Hi

    Thanks. That explained a lot. I am however a bit confused about something. You write cos(pi*(x+1)). I thought it was cos(pi*x+1).
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