Periodic functions

geniusprahar_21
Is there a continuous periodic function which is not trigonometric. if yes, what?

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geniusprahar_21 said:
Is there a continuous periodic function which is not trigonometric. if yes, what?
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)).

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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?

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I don't know any functions that aren't "man-made"!

Muzza
...f(x) = Arccos(cos(x)) = x...

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.

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Muzza said:
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.
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.

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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)

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HallsofIvy said:
I don't know any functions that aren't "man-made"!