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 Post subject: Non periodic function!Posted: Thu Jul 07, 2016 12:16 pm

Joined: Wed Nov 11, 2015 12:47 pm
Posts: 13
Prove that $\sin\left(x^3\right)$ is a non-periodic function.

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 Post subject: Re: Non periodic function!Posted: Thu Jul 07, 2016 12:17 pm

Joined: Mon Nov 09, 2015 11:52 am
Posts: 77
Location: Limassol/Pyla Cyprus
Let $f(x) = \sin(x^3)$, suppose it is periodic, and let $T$ be a period of $f$. Since $f$ is differentiable it follows that $T$ is a period of $f'$ as well. Observe that $f'(x) = 3x^2 \cos(x^3)$ is continuous and so is bounded on $[0,T]$. Since $T$ is a period of $f'$, it follows that $f'$ is bounded. But this is absurd as $f'(a_n) \to \infty$ when $a_n = \sqrt[3]{2n\pi}$. So $f$ is not periodic.

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 Post subject: Re: Non periodic function!Posted: Thu Jul 07, 2016 12:18 pm

Joined: Tue Nov 10, 2015 5:35 pm
Posts: 10
After Demetres' answer, let us generalize this problem:

Find all polymomials $p(x)$ such that $\sin (p(x))$ is a periodic function.

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 Post subject: Re: Non periodic function!Posted: Thu Jul 07, 2016 12:19 pm

Joined: Mon Nov 09, 2015 11:52 am
Posts: 77
Location: Limassol/Pyla Cyprus
I claim that $\sin(p(x))$ is periodic if and only if $p$ is linear (or constant). The "if" part is obvious. For the "only if" part we work similarly as my answer above. If it was periodic then its derivative $p'(x)\cos(p(x))$ would also be periodic and since it is continuous it would also be bounded. But this is not the case. For each $K,$ since $p'$ is a non-constant polynomial we can find $N$ such that $|p'(x)| \geqslant K$ for each $x \geqslant N$. Furthermore we can find $x \geqslant N$ such that $|\cos(p(x))| = 1$ and therefore for that $x$ we would have $|p'(x)\cos(p(x))| \geqslant K$. This contradicts the boundedness of the derivative and therefore the periodicity of the function.

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