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Odd/even functions and periodicity

  1. Aug 17, 2014 #1
    You can prove if f(x) is an odd function and f(x+ t) is an even function then f(x) is periodic with period at most 4t. Are there other theorems like that?i know this is a somewhat open ended and general question, it's just i would like to squeeze some more results from this angle and can not.
  2. jcsd
  3. Aug 17, 2014 #2

    Simon Bridge

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    ... what "more results"? "more" suggests you have already got some results. What results have you got so far?

    Consider the specific example where f(x)=sin(x).
    The general approach would start with the definitions of both.

    (modify slightly so that f(x-t) is even, makes it easier to write...)

    if f(-x)=-f(x) and f(t-x)=f(x-t) then f(x)=f(x-nt): n in Z (?)

    for f(x)=sin(x), t=pi/2, n=4.
  4. Aug 17, 2014 #3
    The result so far is that if f(x) is odd and f(x+t) is even then f(n2t) =0 for all integer n, f(x) is periodic, and the minimum period is no greater than 4t.
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