# Why is sin(x^2) not a periodic function?

Mar 19, 2018

While $\sin \left(x\right)$ repeats every time $x$ changes by a fixed amount ($2 \pi$), $\sin \left({x}^{2}\right)$ repeats only when ${x}^{2}$ changes by $2 \pi$, and this does not happen at uniform intervals of $x$.

#### Explanation:

For a function $f \left(x\right)$ to be periodic with a period $X$, we must have

$f \left(x + X\right) = f \left(x\right) q \quad \forall x$

This immediately implies that $f \left(x\right) , f \left(x + X\right) , f \left(x + 2 x\right) , f \left(x + 3 X\right) , \ldots$ etc. all have the same value. So, a periodic function of $x$ repeats after equally spaced values of $x$.

Now, $\sin \left({x}^{2}\right)$ attains the value 0, for example, when ${x}^{2}$ attains the values 0, $\pi$, $2 \pi$, $3 \pi$, etc. While these values are equally spaced, the corresponding $x$ values are 0, $\sqrt{\pi}$, $\sqrt{2 \pi}$, $\sqrt{3 \pi}$ etc. - and these are not equally spaced.

Mar 19, 2018

While $\sin \left(x\right)$ repeats every time $x$ changes by a fixed amount ($2 \pi$), $\sin \left({x}^{2}\right)$ repeats only when ${x}^{2}$ changes by $2 \pi$, and this does not happen at uniform intervals of $x$.

#### Explanation:

For a function $f \left(x\right)$ to be periodic with a period $X$, we must have

$f \left(x + X\right) = f \left(x\right) q \quad \forall x$

This immediately implies that $f \left(x\right) , f \left(x + X\right) , f \left(x + 2 x\right) , f \left(x + 3 X\right) , \ldots$ etc. all have the same value. So, a periodic function of $x$ repeats after equally spaced values of $x$.

Now, $\sin \left({x}^{2}\right)$ attains the value 0, for example, when ${x}^{2}$ attains the values 0, $\pi$, $2 \pi$, $3 \pi$, etc. While these values are equally spaced, the corresponding $x$ values are 0, $\sqrt{\pi}$, $\sqrt{2 \pi}$, $\sqrt{3 \pi}$ etc. - and these are not equally spaced.