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I was wondering if a periodic function could have 2 different fundamental periods? If so, could you give an example? And If not, how can I prove that?
I don't see how it could. The sine function has a period of ##2\pi##, since ##\sin(x) = \sin(x + 2\pi)##, for any real x. It's also true that ##\sin(x) = \sin(x + k \cdot 2\pi)## for any positive integer k, but unless I'm mistaken, the fundamental period is the smallest number p for which ##\sin(x) = \sin(x + p)##.jaumzaum said:I was wondering if a periodic function could have 2 different fundamental periods? If so, could you give an example? And If not, how can I prove that?
jaumzaum said:What if one of the fundamental periods is irrational (i.e. a integer and pi for example), as @FactChecker pointed out? Is it possible to have a function like that? Would there still be a smaller fundamental period?
jaumzaum said:Sorry by that misunderstanding. What I meant by two different fundamental periods was two periods that are not multiples of each other, such that, for each one, there is no period that is smaller than them.
For example, if we consider that we have a function that has fundamental periods 2 and 3. We can prove that this is impossible:
$$f(x)=f(x+2)$$
$$f(x)=f(x+3)$$
So
$$f(x+2)=f(x+3)$$
And the function has a unique fundamental period of 1.
I think we probable could prove that if a and b are rationals and periods of the function, not multiples of each other, there is a smaller rational that is the fundamental period. I think that @Office_Shredder was trying to prove that, but I didn't understand the proof.
No, I don’t want the lowest common multiple.pbuk said:I change my socks every other day. Every Sunday I go to church to atone for my lack of cleanliness. How often do I have clean socks on at church?
The concept you are looking for is the lowest common multiple.
Office_Shredder said:Sorry, my main point was to point out the existence of these two functions:
1.) One which is 1 is x is rational, 0 otherwise
2.) One which is 1 if ##x=m\sqrt{2}+n\sqrt{3}## for integers m and n, and 0 otherwise
both are periodic.
Neither of them have a fundamental period. You are probably imagining ##\sqrt{2}## and ##\sqrt{3}## are fundamental periods of the second one, and in some philosophical sense you are right, but by the definition of fundamental period you are wrong.
jaumzaum said:Thanks @Office_Shredder!
Now I understood the examples. Can we say that the square roots of 2 and 3 are the two smallest periods of the function?
jaumzaum said:Also, is there a continuous function like that?
Do you mean something like that (red curve)?jaumzaum said:I was wondering if a periodic function could have 2 different fundamental periods? If so, could you give an example? And If not, how can I prove that?
What about them? That's two separate functions each with a well defined period.Svein said:What about periodic functions where the ratio between the periods is irrational - such as sin(3x) and sin(πx)?
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Hearing a beat frequency isn't the same as periodicity.Svein said:I meant something like "modulate the first sine withe the second sine using FM like in the Yamaha DX7 synth". I have experienced a phenomenum called "phantom bass" where two exactly like recorders played by two experience players would create a very low-frequency "beat" that sounde like a very low bass recorder....