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Periodic functions

  1. Jul 6, 2005 #1
    Is there a continuous periodic function which is not trigonometric. if yes, what?
     
  2. jcsd
  3. Jul 6, 2005 #2

    lurflurf

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    Yes there are very many. Define a continuous function on [a,b] where f(a)=f(b) then define f outside of [a,b] so that f(x+(b-a))=f(x). A simple example that is not trigonometric (even though it looks like it is) is Arccos(cos(x)).
     
  4. Jul 6, 2005 #3

    quasar987

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    f(x) = Arccos(cos(x)) = x, the identity function is periodic. Now besides this one and the trig functions, are there other non "man-made" (i.e. cut and pasted according to the process described by lurflurf) that are periodic?
     
  5. Jul 6, 2005 #4

    HallsofIvy

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    I don't know any functions that aren't "man-made"!
     
  6. Jul 6, 2005 #5
    No, the inverse cosine function returns values in a specific interval (which I can't remember atm), so you can't have arccos(cos(x)) = x for all x.
     
  7. Jul 6, 2005 #6

    lurflurf

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    That is right Arccos(cos(x))=x on [0,pi], it is also periodic with period pi, so it repeats all those values. I use Arccos with the capital A to make clear that I am using the principle value of Arccos not just any value that gives the needed value. This is a general way to write periodic functions. let f(x) be diffined and continuous on [a,b] with f(a)=f(b) then
    g(x)=f(a+(b-a)(1+(1/pi)Arccos(cos(pi(x-a)/(b-a)))))
    is a periodic extension of f that is f=g on [a,b] and g(x+2n(b-a))=g(x)
    when n is an integer.
    remenber the definition of a periodic function is a function is periodic with period p if
    f(x+p)=f(x) for all x.
     
    Last edited: Jul 6, 2005
  8. Jul 6, 2005 #7

    Alkatran

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    Modulus is periodic, any real number to the power of any other real number + an imaginary variable is periodic.

    For example, there is:
    e^(2+x*i)
     
  9. Jul 6, 2005 #8

    quasar987

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    Every function is man-made. Not every function is "man-made". :wink:
     
  10. Jul 13, 2005 #9
    every function is man-made yaar....mathematics itself is man-made ;) functions are infinite...i can define a function rite now 2 suit ur needs...lemme see...
    f(x)=x-2n for x belonging to [2n, 2n+1) where n is any integer
    = (2n+2)-x for x belonging to [2n+1, 2n+2]
    check this out...if i havent made any silly mistakes...it shud come out 2 be continuous and periodic...ive modelled it on the sin graph + on the [x] graph..lol...cudnt think of a better example sorry....cheers! ;)
     
  11. Jul 13, 2005 #10

    Galileo

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    Constant functions.
     
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